Inferential Semantics for First-Order Logic: Motivating Rules of Inference from Rules of Evaluation by Neil Tennant∗ Department of Philosophy The Ohio State University Columbus, Ohio 43210 email [email protected] June 17, 2009

I am grateful to Tim, as my teacher in logic, for emphasizing that its main notions are relational: ‘ϕ is deducible from ∆’ and ‘ϕ is a logical consequence of ∆’. The notions ‘ϕ is a theorem’ and ‘ϕ is logically true’ are special cases. In the early 1970s Tim gave a formative series of lectures emphasizing how proofs are to be understood as perfected arguments, in Aristotle’s sense. The present discussion of verification and falsification is fully in the inferentialist spirit of Tim’s emphases. The aim is to render even the notions ‘ϕ is true’ and ‘ϕ is false’ as essentially relational and inferential. A sentence’s truth-value is determined relative to collections of rules of inference that constitute an interpretation. Moreover, truthmakers and falsity-makers are themselves proof-like objects, encoding the inferential process of evaluation involved. The inference rules involved in the determination of truth-value are almost identical to those involved in securing the transmission of truth from premises to conclusion of a valid argument. We shall see how smoothly one can ‘morph’ the former into the latter. ∗

Written for the Festschrift for Timothy Smiley, edited by Jonathan Lear and Alex Oliver. Please do not cite or circulate without the author’s permission.

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Rules of evaluation

1.1

Verification and falsification of sentences in models

We shall first describe, in general terms, interpretations, or models, of a firstorder language. Then we shall provide a ‘toy example’ that will illustrate the ingredients of the definition. A model consists of (i) a domain of individuals; (ii) a denotation mapping for names (if there are any names in the objectlanguage); (iii) the structure that consists in primitive predicate-extensions; and (iv) the structure that consists in the mappings represented by primitive function-signs. The names, predicates and function-signs make up the extra-logical vocabulary that is being interpreted by the model in question. Because we are dealing with a toy example, we shall confine ourselves to one-place predicates and function signs. Towards the end of this discussion we shall have occasion to consider a two-place predicate. So our expressive resources will be quite modest. Still, we shall be painstaking in illustrating what is involved in these different components (i)–(iv) of models. We shall build up our chosen model M in stages. To the left will be a diagram, which can be thought of as the model M itself. The large dots will be the individuals; each one-place predicate extension will be represented by an enclosure; and each one-place mapping will be represented by arrows (a different style of arrow for each mapping). These diagrammatic components will be added in sequence, as the model is built up. So, for our toy example, we first choose a domain of individuals (here, three). They are labeled α, β and γ in the metalanguage. To the right of the diagram are three ‘M -relative’ rules of inference. These rules ensure that the individuals are pairwise distinct. Their conclusions are ⊥ and their premises are what we shall call saturated identity formulae (see below) involving all possible pairs of our chosen individuals. Such rules are a necessary part of an eventual set of rules that will completely capture the diagram that will be on the left once the model has been fully constructed. Indeed, the eventual set of rules on the right can be thought of as an adequate substitute for the model itself. 2

M • α

1.2

• β

α = βM ⊥

• γ

α = γM ⊥

β = γM ⊥

A digression on saturated terms and formulae

In general, terms and formulae of the object-language may contain free variables. If they do, then they are called open. Those that are not open are called closed. A closed formula is called a sentence. A closed term may be called a (simple or complex) name. The semantic value of a name, when it has one, is an individual, which the name is then said to denote. The semantic value of a sentence, when it has one, is a truth-value, and the sentence is said to be true or false according as that value is T or F . Closing an open term or formula involves substituting closed terms (of the object-language) for free occurrences of variables. Thus one could substitute the object-linguistic name j for the free occurrence of the variable x in the open term f (x), to obtain the closed term f (j) (‘the father of John’). Or, to complicate the example slightly, one could substitute the closed objectlinguistic term m(j) for that free occurrence, to obtain the closed term f (m(j)) (‘John’s maternal grandfather’). Likewise, an open formula, say L(x, y), may be closed by substituting closed terms for its free occurrences of variables. One such closing would result in the sentence L(m(f (j)), f (j)) (‘John’s paternal grandmother loves his father’). Here we shall introduce an operation on open terms and formulae analogous to the operation of closing, but importantly different from it. The new operation will be called saturation. Like the operation of closing, the operation of saturation gets rid of all free occurrences of variables within an object-linguistic term or formula. But the way it does so is importantly different. Instead of substituting closed object-linguistic terms for free occurrences of variables, saturation is effected by substituting individuals from the domain for those free occurrences. Thus if α and β are individuals from the domain, one saturation of the open formula L(x, y) would be L(α,β). Another one would be L(m(f (α)),f (α)), where the saturation is effected by substituting the saturated terms m(f (α)) and f (α) for the free occurrences of the variables x and y respectively. When the domain D supplies all the individuals involved in a saturation

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operation, the resulting saturated terms are called saturated D-terms, and the resulting saturated formulae are called saturated D-formulae. Saturated terms and formulae are object-linguistic and metalinguistic hybrids. But, as mathematical objects, they are well-defined. When one treats, in standard Tarskian semantics, of assignments of individuals to variables, one is assuming such well-defined status for ordered pairs of the form hx,αi, where x is a free variable of the object-language, and α is an individual from the domain of discourse.1 Since standard semantics is already committed to the use of such hybrid entities, it may as well take advantage of similar hybrid entities such as saturated terms and formulae. We shall be taking advantage of them by having them feature in the rules of inference on the right in our description of models. Indeed, such rules will form a constitutive part of the model in question, as will emerge in due course.

1.3

Back to the exposition of models

With all subsequent displays of rules on the right, we shall omit the subscripts M next to their inference strokes, as they will always be understood from the context. (Occasionally we shall restore them, for appropriate emphases.) We shall also suppress the label M at the top and left of the diagram. The model M is not yet completely specified. For so far we have specified only its domain; we have yet to specify its predicational and functional structure. Note how we have labelled the individuals as α, β and γ by placing these labels right next to them, within the outer box that represents the ‘boundary’ of the domain. By means of this convention we indicate that the labels are metalinguistic. As far as the object-language is concerned, the individuals could be nameless. Suppose, however, that the individuals happened to possess names in the object-language—say, a, b and c respectively. Then this circumstance will be conventionally indicated by placing those names outside the outer box, and indicating, with arrows, what their respective denotations are. To the right we would specify the denotation 1

This is true not only of Tarski’s original treatment [1], which invoked infinite sequences of individuals correlated with object-linguistic variables, but also of the treatment (in [2]) of Tarski’s approach that appeals, more modestly, to finitary assignments of individuals to the free variables in a formula.

4

mapping d: a

b

c

?

• β

?

• γ

• α

?

d(a) = α; d(b) = β; d(c) = γ

Our toy example will not, however, involve any names in the object-language, so we can set these details aside. Next, we add the structure of a one-place predicate P , by showing its extension in the diagram, and also by supplementing our rules of inference in order to show which individuals are, and which are not, in the extension of the predicate P . (If we failed to specify, by means of our rules, which individuals are not in the extension of P , then the rules themselves would fail, collectively, to specify the intended diagram, in which certain individuals definitely lie outside the enclosure indicating the extension of P .)

α=β α=γ β=γ ⊥ ⊥ ⊥

P • α

• β

• γ

P (β) P (γ) P (α) ⊥ ⊥

Finally, we add the structure of a one-place function f , supplementing again on the right with the relevant ‘axioms of function-values’:

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P • α

- •

β

α=β α=γ β=γ ⊥ ⊥ ⊥

n - •

γ

P (β) P (γ) ⊥ ⊥

P (α)

f (α) = β

f (β) = γ

f (γ) = γ

The axioms and rules on the right re-state what one can read off from the diagram on the left. There is, however, one aspect of the diagram that is only ‘shown’, and not yet ‘said’, by the rules on the right. This is that the individuals α, β and γ are all the individuals there are. This feature of the diagram will find expression in the following ‘M -relative’ rules of evaluation: (i)

ψ(α) ψ(β) ψ(γ) M ∀xψ(x)

∃xψ(x)

(i)

(i)

ψ(α) .. .

ψ(β) .. .

ψ(γ) .. .

⊥

⊥

⊥

(i) M

⊥ The first of these rules, which allows one to evaluate as true any universal claim ∀xψ(x), requires only that ψ should hold of each of the individuals α, β and γ. The second rule, which allows one to evaluate as false any existential claim ∃xψ(x), requires only that ψ should not hold of any of the individuals α, β and γ. (One shows that a claim does not hold by assuming that it does, and deriving ⊥ from that assumption.) It is clear that these M -relative rules for the verification of universals and the falsification of existentials constrain the domain D to consist of exactly the individuals α, β and γ. And so it will be with any finite model. Let N be such, and suppose its domain consists of exactly the n individuals α1 , . . . , αn . Then the N -relative rules for the verification of universals and the falsification of existentials will be

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(i)

ψ(α1 ) . . . ψ(αn ) N ∀xψ(x)

∃xψ(x)

(i)

ψ(α1 ) ψ(αn ) .. .. ... . . ⊥ ⊥

(i) N

⊥ Let us now return to our toy model M , and consider some simple sentences that can be verified, or falsified, in M . We shall choose sentences simple enough for easy, intuitive determination of their truth-values in M . The point of the exercise is to show how that intuitive determination can be represented formally as a certain kind of M -relative ‘evaluation proof’ or ‘evaluation disproof’ of the sentence in question. Here again is our model M : M

The M -rules:

P • α

- •

n - •

β

γ

α=β α=γ β=γ ⊥ ⊥ ⊥ P (α)

P (β) P (γ) ⊥ ⊥

f (α) = β

f (β) = γ

f (γ) = γ

Intuitively it is easy to see that the following sentences are true-in-M , or false-in-M , as indicated: 1. 2. 3. 4. 5.

True-in-M ∃xP (x) ∃x f (x) = x ∃x(P (x) ∧ ¬f (x) = x) ∀x(¬P (x) ∨ ¬f (x) = x) ∀x(f (f (x)) = x → ¬P (x))

False-in-M ∀xP (x) ∀x ¬f (x) = x ∀x(¬P (x) → f (x) = x) ∀x∃y f (y) = x P (f (α)) ∨ P (f (β))

The first two truths are easy to verify: M

P (α) ∃xP (x)

M

f (γ) = γ ∃x f (x) = x

In both these evaluation proofs, the final step verifies an existential conclusion on the basis of an appropriate instance—α in the first, γ in the second. 7

The general form for the verification of an existential claim ∃xψ(x) is: .. . ψ(α) ∃xψ(x)

where α is an individual in the domain.

The vertical dots indicate the presence of some verification of some instance ψ(α), for some individual α in the domain of the model. For the third truth in the list above, ∃x(P (x) ∧ ¬f (x) = x), we make use of the following rule (for the construction of verifications and/or falsifications) governing identity: (i)

α1 = α2 M ⊥

t = α1 t = α2

(i)

where t is any saturated D-term

⊥ The M -relative evaluation-proof, or verification, of ∃x(P (x) ∧ ¬f (x) = x) is as follows: (1)

α=β ⊥

(2)

f (α) = α

M

f (α) = β

(1)

⊥ M (2) P (α) ¬f (α) = α P (α) ∧ ¬f (α) = α ∃x(P x ∧ ¬f (x) = x) Note that the penultimate step verifies the conjunction P (α)∧¬f (α) = α by verifying the left conjunct P (α) and verifying the right conjunct ¬f (α) = α. The general form of such a step is ϕ ψ ϕ∧ψ The step labelled (2) is an application of the following general rule for verifying negations: (i)

ϕ .. . ⊥ ¬ϕ

(i)

8

where the vertical dots indicate the presence of some falsification of ϕ. The fourth truth on our list, ∀x(¬P (x) ∨ ¬f (x) = x), has the following M -relative verification: (4)

α=β ⊥

(1)

M

(2)

f (α) = β

f (α) = α

⊥ (1) ¬f (α) = α ¬P (α) ∨ ¬f (α) = α

(3)

P (β) M P (γ) M (4) ⊥ (2) ⊥ (3) ¬P (β) ¬P (γ) ¬P (β) ∨ ¬f (β) = β ¬P (γ) ∨ ¬f (γ) = γ ∀x(¬P (x) ∨ ¬f (x) = x)

Here, at the final step, we see in action the rule for verifying universal claims in any model whose only individuals are α, β and γ. One has to verify each of the three instances of the universal claim. Each of the three immediately preceding steps involved verifying a disjunction by verifying one or the other of its disjuncts. The general form of such a step is one or the other of ϕ ϕ∨ψ

ψ ϕ∨ψ

Even when both disjuncts are true, the most economical route to the verification of the disjunction is to focus on the disjunct with the easiest verification. Hence there is no need to add to, or alter, the form of these last two rules for the verification of disjunctions. Let us turn our attention now to the first four false sentences in the list above. For these we need to construct M -relative falsifications. It takes only one counterinstance to render a universal claim false. To falsify ∀xP (x), we have a choice between β and γ as counterinstance. (Since α lies in the extension of P , we obviously cannot appeal to α in order to falsify ∀xP (x).) Let us take β for this purpose. The falsification is then as follows: (1)

∀xP (x)

P (β) M ⊥ (1)

⊥ Any falsification of a saturated formula ϕ will use ϕ as an assumption— indeed, its sole undischarged assumption—and derive ⊥ therefrom, using as its remaining premises only the ‘basic claims’ about the model in question. (A ‘basic claim’ can of course be an inference from a primitive saturated formula to ⊥, as we have seen.) Note how the final step of this last two-step 9

M

falsification discharges the assumption P (β) within the subordinate (onestep) falsification, on the right, of P (β) itself. The final occurrence of ⊥ is thereby made to depend only on the undischarged assumption ∀xP (x), rather than on the (false!) instance P (β) of that universal claim. The general evaluation rule being applied here for the falsification of universal claims is the following: (i)

ψ(α) .. . ∀xψ(x)

⊥

(i)

⊥ where the vertical dots indicate the presence of some falsification of some instance ψ(α). (In our last example, this was a one-step falsification. But in general, of course, there can be more than one step, the exact number of steps depending both on the complexity of ψ and on the chosen counterinstance α.) The second falsehood on our list above, ∀x ¬f (x) = x, enjoys the following falsification: (1)

M

¬f (γ) = γ ∀x ¬f (x) = x

f (γ) = γ ⊥

(1)

⊥ The upper step is an application of the general evaluation rule .. . ϕ

¬ϕ ⊥

for the falsification of negations. The vertical dots indicate the presence of some verification of ϕ. The third falsehood on our list, ∀x(¬P (x) → f (x) = x), can be shown to be false as follows. The falsification involves choosing β as the counterinstance to the universal claim. This choice generates the task of falsifying the conditional claim ¬P (β) → f (β) = β. A falsification of a conditional θ → χ consists in a verification of the antecedent θ and a falsification of the consequent χ:

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(i)

θ→χ

χ .. .

.. . θ

⊥

(i)

⊥ So we need in the case at hand to verify ¬P (β) and to falsify f (β) = β. The details are as follows: (2)

(4)

(3)

¬P (β) → f (β) = β

β=γM P (β) ⊥ ⊥ (4) ¬P (β)

∀x(¬P (x) → f (x) = x)

⊥

(1)

⊥

M

f (β) = γ

f (β) = β (1)

(3)

⊥ The step labelled (2) is an application of the rule for identity explained earlier. We come to our fourth illustration of falsifications, that of the sentence ∀x∃y f (y) = x. A glance at the diagram for M reveals that α is the counterinstance we want—for it is the only individual that is not the f -value of any individual in the domain. Our task is now to fill in the dots in the falsification schema (1)

∃y f (y) = α .. . ⊥

∀x∃y f (y) = x

(1)

⊥ In order to falsify the existential ∃y f (y) = α, we need to falsify each of its three instances f (α) = α, f (β) = α, and f (γ) = α. The task accordingly reduces to that of filling in the dots in the schema (2)

(1)

(2)

(2)

f (α) = α .. .

f (β) = α .. .

f (γ) = α .. .

⊥

⊥

⊥

∃y f (y) = α ∀x∃y f (y) = x

⊥ ⊥

11

(1)

(2)

(2)

The desired M -relative falsifications of f (α) = α, f (β) = α and f (γ) = α (to fill in the respective vertical dots) are as follows: (3)

α=β ⊥

M

f (α) = α

f (α) = β

(3)

⊥ (4)

α=γ ⊥

M

f (β) = α

f (β) = γ

(4)

⊥ (5)

α=γ ⊥

M

f (γ) = α

f (γ) = γ

(5)

⊥ We refrain here from inserting these into the previous schema, since the sideways spread would be too wide for the page to contain. The reader will appreciate that these insertions would be completely straightforward, if one had paper wide enough. Our exposition of the rules for verifying and/or falsifying sentences (or saturated formulae) is not yet complete. We have not yet seen how to verify a conditional claim (one of the form ϕ → ψ), nor how to falsify a disjunctive claim (one of the form ϕ ∨ ψ). In order to see these rules in action, we turn to the fifth truth and the fifth falsehood in our list. The fifth truth is the claim ∀x(f (f (x)) = x → ¬P (x)). As we inspect the diagram M , we realize that this universal generalization is true for want of a counterinstance. No individual in the domain is such that it is identical to its own f f -image, yet has the property P . Each of α, β and γ fails to be such a counterinstance, and in interesting ways. First, α is not its own f f -image. (It is, however, in the extension of P .) Secondly, β, likewise, is not its own f f -image; but, also, β is not in the extension of P . Indeed, it is easier to tell that the latter is the case, than that the former is the case. So, as far as β is concerned, its failure to be a counterinstance to the claim ∀x(f (f (x)) = x → ¬P (x)) is more easily shown by showing ¬P (β) than by deriving ⊥ from f (f (β)) = β. Thirdly, γ fails to be a counterinstance to the claim ∀x(f (f (x)) = x → ¬P (x)) for a similar reason: P (γ) is false. Our verification of the claim ∀x(f (f (x)) = x → ¬P (x)) of course pro-

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ceeds by verifying each of its three instances f (f (α)) = α → ¬P (α) f (f (β)) = β → ¬P (β) f (f (γ)) = γ → ¬P (γ) and we are proposing to do this, instance by instance, as follows: (1)

f (f (α)) = α .. .

.. .. . . ¬P (β) ¬P (γ) f (f (β)) = β → ¬P (β) f (f (γ)) = γ → ¬P (γ)

⊥ (1) f (f (α)) = α → ¬P (α) ∀x(f (f (x)) = x → ¬P (x)) The missing falsification is as follows: M

(2)

α=γM ⊥

f (f (α)) = α

M

f (β) = γ f (α) = β f (f (α)) = γ (2)

⊥ The missing verification in the middle we have seen before, and the rightmost one is similar to it. They are as follows: (4)

P (β) ⊥ (4) ¬P (β)

(5)

P (γ) ⊥ (5) ¬P (γ)

Once again, it is left to the reader to insert these last three bits of detailed working into the appropriate places (indicated by the vertical dots) in the preceding verification-schema. Some comments are now in order on the new rules that have just found application within the last example. First, we have seen that there are two ways to verify a conditional ϕ → ψ: one can falsify its antecedent ϕ, or one can verify its consequent ψ. Both these methods found application in the verification above of the claim ∀x(f (f (x)) = x → ¬P (x)). Its α-instance was verified by falsifying the antecedent; while the β- and γ-instances were verified by verifying their consequents. The general form of the rule for verifying a conditional is accordingly

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M

(i)

.. . ψ ϕ→ψ

ϕ .. . ⊥ ϕ→ψ

(i)

where the missing dots on the left indicate the presence of a verification of the consequent ψ, while those on the right indicate the presence of a falsification of the antecedent ϕ. Our manipulations of identity statements in the last example were also novel. The step f (β) = γ f (α) = β f (f (α)) = γ is a special case of the rule f (α1 ) = α u1 = α1 f (u1 ) = α where u1 is a saturated D-term and α, α1 are individuals in the domain A moment’s reflection reveals that this is the canonical way to establish a conclusion of the form f (u1 ) = α. How does one work out that f (u1 ) is indeed the object α? One first finds the object α1 denoted by the contained (saturated) term u1 . (See the second premise of the rule just stated.) Then one finds the object (call it α) denoted by the (saturated) term f (α1 ). (See the first premise.) That object α is thereby shown to be (the denotation of the term) f (u1 ). (See the conclusion.) So this rule for identity simply teases out what is involved in computing function-values stage-by-stage. The rule just stated can be generalized to n-place functions as follows: f (α1 , . . . , αn ) = α u1 = α1 ... f (u1 , . . . un ) = α

un = αn

where f is a primitive n-place function sign, u1 , . . . , un are saturated D-terms and α, α1 , . . . , αn are individuals in the domain The computation of the values of the (saturated) terms u1 , . . . , un proceeds in parallel.

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We turn now to our final example, which is the fifth falsehood: P (f (α)) ∨ P (f (β)). In order to falsify a disjunction ϕ ∨ ψ, one needs to falsify each disjunct: (i)

ϕ∨ψ

(i)

ϕ .. .

ψ .. .

⊥

⊥

(i)

⊥ Applied to the case at hand, we obtain the falsification schema (1)

(1)

P (f (α)) .. .

P (f (β)) .. .

⊥

⊥

P (f (α)) ∨ P (f (β))

(1)

⊥ The missing falsifications to be inserted here are (2)

P (β) M ⊥

M

f (α) = β

P (f (α)) (2)

⊥ and (3)

P (γ) M ⊥

M

f (β) = γ

P (f (β)) (3)

⊥ The final steps of the last two falsifications were applications, for the case n = 1, of the identity rule (i)

A(α1 , . . . , αn ) M ⊥

t1 = α1 . . . tn = αn

A(t1 , . . . , tn )

(i)

⊥ where A is a primitive n-place predicate, t1 , . . . , tn are saturated 15

D-terms, and α1 , . . . , αn are individuals in the domain In order to motivate the formulation of a few more rules governing the construction of verifications and falsifications, let us consider a slight embellishment of our earlier model M . Let M 0 be the following model, identical to M on the vocabulary of M , but with a new, two--place predicate Lxy (to be read as ‘x is to the left of y’). The extension of L is represented by dashed arrows in the diagram for M 0 . α=β α=γ β=γ ⊥ ⊥ ⊥ M0

P (α) P - • • P P α P Pβ P

n - • 1 γ

P (β) P (γ) ⊥ ⊥

f (α) = β

f (β) = γ

f (γ) = γ

L(α, α) L(α, β) L(α, γ) ⊥ L(β, α) L(β, β) L(β, γ) ⊥ ⊥ L(γ, α) L(γ, β) L(γ, γ) ⊥ ⊥ ⊥

Note that all the rules of the model M are rules of the model M 0 . We extend our earlier list of sentences whose truth-values are to be determined by means of the rules framed here. 1. 2. 3. 4. 5. 6. 7.

True-in-M 0 ∃xP (x) ∃x f (x) = x ∃x(P (x) ∧ ¬f (x) = x) ∀x(¬P (x) ∨ ¬f (x) = x) ∀x(f (f (x)) = x → ¬P (x)) f (f (f (α))) = f (f (α)) L(f (α), f (β))

False-in-M 0 ∀xP (x) ∀x ¬f (x) = x ∀x(¬P (x) → f (x) = x) ∀x∃y f (y) = x P (f (α)) ∨ P (f (β)) f (f (f (α))) = α ∃xP (x) ∧ ∀x f (f (x)) = x

16

The seventh sentence on the left, L(f (α), f (β)), is true in M 0 . Its M 0 relative verification is: M

L(β, γ)

M

f (α) = β L(f (α), f (β))

M

f (β) = γ

The final step here is an instance, for the case n = 2, of the following identity rule: A(α1 , . . . , αn ) t1 = α1 . . . A(t1 , . . . , tn )

tn = αn

where A is a primitive n-place predicate, t1 , . . . , tn are saturated D-terms, and α1 , . . . , αn are individuals in the domain. The careful reader will have noticed, from the three substitution rules already stated for identity, that there is a pattern to them that would be completed by having the following rule: (i)

f (α1 , . . . , αn ) = α M ⊥

u1 = α1 . . . un = αn

f (u1 , . . . , un ) = α

(i)

⊥ where f is a primitive n-place function sign, u1 , . . . , un are saturated D-terms, and α, α1 ,. . . ,αn are individuals in the domain. An example in which this rule finds application is the falsification of the sixth sentence on the right in the list above: f (f (f (α))) = α. The rule is applied in the final step of the following falsification, whose rightmost premise is the claim being falsified: (1)

f (γ) = α M ⊥

M

M

f (β) = γ f (α) = β f (f (α)) = γ

f (f (f (α))) = α (1)

⊥ There is only one more ‘model-relative’ rule (or axiom) that needs to be specified in order to have an adequate set of rules for the verification and 17

falsification of claims in a first-order language with identity. It is the rule of reflexivity of identity, stated only for individuals α in the domain of the model in question: where α is an individual in the domain

α=α

This rule enables one to derive the identity claim t = u (where t and u are saturated D-terms of any complexity) when each of those terms has been verified as denoting some same individual α: α=α

t=α u=α t=u

Note that this inference is an application of our earlier substitutivity rule for n-place atomic predicates A—identity being a two-place atomic predicate that can take the place of A in the statement of that rule. In our model M , for example, the claim f (f (f (α))) = f (f (α)) is true. Its verification, whose final step is of the form just displayed, is M

γ=γ

f (β) = γ f (α) = β M f (γ) = γ f (f (α)) = γ f (f (f (α))) = γ f (f (f (α))) = f (f (α))

M M

f (β) = γ f (α) = β f (f (α)) = γ

Consider a conjunction that is false in M , such as ∃xP (x) ∧ ∀x f (f (x)) = x. (This is the seventh sentence on the right in the list above. It is the righthand conjunct that is false.) A falsification of the whole conjunction in such a case must proceed by falsifying the culprit conjunct. The general form for falsification of a conjunction ϕ ∧ ψ via its right conjunct ψ is as follows: (i)

ψ .. . ϕ∧ψ

⊥

(i)

⊥ Of course there must also be a way of falsifying a conjunction when it is only its left conjunct that is false:

18

M

(i)

ϕ .. . ⊥

ϕ∧ψ

(i)

⊥ Even when both conjuncts are false, the most economical route to the falsification of the conjunction is to focus on the conjunct with the easiest falsification. Hence there is no need to add to, or alter, the form of these last two rules for the falsification of conjunctions. We are now in a position to take stock by listing all the rules to which we have had recourse in verifying or falsifying our chosen example sentences above. It is worth stressing that these are rules for determination of the truth-value of an arbitrary sentence, using as synthetic ‘first principles’ the (inferentially coded) basic information in the model in question. The analytic ‘first principles’ are the rules themselves. They characterize the meanings of logical expressions in terms of their roles in determining sentences as true, or as false, in any model. Rules for verification and falsification of primitive saturated formulae.

A(α1 , . . . , αn ) t1 = α1 . . . A(t1 , . . . , tn )

tn = αn

where A is a primitive n-place predicate, t1 , . . . , tn are saturated D-terms, and α1 , . . . , αn are individuals in the domain.

(i)

A(α1 , . . . , αn ) M ⊥

t1 = α1 . . . tn = αn

A(t1 , . . . , tn )

(i)

⊥ where A is a primitive n-place predicate, t1 , . . . , tn are saturated D-terms, and α1 , . . . , αn are individuals in the domain.

19

f (α1 , . . . , αn ) = α u1 = α1 ... f (u1 , . . . un ) = α

un = αn

where f is a primitive n-place function sign, u1 , . . . , un are saturated D-terms and α, α1 , . . . , αn are individuals in the domain

(i)

f (α1 , . . . , αn ) = α M ⊥

u1 = α1 . . . un = αn

f (u1 , . . . , un ) = α

⊥ where f is a primitive n-place function sign, u1 , . . . , un are saturated D-terms, and α, α1 ,. . . ,αn are individuals in the domain.

α=α

where α is an individual in the domain

Rules for verification and falsification of saturated formulae with a connective dominant

(i)

ϕ .. . ⊥ ¬ϕ

.. . ϕ

¬ϕ (i)

⊥

(i)

ϕ ψ ϕ∧ψ

(i)

ϕ .. . ϕ∧ψ ⊥

20

⊥

ψ .. . (i)

ϕ∧ψ ⊥

⊥

(i)

(i)

(i)

ϕ ϕ∨ϕ

ψ ϕ∨ψ

ϕ∨ψ

(i)

ϕ .. .

ψ .. .

⊥

⊥

(i)

⊥

(i)

(i)

.. . ψ ϕ→ψ

ϕ .. . ⊥ ϕ→ψ

(i)

ψ .. .

.. . ϕ

ϕ→ψ

⊥

(i)

⊥

Rules for verification and falsification of saturated formulae with a quantifier dominant

(i)

ψ(α1 ) . . . ψ(αn ) . . . ∀xψ(x)

ψ(α) .. . ⊥

∀xψ(x)

(i)

⊥

(i)

.. . ψ(α) ∃xψ(x)

∃xψ(x)

(i)

ψ(α1 ) ψ(αn ) .. .. ... ... . . ⊥ ⊥ (i)

⊥

We have noted above how quickly one can incur sideways spread in writing down a detailed verification or falsification. This feature militates against the actual construction of these otherwise very illuminating and 21

detailed constructions for sentences (and saturated formulae) relative to a given model M . As soon as one has three or more individuals in the domain of M , along with nested quantifiers (especially when they occasion the use of the two rules that require investigation of all instances of a quantified claim), the blow-up, in the form of sideways spread, is prohibitive. But the resulting construction is only ever as deep as the longest branch within the analysis tree of the sentence (or saturated formula) being evaluated. Moreover, in cases where the domain is infinite, some of these verifications and falsifications will contain steps (for the verification of a universal, or the falsification of an existential) that require infinitely many premises (in the form of instances of the quantified claims in question). In such cases the constructions cannot be written down. Instead, they exist only as infinitary mathematical objects: labelled trees (where the labels are at least finite!) that can have infinite branching, albeit only with branches of finite length. Ultimately, the present ‘inferentialist’ approach to formal semantics via the verifications and falsifications illustrated above requires no more powerful mathematical machinery than is needed in order to vouchsafe the existence of these (rather modest) kinds of infinitary object.

2 2.1

General consequences of the rules of evaluation Identity

Our rules of evaluation, which are framed in the metalanguage, allow for reflexivity of identity only in the form

α=α

where α is an individual in the domain

This is not the same as saying that the rule

t=t

where t is a term of the object-language

is valid. But its validity is not hard to establish, using constructive reasoning in the metalanguage, for models in which all names in the object language denote, and all functions signs represent total functions. Every such model M for any language containing the extra-logical expressions involved in the term t has in its domain some individual α for which there is a verification V , say, of the claim t = α. Applying the first rule for verification of primitive saturated formulae, one obtains the verification

22

V V t=α t=α t=t Thus it follows from our rules of evaluation in the metalanguage that reflexivity of identity holds generally for terms t of the object-language. A similar account can be given of the validity of the rule of substitutivity of identicals for terms t, u in the object-language: ϕ

t=u ψ

where ϕut = ψtu

Reflexivity of identity and substitutivity of identicals therefore become available as general rules of deduction. They preserve truth from their premises to their conclusions. (In the case of reflexivity, of course, the set of premises is empty. Its conclusion is true in every model of any language containing the extra-logical expressions in t .)

2.2

Non-contradiction

It is clear that the basic axioms and rules in the atomic diagram of a model are coherent, in the sense that for no primitive saturated formula does the model contain both the axiom allowing one to infer it from no assumptions, and the rule allowing one to infer ⊥ from it. This point generalizes. Lemma 1 Let M be a model for the extra-logical vocabulary in an M saturated formula ϕ. Then there cannot be both an M -relative verification of ϕ and an M -relative falsification of ϕ. Proof. By induction on the complexity of ϕ. The basis step is obvious. Inductive hypothesis: Assume that the result holds for all M -saturated subformulae of ϕ. Inductive step: By cases, according as ϕ is of the form (i) ¬ψ, (ii) ψ1 ∧ ψ2 , (iii) ψ1 ∨ ψ2 , (iv) ψ1 → ψ2 , (v) ∃xψ, or (vi) ∀xψ. In what follows, λ and µ (with or without numerical subscripts) will be parts of the atomic diagram of M , and V and F (with or without numerical subscripts) will be M -relative verifications and falsifications. We shall also assume that the individuals in the domain of M are α1 , . . . , αn , . . . . Case (i). Any M -relative verification of ¬ψ would take the form

23

(i)

λ, ψ F ⊥ (i) ¬ψ and any M -relative falsification of ψ would take the following form µ V ψ

¬ψ ⊥

But then ψ would have both an M -relative verification and an M -relative falsification, contrary to the inductive hypothesis. Case (ii). Any M -relative verification of ψ1 ∧ ψ2 would take the form µ1 µ2 V1 V2 ψ1 ψ2 ψ1 ∧ ψ2 and any M -relative falsification of ψ1 ∧ ψ2 would take one of the following two forms: (i)

ψ1 ∧ ψ2 ⊥

λ1 , ψ1 F1 ⊥

(i)

(i)

ψ1 ∧ ψ2 ⊥

λ2 , ψ2 F2 ⊥

(i)

Either way, we would have both an M -relative verification and an M -relative falsification of one of the conjuncts, contrary to the inductive hypothesis. Case (iii) and Case (iv) are similar. Case (v). Recall that the individuals in the domain of M are α1 , . . . , αn , . . . . So any M -relative verification of ∀xψ(x) would take the form

24

µ1 V1 ψ(α1 )

µv ... Vn ψ(αn ) ∀xψ(x)

...

and any M -relative falsification of ∀xψ(x) would take the form (i)

∀xψ(x) ⊥

λi , ψ(αi ) Fi ⊥ (i)

The instance ψ(αi ) would therefore have both an M -relative verification and an M -relative falsification, contrary to the inductive hypothesis. Case (vi) is similar.

3

QED

Special features of the rules of evaluation

The foregoing rules of evaluation permit the construction of proof-like objects (verifications and falsifications; or, evaluation proofs and evaluation disproofs). They do, however, have some special limiting characteristics.

3.1

The undischarged assumptions

First, the ‘undischarged assumptions’ of a verification are always either (saturated) primitive formulae, or rules equivalent to negations thereof. When one constructs a verification using the primitive (positive or negative) information in a model (its atomic diagram Λ), there is no complexity in the undischarged assumptions involved (apart from the negation signs in negative literals). The same holds for a falsification, except that the (saturated) formula being falsified may itself be complex. But it will be the only complex formula among the undischarged assumptions of the falsification. So: apart from the complex formula being falsified (when the construction in question is a falsification), construction by means of our rules allows us to ‘reason away from’ at best primitive formulae (and negations thereof). We can emphasize this point by adding mention, within the statement of our rules, of the primitive information upon which the evaluation rests. We shall use λ, λ1 , λ2 as variables ranging over subsets of the atomic diagram Λ.

25

(We use the Greek letter lambda to suggest ‘literals’.) We shall illustrate the point by reference to the rules for the connectives and the quantifiers.

(i)

λ .. .

λ, ϕ .. . ⊥ ¬ϕ

¬ϕ

(i)

λ1 .. .

ϕ ⊥

(i)

λ2 .. .

(i)

λ, ϕ .. .

ϕ ψ ϕ∧ψ

⊥

ϕ∧ψ

λ, ψ .. .

⊥

λ .. .

λ .. .

ϕ ϕ∨ψ

ψ ϕ∨ψ

⊥

(i)

ϕ∨ψ

(i)

λ1 , ϕ .. .

λ2 , ψ .. .

⊥

⊥

λ, ϕ .. .

ψ ϕ→ψ

⊥ ϕ→ψ

(i)

⊥

(i)

(i)

λ .. .

(i)

ϕ→ψ

λ1 .. .

λ2 , ψ .. .

ϕ

⊥

⊥

26

⊥

ϕ∧ψ

(i)

(i)

(i)

λ1 .. . ψ(α1 )

...

λn .. .

ψ(αn ) ∀xψ(x)

3.2

...

⊥

∀xψ(x)

(i)

⊥

(i)

λ .. . ψ(α) ∃xψ(x)

(i)

λ , ψ(α) .. .

λ1 , ψ(α1 ) .. . ∃xψ(x)

(i)

...

λn , ψ(αn ) .. .

⊥

⊥

... (i)

⊥

Conclusions

The second limiting characteristic is that the rules for falsification have ⊥ as their main conclusions, and as conclusions of their subordinate ‘disproofs’. So: the only way to ‘reason away from’ a complex formula (by means of our rules of evaluation) is to reason towards absurdity.

3.3

Domain-dependence of quantifier rules

Thirdly, the rules for verification of universals and for falsification of existentials call for as many subordinate proofs as there are individuals in the domain (one subordinate proof for each individual). And this involves infinite sideways branching when the domain of the model is infinite. (Remember, our verifications and falsifications are model-relative. The are not like deductions in general. The job of a deduction—which is always finitary—is to preserve M -relative truth from its premises to its conclusion, for all models M .)

3.4

The contrast with rules of deduction in general

General rules of deduction allow one in general to reason away from (finite) sets of sentences of any complexity to sentences of any complexity. Of course, primitive sentences (and negations thereof) can stand as assumptions of deductions; and absurdity can stand as a conclusion (in which case the 27

deduction is called a reductio ad absurdum, or refutation, of its set of undischarged assumptions). But deduction in general involves reasoning from a (finite) set of complex sentences as assumptions, to a complex conclusion. We are now in pursuit of rules of inference governing such reasoning, rules in accordance with which more general proofs can be constructed—more general, that is, than our model-relative evaluation proofs (i.e., verifications) and evaluation disproofs (i.e., falsifications). When deducing a sentence ϕ from a set ∆ of sentences, we are no longer working with the atomic diagram of a particular model. Rather, the sentences in the set ∆ (the premises of our sought proof) might all be true ‘simultaneously’ in many different models. The task is to show that in any such model, the sentence ϕ will be true too. That is the job of proof in general. When one has a proof of ϕ whose undischarged assumptions form the set ∆, one must be able to say: any model that verifies every member of ∆ verifies ϕ. This means that we cannot use the present rule for verification of universals when it comes to deductive reasoning towards a universal (trying to establish it as a conclusion); nor can we use the present rule for falsification of existentials when it comes to deductive reasoning away from an existential (trying to use it as a premise). For, both these rules call for a specific number of subordinate deductions, one for each individual in the domain of a specific model (relative to which truth-value determination takes place according to the evaluation rule in question). Deductive reasoning, however, is undertaken without any specific model in mind. What is important is only preservation of truth-value from premises to conclusion—so that every model for the premises is a model for the conclusion.

4

From rules of evaluation to rules for deduction in general

Our task now is to find suitable generalizations or analogues of our rules for verification and falsification that can serve as rules governing deduction towards, and deduction away from, complex sentences. As we survey our rules of verification and falsification, certain of these analogues are immediate. A box subscript on a discharge stroke indicates that the assumption in question must have been used, and therefore be eligible to be discharged. (This was obvious in the case of verifications and falsifications, but now needs to be emphasized, since we are moving towards a statement of rules of inference in general.) 28

We shall take the connectives in turn, as we morph the rules for verification and falsification of sentences with a given connective dominant into more general rules of deduction. These are introduction rules (which tell one how to introduce a dominant occurrence of that connective into the conclusion of an inference), and elimination rules (which tell one how to eliminate a dominant occurrence of the connective from the major premise of an inference).

4.1

Negation

The sought generalizations of the rules for negation are straightforward. 4.1.1

Introduction

The rule for verifying a negation becomes the negation introduction rule upon allowing for more general sets ∆ of side-assumptions in the subordinate reductio:

(i)

∆, ϕ .. . ⊥ ¬ϕ

(i)

Note that the conclusion ¬ϕ depends only on the assumptions in ∆; the assumption ϕ (for reductio ad aburdum) is discharged by applying the rule. Moreover, as indicated by the box subscript on the discharge stroke, the assumption ϕ must have been used, and be undischarged within the subordinate reductio, in order that the rule be applicable. 4.1.2

Elimination

The rule for falsifying a negation becomes the negation elimination rule upon allowing for more general sets ∆ of assumptions in the subordinate proof of the minor premise: ∆ .. . ¬ϕ

ϕ ⊥

29

The conclusion rests both on ¬ϕ and on the assumptions in ∆. Note also that we are not allowing the major premise ¬ϕ to stand, itself, as the conclusion of any proof-work above it. Rather, ¬ϕ stands proud as an undischarged assumption. (It could, however, be discharged by subsequent applications of rules of inference, as the proof-work proceeded further in a downward direction).

4.2

Conjunction

4.2.1

Introduction

The morphing of the rule for model-relative verification of conjunctions into the introduction rule (for inferring conjunctions from arbitrary sets of premises) is straightforward: ∆1 .. .

∆2 .. .

ϕ ψ ϕ∧ψ Note that we allow for ∆1 to be distinct from ∆2 . The conclusion ϕ ∧ ψ depends on their union. 4.2.2

Elimination

Now consider the rule for model-relative falsification of a conjunction: (i)

(i)

∆1 , ϕ .. . ⊥

ϕ∧ψ

∆2 , ψ .. . ⊥

ϕ∧ψ

(i)

⊥

(i)

⊥

We want the elimination rule to allow for the derivation of general conclusions θ in place of ⊥: (i)

(i)

∆1 , ϕ .. . ϕ∧ψ

θ θ

(i)

∆2 , ψ .. . ϕ∧ψ

θ θ 30

(i)

and we may as well economize by allowing for simultaneous discharge of the dischargeable assumptions. At the same time we require that at least one such assumption should have been used, and therefore be eligible to be discharged—this requirement being indicated by a box affixed to the inference strokes: (i)

(i)

∆ ,ϕ, ψ .. . ϕ∧ψ

θ

(i)

θ The conclusion θ depends only on ϕ ∧ ψ and the assumptions in ∆. The major premise ϕ ∧ ψ stands proud.

4.3 4.3.1

Disjunction Introduction

As with conjunction, the introduction rule for disjunction is a straightforward generalization of the rule of model-relative verification:

4.3.2

∆ .. .

∆ .. .

ϕ ϕ∨ψ

ψ ϕ∨ψ

Elimination

The rule for reasoning away from a disjunctive premise ϕ ∨ ψ needs likewise to be generalized so as to permit the deduction of a sentence θ in general, rather than just ⊥. But to this end it would suffice to deduce θ from but one of the cases ϕ and ψ. If the other case closes off with ⊥, then we know the truth does not lie there; hence, lies with the case that leads to θ. So permissible deductive moves would be:

31

(i)

ϕ∨ψ

(i)

∆1 , ϕ .. .

∆2 , ψ .. .

θ

⊥

(i)

ϕ∨ψ

(i)

⊥

(i)

∆1 , ϕ .. .

∆2 , ψ .. .

⊥

θ

(i)

⊥

Naturally also if θ is deducible from each case-assumption, then θ should be deducible overall: (i)

ϕ∨ψ

(i)

∆1 , ϕ .. .

∆2 , ψ .. .

θ

θ

(i)

θ And a special case of θ in this last rule is of course ⊥ itself, as with the rule of falsification with which we began. We can sum up the possibilities just canvassed as follows:

ϕ∨ψ

(i)

∆1 , ϕ .. .

∆2 , ψ .. .

θ/⊥

θ/⊥

(i)

(i)

θ/⊥ The conclusion (θ or ⊥) depends on ϕ ∨ ψ, the members of ∆1 and the members of ∆2 . The sets ∆1 and ∆2 could be distinct. The major premise ϕ ∨ ψ stands proud. The rule of ∨-Elimination is also known as proof by cases. The two subproofs indicated are called the case-proofs. The rule of thumb is: if either one of the case-proofs ends with ⊥, one may bring down the conclusion of the other case-proof as the overall conclusion.

4.4 4.4.1

The conditional Introduction

A refutation of ϕ modulo ∆:

32

∆, ϕ .. . ⊥ guarantees that any model of ∆ falsifies ϕ.2 Thus, by the first half of the model-relative verification rule for the conditional, any model of ∆ verifies ϕ → ψ. So the first half of our model-relative verification rule generalizes into the first half of the sought introduction rule as follows:

(i)

∆, ϕ .. . ⊥ ϕ→ψ

(i)

The second half of our model-relative verification rule needs, however, to be generalized more carefully. We need to allow for the distinct possibility that one might not be in a position to deduce the consequent, given the under-specific information ∆ at hand (as opposed to the highly specific information about a model, which will tell one whether the consequent holds). Since we are now allowing deductions from arbitrary sets of complex sentences, one can imagine a situation in which one has a deduction of the consequent ψ from the antecedent ϕ along with other assumptions forming a set ∆, say: ∆, ϕ .. . ψ Remember that deductions are to be truth-preserving. So, every model of ∆ that verifies ϕ verifies ψ. Therefore one can say of every model M of ∆: if M verifies ϕ, then M verifies ψ—whence, M verifies ϕ → ψ. That justifies the following second half of the introduction rule for the conditional: ∆, ϕ .. .

(i)

ψ (i) ϕ→ψ 2

This innocuous-seeming claim requires, and admits of, proof.

33

The use of the diamond here indicates that the subordinate proof need not have used ϕ as an assumption. But if it did, then that assumption will be discharged by application of the rule. In a case where ϕ is not used as an assumption, the justification of ϕ → ψ is immediate by the verification rule: for every model of ∆ would, ex hypothesi, verify ψ, hence also (by the verification rule) verify ϕ → ψ. 4.4.2

Elimination

The elimination rule for the conditional is obtained by straightforward morphing of the rule for model-relative falsification of conditionals:

ϕ→ψ

(i)

∆1 .. .

∆2 , ψ .. .

ϕ

θ

(i)

θ The conclusion θ depends on ϕ → ψ, all the members of ∆1 and all the members of ∆2 . The sets ∆1 and ∆2 can be distinct. The major premise ϕ → ψ stands proud. The proof of ϕ from ∆1 is called the minor proof; that of θ from ∆2 , ψ is called the major proof (for →-Elimination).

5 5.1

The universal quantifier Introduction

Recall the model-relative rule for verifying a universal claim (without loss of generality here we shall assume that the domain is finite): .. .. . . ... ψ(α1 ) ψ(αn ) ∀xψ(x)

where α1 , . . . , αn are all the individuals in the domain

In the presence of the indicated verifications, one would have a proof of ψ(a) from the assumption a = α1 ∨ . . . ∨ a = αn , to the effect that a is a member of the domain:3 3 In the infinitary case, the disjunctive major premise would have to be infinitary. That would pose no problems in principle, however, since the disjunction in question is being

34

.. . ψ(α1 ) a = α1 ∨ . . . ∨ a = αn

(1)

a = α1 ψ(a)

...

.. . ψ(αn )

(1)

a = αn ψ(a) (1)

ψ(a) The idea behind the introduction rule for ∀ is that any ‘parametric’ proof to the effect a is in the domain .. . ψ(a) should justify one in drawing the conclusion ∀xψ(x): (i)

a is in the domain .. . ψ(a)

(i)

∀xψ(x) Standard logic is the logic of a logically perfect language. The perfection assumption is that every well-formed singular term denotes. In free logic, one gives up this assumption. One allows for empty names, such as ‘Pegasus’. One allows also for partial functions, that is, functions that are ‘not everywhere defined’, such as division (which is not defined when the divisor is 0). And one needs a free logic if one aims to accommodate the definite description operator as a primitive variable-binding term-forming operator. This is because any term of the form ιx(ϕx ∧ ¬ϕx) fails to denote. The same need arises with the set abstraction operator. For, as Russell’s Paradox shows, the term {x | ¬x ∈ x} fails to denote. In free logic, the assumption that a is in the domain is expressed by the formal sentence ∃x x = a, often abbreviated as ∃!a. Thus the rule of ∀-Introduction in free logic is (i)

∃!a .. . ψ(a)

where ∃!a is the only assumption containing a on which ψ(a) depends (i)

∀xψ(x) invoked only by way of motivation of the main idea.

35

In standard logic, however, the assumption that a is in the domain does not need to find expression. One can limit oneself to proofs of ψ(a) from assumptions making no mention of a: .. . ψ(a) ∀xψ(x)

5.2

where ψ(a) depends on no assumptions containing a

Elimination

The elimination rule for ∀ can be stated as a rule allowing for multiple discharge (of assumption-instances of the predicate involved). This is on grounds analogous to those on which we allowed the ∧-Elimination rule simultaneously to discharge all assumption-occurrences of either of its conjuncts in the subordinate proof. The rule of ∀-Elimination is accordingly as follows: (i)

(i)

... {z .. .

ψ(t1 ) | ∀xψ(x)

θ

ψ(tn ) } (i)

θ In free logic one would invoke extra premises to the effect that one’s chosen terms t1 , . . . , tn do indeed denote: (i)

(i)

... {z .. .

ψ(t1 ) | ∀xψ(x) ∃!t1 . . . ∃!tn

θ

ψ(tn ) } (i)

θ

6 6.1

The existential quantifier Introduction

The introduction rule for ∃ closely resembles the model-relative rule for verifying existentials. In the latter, it sufficed to verify a single instance: 36

.. . ψ(α) ∃xψ(x) When deducing conclusions from arbitrary sets of sentences, the available instances are no longer saturated formulae using particular individuals α from a domain. Rather, one might prove, for some term t, that ψ(t) holds (conditionally on whatever assumptions are being used). Then—on the background assumption that said term t denotes an individual—one would be able to conclude ∃xψ(x): ψ(t) ∃xψ(x) In free logic one would invoke an extra premise to the effect that one’s chosen term does indeed denote: ψ(t) ∃!t ∃xψ(x)

6.2

Elimination

Recall the model-relative rule for falsifying an existential claim (in the finite case): (i)

∃xψ(x)

(i)

ψ(α1 ) ψ(αn ) .. .. ... . . ⊥ ⊥

where α1 , . . . , αn are all the individuals in the domain

(i)

⊥ Similar considerations apply here as applied in our formulation of the rule of ∀-Introduction above. In the presence of the indicated falsifications, one would have a disproof of ψ(a), using the assumption a = α1 ∨ . . . ∨ a = αn , to the effect that a is a member of the domain:4 4

See footnote 3.

37

(1)

ψ(a)

a = α1 ∨ . . . ∨ a = αn

a = α1 ψ(α1 ) .. .

(1)

ψ(a) ...

⊥

a = α1 ψ(α1 ) .. . ⊥

(1)

⊥ Accordingly, in free logic the rule of ∃-Elimination would be (i)

(i)

∃!a ψ(a) | {z } .. . ∃xψ(x)

θ

(i)

where the parameter a does not occur in any assumption, other than ∃!a and ψ(a), on which the upper occurrence of θ depends; and does not occur in either ∃xψ(x) or θ

θ while in standard logic it would be (i)

ψ(a) .. . ∃xψ(x)

θ

(i)

where the parameter a does not occur in any assumption, other than ψ(a), on which the upper occurrence of θ depends; and does not occur in either ∃xψ(x) or θ

θ

7

Summary of introduction and elimination rules

We collect together the introduction and elimination rules for standard, unfree logic. Where sets of assumptions (other than those being discharged) are permitted within subproofs, we indicate this by means of ∆ (with or without a numerical subscript). In the rule ∧-E, the box indicates that at least one of the conjuncts ϕ, ψ must appear as an (undischarged) assumption in the subordinate proof. Likewise, in the rule ∀-E, the box at the level of the discharge strokes indicates that at least one undischarged assumption of the form ψ(t) must appear in the subordinate proof. Other boxes attached to discharge strokes indicate that an assumption of the form in question must appear undischarged in the subordinate proof. The diamond in the second half of →, however, indicates that no such assumption need appear; if it does, however, it is discharged by the application of the rule.

38

Introduction

¬

(i)

∨

¬ϕ

(i)

∆1 .. .

ϕ ⊥

(i)

∆2 .. . ϕ∧ψ

ϕ ϕ∨ψ

ψ ϕ∨ψ

(i)

∆2 , ψ .. .

θ/⊥

θ/⊥

∀

ψ(a) ∀xψ(x)

(i)

(i)

θ/⊥

(i)

∆2 , ψ .. .

ϕ

θ

ϕ→ψ

ψ (i) ϕ→ψ

(i)

∆1 .. .

(i)

θ ...

(i)

∆ .. .

(i)

∆1 , ϕ .. . ϕ∨ψ

∆, ϕ .. .

∆, ϕ .. .

(i)

∆ .. .

⊥ ϕ→ψ

θ θ

∆ .. .

(i)

(i)

∆ ,ϕ, ψ .. .

ϕ ψ ϕ∧ψ

→

∆ .. .

∆, ϕ .. . ⊥ ¬ϕ

∧

Elimination

∆ |

where a does not occur in any member of ∆

ψ(t1 ) {z .. .

∀xψ(x)

θ θ

39

2... ...

(i)

(i)

ψ(tn ) }

2

∆ .. .

∃

ψ(t) ∃xψ(x)

∃xψ(x)

θ θ

ϕ

=

(i)

∆, ψ(a) | {z } .. .

t=t

t=u ψ

(i)

where a does not occur in ∃xψ(x), θ or any member of ∆

where ϕut = ψtu

Important note: Major premises for eliminations stand proud. They are not drawn as conclusions of any proof-work above them.

8

Conclusion

8.1

Summary

The foregoing rules are those of core logic.5 The aim of this study has been to reveal the natural way in which core logic emerges from reflections on how one establishes sentences as true or as false under interpretations, and how one can generalize those movements in thought so as to deal with complex sentences in general, as starting points and as endpoints of trains of reasoning. Elsewhere, I have argued that core logic is the correct logic according to an anti-realist account of meaning;6 that it suffices for constructive mathematics;7 that it suffices for hypothetico-deductive testing of theories;8 that it enables efficient automated proof-search;9 and that it is the minimal canon invulnerable to revision, every part of which is indispensable for the process of rational belief-revision.10 On this occasion, however, the aim has been to describe a natural conceptual route to core logic, beginning with one’s rudimentary grasp of verification- and falsification-conditions, construed in a suitably inferentialist fashion. 5

Formerly called IR, or intuitionistic relevant logic, as in [4], [5] and [7]. See [4], [7]. 7 See [6]. 8 See [3]. 9 See [5]. 10 See [9]. 6

40

8.2

Further developments

Model-relative verifications and falsifications, as introduced here, are rigorously definable as well-understood mathematical objects: they are trees of finite depth, whose nodes are labeled by saturated formulae. When the domain of the model is infinite, the sideways branchings corresponding to verifications of universals and falsifications of existentials will be infinite. But all branches will be of finite length. In a reasonably weak metamathematical theory, one can prove the following. Theorem 1 For any model M , and any saturated formula ϕ, ϕ is true in M (in Tarski’s sense) ⇔ there is an M -relative verification of ϕ and ϕ is false in M (in Tarski’s sense) ⇔ there is an M -relative falsification of ϕ The theory of model-relative verifications and falsifications is a structuretheory for truth-makers and falsity-makers. The current philosophical literature on truth-makers appears to be bereft of such a structure-theory. In future work I intend to demonstrate some advantages in conceiving of philosophers’ ‘truth-makers’ as these (appropriately structured) model-relative verifications. Other results, to be presented in detail elsewhere,11 are the following. Theorem 2 Given any two proofs in core logic, where the conclusion of the first proof is a premise of the second proof (call it the ‘cut-sentence’), one can effectively find a core proof of the second proof ’s conclusion, or of absurdity, from premises of the given proofs other than the cut sentence. Theorem 3 Proofs in core logic provide an ‘effective’ means of transforming (for any model M ), M -relative verifications of their undischarged assumptions into M -relative verifications of their conclusions.

11

These results were stated in [8].

41

References [1] Alfred Tarski. The Concept of Truth in Formalized Languages. In J. H. Woodger, editor, Logic, Semantics, Metamathematics, pages 152–278. Clarendon Press, Oxford, 1956. [2] Neil Tennant. Natural Logic. Edinburgh University Press, 1978. [3] Neil Tennant. Minimal logic is adequate for Popperian science. British Journal for Philosophy of Science, 36:325–329, 1985. [4] Neil Tennant. Anti-Realism and Logic: Truth as Eternal. Clarendon Library of Logic and Philosophy, Oxford University Press, 1987. [5] Neil Tennant. Autologic. Edinburgh University Press, 1992. [6] Neil Tennant. Intuitionistic mathematics does not need ex falso quodlibet. Topoi, pages 127–133, 1994. [7] Neil Tennant. The Taming of The True. Oxford University Press, 1997. [8] Neil Tennant. Cut for Core Logic. Paper presented to the Sixth Quadrennial Fellows Conference of the Pittsburgh Center for Philosophy of Science, July 2008, Ohio University, Athens and to the NYU Conference in Philosophy of Mathematics, April 2009, Unpublished typescript. [9] Neil Tennant. Rational Belief Revision. Unpublished typescript.

42

I am grateful to Tim, as my teacher in logic, for emphasizing that its main notions are relational: ‘ϕ is deducible from ∆’ and ‘ϕ is a logical consequence of ∆’. The notions ‘ϕ is a theorem’ and ‘ϕ is logically true’ are special cases. In the early 1970s Tim gave a formative series of lectures emphasizing how proofs are to be understood as perfected arguments, in Aristotle’s sense. The present discussion of verification and falsification is fully in the inferentialist spirit of Tim’s emphases. The aim is to render even the notions ‘ϕ is true’ and ‘ϕ is false’ as essentially relational and inferential. A sentence’s truth-value is determined relative to collections of rules of inference that constitute an interpretation. Moreover, truthmakers and falsity-makers are themselves proof-like objects, encoding the inferential process of evaluation involved. The inference rules involved in the determination of truth-value are almost identical to those involved in securing the transmission of truth from premises to conclusion of a valid argument. We shall see how smoothly one can ‘morph’ the former into the latter. ∗

Written for the Festschrift for Timothy Smiley, edited by Jonathan Lear and Alex Oliver. Please do not cite or circulate without the author’s permission.

1

Rules of evaluation

1.1

Verification and falsification of sentences in models

We shall first describe, in general terms, interpretations, or models, of a firstorder language. Then we shall provide a ‘toy example’ that will illustrate the ingredients of the definition. A model consists of (i) a domain of individuals; (ii) a denotation mapping for names (if there are any names in the objectlanguage); (iii) the structure that consists in primitive predicate-extensions; and (iv) the structure that consists in the mappings represented by primitive function-signs. The names, predicates and function-signs make up the extra-logical vocabulary that is being interpreted by the model in question. Because we are dealing with a toy example, we shall confine ourselves to one-place predicates and function signs. Towards the end of this discussion we shall have occasion to consider a two-place predicate. So our expressive resources will be quite modest. Still, we shall be painstaking in illustrating what is involved in these different components (i)–(iv) of models. We shall build up our chosen model M in stages. To the left will be a diagram, which can be thought of as the model M itself. The large dots will be the individuals; each one-place predicate extension will be represented by an enclosure; and each one-place mapping will be represented by arrows (a different style of arrow for each mapping). These diagrammatic components will be added in sequence, as the model is built up. So, for our toy example, we first choose a domain of individuals (here, three). They are labeled α, β and γ in the metalanguage. To the right of the diagram are three ‘M -relative’ rules of inference. These rules ensure that the individuals are pairwise distinct. Their conclusions are ⊥ and their premises are what we shall call saturated identity formulae (see below) involving all possible pairs of our chosen individuals. Such rules are a necessary part of an eventual set of rules that will completely capture the diagram that will be on the left once the model has been fully constructed. Indeed, the eventual set of rules on the right can be thought of as an adequate substitute for the model itself. 2

M • α

1.2

• β

α = βM ⊥

• γ

α = γM ⊥

β = γM ⊥

A digression on saturated terms and formulae

In general, terms and formulae of the object-language may contain free variables. If they do, then they are called open. Those that are not open are called closed. A closed formula is called a sentence. A closed term may be called a (simple or complex) name. The semantic value of a name, when it has one, is an individual, which the name is then said to denote. The semantic value of a sentence, when it has one, is a truth-value, and the sentence is said to be true or false according as that value is T or F . Closing an open term or formula involves substituting closed terms (of the object-language) for free occurrences of variables. Thus one could substitute the object-linguistic name j for the free occurrence of the variable x in the open term f (x), to obtain the closed term f (j) (‘the father of John’). Or, to complicate the example slightly, one could substitute the closed objectlinguistic term m(j) for that free occurrence, to obtain the closed term f (m(j)) (‘John’s maternal grandfather’). Likewise, an open formula, say L(x, y), may be closed by substituting closed terms for its free occurrences of variables. One such closing would result in the sentence L(m(f (j)), f (j)) (‘John’s paternal grandmother loves his father’). Here we shall introduce an operation on open terms and formulae analogous to the operation of closing, but importantly different from it. The new operation will be called saturation. Like the operation of closing, the operation of saturation gets rid of all free occurrences of variables within an object-linguistic term or formula. But the way it does so is importantly different. Instead of substituting closed object-linguistic terms for free occurrences of variables, saturation is effected by substituting individuals from the domain for those free occurrences. Thus if α and β are individuals from the domain, one saturation of the open formula L(x, y) would be L(α,β). Another one would be L(m(f (α)),f (α)), where the saturation is effected by substituting the saturated terms m(f (α)) and f (α) for the free occurrences of the variables x and y respectively. When the domain D supplies all the individuals involved in a saturation

3

operation, the resulting saturated terms are called saturated D-terms, and the resulting saturated formulae are called saturated D-formulae. Saturated terms and formulae are object-linguistic and metalinguistic hybrids. But, as mathematical objects, they are well-defined. When one treats, in standard Tarskian semantics, of assignments of individuals to variables, one is assuming such well-defined status for ordered pairs of the form hx,αi, where x is a free variable of the object-language, and α is an individual from the domain of discourse.1 Since standard semantics is already committed to the use of such hybrid entities, it may as well take advantage of similar hybrid entities such as saturated terms and formulae. We shall be taking advantage of them by having them feature in the rules of inference on the right in our description of models. Indeed, such rules will form a constitutive part of the model in question, as will emerge in due course.

1.3

Back to the exposition of models

With all subsequent displays of rules on the right, we shall omit the subscripts M next to their inference strokes, as they will always be understood from the context. (Occasionally we shall restore them, for appropriate emphases.) We shall also suppress the label M at the top and left of the diagram. The model M is not yet completely specified. For so far we have specified only its domain; we have yet to specify its predicational and functional structure. Note how we have labelled the individuals as α, β and γ by placing these labels right next to them, within the outer box that represents the ‘boundary’ of the domain. By means of this convention we indicate that the labels are metalinguistic. As far as the object-language is concerned, the individuals could be nameless. Suppose, however, that the individuals happened to possess names in the object-language—say, a, b and c respectively. Then this circumstance will be conventionally indicated by placing those names outside the outer box, and indicating, with arrows, what their respective denotations are. To the right we would specify the denotation 1

This is true not only of Tarski’s original treatment [1], which invoked infinite sequences of individuals correlated with object-linguistic variables, but also of the treatment (in [2]) of Tarski’s approach that appeals, more modestly, to finitary assignments of individuals to the free variables in a formula.

4

mapping d: a

b

c

?

• β

?

• γ

• α

?

d(a) = α; d(b) = β; d(c) = γ

Our toy example will not, however, involve any names in the object-language, so we can set these details aside. Next, we add the structure of a one-place predicate P , by showing its extension in the diagram, and also by supplementing our rules of inference in order to show which individuals are, and which are not, in the extension of the predicate P . (If we failed to specify, by means of our rules, which individuals are not in the extension of P , then the rules themselves would fail, collectively, to specify the intended diagram, in which certain individuals definitely lie outside the enclosure indicating the extension of P .)

α=β α=γ β=γ ⊥ ⊥ ⊥

P • α

• β

• γ

P (β) P (γ) P (α) ⊥ ⊥

Finally, we add the structure of a one-place function f , supplementing again on the right with the relevant ‘axioms of function-values’:

5

P • α

- •

β

α=β α=γ β=γ ⊥ ⊥ ⊥

n - •

γ

P (β) P (γ) ⊥ ⊥

P (α)

f (α) = β

f (β) = γ

f (γ) = γ

The axioms and rules on the right re-state what one can read off from the diagram on the left. There is, however, one aspect of the diagram that is only ‘shown’, and not yet ‘said’, by the rules on the right. This is that the individuals α, β and γ are all the individuals there are. This feature of the diagram will find expression in the following ‘M -relative’ rules of evaluation: (i)

ψ(α) ψ(β) ψ(γ) M ∀xψ(x)

∃xψ(x)

(i)

(i)

ψ(α) .. .

ψ(β) .. .

ψ(γ) .. .

⊥

⊥

⊥

(i) M

⊥ The first of these rules, which allows one to evaluate as true any universal claim ∀xψ(x), requires only that ψ should hold of each of the individuals α, β and γ. The second rule, which allows one to evaluate as false any existential claim ∃xψ(x), requires only that ψ should not hold of any of the individuals α, β and γ. (One shows that a claim does not hold by assuming that it does, and deriving ⊥ from that assumption.) It is clear that these M -relative rules for the verification of universals and the falsification of existentials constrain the domain D to consist of exactly the individuals α, β and γ. And so it will be with any finite model. Let N be such, and suppose its domain consists of exactly the n individuals α1 , . . . , αn . Then the N -relative rules for the verification of universals and the falsification of existentials will be

6

(i)

ψ(α1 ) . . . ψ(αn ) N ∀xψ(x)

∃xψ(x)

(i)

ψ(α1 ) ψ(αn ) .. .. ... . . ⊥ ⊥

(i) N

⊥ Let us now return to our toy model M , and consider some simple sentences that can be verified, or falsified, in M . We shall choose sentences simple enough for easy, intuitive determination of their truth-values in M . The point of the exercise is to show how that intuitive determination can be represented formally as a certain kind of M -relative ‘evaluation proof’ or ‘evaluation disproof’ of the sentence in question. Here again is our model M : M

The M -rules:

P • α

- •

n - •

β

γ

α=β α=γ β=γ ⊥ ⊥ ⊥ P (α)

P (β) P (γ) ⊥ ⊥

f (α) = β

f (β) = γ

f (γ) = γ

Intuitively it is easy to see that the following sentences are true-in-M , or false-in-M , as indicated: 1. 2. 3. 4. 5.

True-in-M ∃xP (x) ∃x f (x) = x ∃x(P (x) ∧ ¬f (x) = x) ∀x(¬P (x) ∨ ¬f (x) = x) ∀x(f (f (x)) = x → ¬P (x))

False-in-M ∀xP (x) ∀x ¬f (x) = x ∀x(¬P (x) → f (x) = x) ∀x∃y f (y) = x P (f (α)) ∨ P (f (β))

The first two truths are easy to verify: M

P (α) ∃xP (x)

M

f (γ) = γ ∃x f (x) = x

In both these evaluation proofs, the final step verifies an existential conclusion on the basis of an appropriate instance—α in the first, γ in the second. 7

The general form for the verification of an existential claim ∃xψ(x) is: .. . ψ(α) ∃xψ(x)

where α is an individual in the domain.

The vertical dots indicate the presence of some verification of some instance ψ(α), for some individual α in the domain of the model. For the third truth in the list above, ∃x(P (x) ∧ ¬f (x) = x), we make use of the following rule (for the construction of verifications and/or falsifications) governing identity: (i)

α1 = α2 M ⊥

t = α1 t = α2

(i)

where t is any saturated D-term

⊥ The M -relative evaluation-proof, or verification, of ∃x(P (x) ∧ ¬f (x) = x) is as follows: (1)

α=β ⊥

(2)

f (α) = α

M

f (α) = β

(1)

⊥ M (2) P (α) ¬f (α) = α P (α) ∧ ¬f (α) = α ∃x(P x ∧ ¬f (x) = x) Note that the penultimate step verifies the conjunction P (α)∧¬f (α) = α by verifying the left conjunct P (α) and verifying the right conjunct ¬f (α) = α. The general form of such a step is ϕ ψ ϕ∧ψ The step labelled (2) is an application of the following general rule for verifying negations: (i)

ϕ .. . ⊥ ¬ϕ

(i)

8

where the vertical dots indicate the presence of some falsification of ϕ. The fourth truth on our list, ∀x(¬P (x) ∨ ¬f (x) = x), has the following M -relative verification: (4)

α=β ⊥

(1)

M

(2)

f (α) = β

f (α) = α

⊥ (1) ¬f (α) = α ¬P (α) ∨ ¬f (α) = α

(3)

P (β) M P (γ) M (4) ⊥ (2) ⊥ (3) ¬P (β) ¬P (γ) ¬P (β) ∨ ¬f (β) = β ¬P (γ) ∨ ¬f (γ) = γ ∀x(¬P (x) ∨ ¬f (x) = x)

Here, at the final step, we see in action the rule for verifying universal claims in any model whose only individuals are α, β and γ. One has to verify each of the three instances of the universal claim. Each of the three immediately preceding steps involved verifying a disjunction by verifying one or the other of its disjuncts. The general form of such a step is one or the other of ϕ ϕ∨ψ

ψ ϕ∨ψ

Even when both disjuncts are true, the most economical route to the verification of the disjunction is to focus on the disjunct with the easiest verification. Hence there is no need to add to, or alter, the form of these last two rules for the verification of disjunctions. Let us turn our attention now to the first four false sentences in the list above. For these we need to construct M -relative falsifications. It takes only one counterinstance to render a universal claim false. To falsify ∀xP (x), we have a choice between β and γ as counterinstance. (Since α lies in the extension of P , we obviously cannot appeal to α in order to falsify ∀xP (x).) Let us take β for this purpose. The falsification is then as follows: (1)

∀xP (x)

P (β) M ⊥ (1)

⊥ Any falsification of a saturated formula ϕ will use ϕ as an assumption— indeed, its sole undischarged assumption—and derive ⊥ therefrom, using as its remaining premises only the ‘basic claims’ about the model in question. (A ‘basic claim’ can of course be an inference from a primitive saturated formula to ⊥, as we have seen.) Note how the final step of this last two-step 9

M

falsification discharges the assumption P (β) within the subordinate (onestep) falsification, on the right, of P (β) itself. The final occurrence of ⊥ is thereby made to depend only on the undischarged assumption ∀xP (x), rather than on the (false!) instance P (β) of that universal claim. The general evaluation rule being applied here for the falsification of universal claims is the following: (i)

ψ(α) .. . ∀xψ(x)

⊥

(i)

⊥ where the vertical dots indicate the presence of some falsification of some instance ψ(α). (In our last example, this was a one-step falsification. But in general, of course, there can be more than one step, the exact number of steps depending both on the complexity of ψ and on the chosen counterinstance α.) The second falsehood on our list above, ∀x ¬f (x) = x, enjoys the following falsification: (1)

M

¬f (γ) = γ ∀x ¬f (x) = x

f (γ) = γ ⊥

(1)

⊥ The upper step is an application of the general evaluation rule .. . ϕ

¬ϕ ⊥

for the falsification of negations. The vertical dots indicate the presence of some verification of ϕ. The third falsehood on our list, ∀x(¬P (x) → f (x) = x), can be shown to be false as follows. The falsification involves choosing β as the counterinstance to the universal claim. This choice generates the task of falsifying the conditional claim ¬P (β) → f (β) = β. A falsification of a conditional θ → χ consists in a verification of the antecedent θ and a falsification of the consequent χ:

10

(i)

θ→χ

χ .. .

.. . θ

⊥

(i)

⊥ So we need in the case at hand to verify ¬P (β) and to falsify f (β) = β. The details are as follows: (2)

(4)

(3)

¬P (β) → f (β) = β

β=γM P (β) ⊥ ⊥ (4) ¬P (β)

∀x(¬P (x) → f (x) = x)

⊥

(1)

⊥

M

f (β) = γ

f (β) = β (1)

(3)

⊥ The step labelled (2) is an application of the rule for identity explained earlier. We come to our fourth illustration of falsifications, that of the sentence ∀x∃y f (y) = x. A glance at the diagram for M reveals that α is the counterinstance we want—for it is the only individual that is not the f -value of any individual in the domain. Our task is now to fill in the dots in the falsification schema (1)

∃y f (y) = α .. . ⊥

∀x∃y f (y) = x

(1)

⊥ In order to falsify the existential ∃y f (y) = α, we need to falsify each of its three instances f (α) = α, f (β) = α, and f (γ) = α. The task accordingly reduces to that of filling in the dots in the schema (2)

(1)

(2)

(2)

f (α) = α .. .

f (β) = α .. .

f (γ) = α .. .

⊥

⊥

⊥

∃y f (y) = α ∀x∃y f (y) = x

⊥ ⊥

11

(1)

(2)

(2)

The desired M -relative falsifications of f (α) = α, f (β) = α and f (γ) = α (to fill in the respective vertical dots) are as follows: (3)

α=β ⊥

M

f (α) = α

f (α) = β

(3)

⊥ (4)

α=γ ⊥

M

f (β) = α

f (β) = γ

(4)

⊥ (5)

α=γ ⊥

M

f (γ) = α

f (γ) = γ

(5)

⊥ We refrain here from inserting these into the previous schema, since the sideways spread would be too wide for the page to contain. The reader will appreciate that these insertions would be completely straightforward, if one had paper wide enough. Our exposition of the rules for verifying and/or falsifying sentences (or saturated formulae) is not yet complete. We have not yet seen how to verify a conditional claim (one of the form ϕ → ψ), nor how to falsify a disjunctive claim (one of the form ϕ ∨ ψ). In order to see these rules in action, we turn to the fifth truth and the fifth falsehood in our list. The fifth truth is the claim ∀x(f (f (x)) = x → ¬P (x)). As we inspect the diagram M , we realize that this universal generalization is true for want of a counterinstance. No individual in the domain is such that it is identical to its own f f -image, yet has the property P . Each of α, β and γ fails to be such a counterinstance, and in interesting ways. First, α is not its own f f -image. (It is, however, in the extension of P .) Secondly, β, likewise, is not its own f f -image; but, also, β is not in the extension of P . Indeed, it is easier to tell that the latter is the case, than that the former is the case. So, as far as β is concerned, its failure to be a counterinstance to the claim ∀x(f (f (x)) = x → ¬P (x)) is more easily shown by showing ¬P (β) than by deriving ⊥ from f (f (β)) = β. Thirdly, γ fails to be a counterinstance to the claim ∀x(f (f (x)) = x → ¬P (x)) for a similar reason: P (γ) is false. Our verification of the claim ∀x(f (f (x)) = x → ¬P (x)) of course pro-

12

ceeds by verifying each of its three instances f (f (α)) = α → ¬P (α) f (f (β)) = β → ¬P (β) f (f (γ)) = γ → ¬P (γ) and we are proposing to do this, instance by instance, as follows: (1)

f (f (α)) = α .. .

.. .. . . ¬P (β) ¬P (γ) f (f (β)) = β → ¬P (β) f (f (γ)) = γ → ¬P (γ)

⊥ (1) f (f (α)) = α → ¬P (α) ∀x(f (f (x)) = x → ¬P (x)) The missing falsification is as follows: M

(2)

α=γM ⊥

f (f (α)) = α

M

f (β) = γ f (α) = β f (f (α)) = γ (2)

⊥ The missing verification in the middle we have seen before, and the rightmost one is similar to it. They are as follows: (4)

P (β) ⊥ (4) ¬P (β)

(5)

P (γ) ⊥ (5) ¬P (γ)

Once again, it is left to the reader to insert these last three bits of detailed working into the appropriate places (indicated by the vertical dots) in the preceding verification-schema. Some comments are now in order on the new rules that have just found application within the last example. First, we have seen that there are two ways to verify a conditional ϕ → ψ: one can falsify its antecedent ϕ, or one can verify its consequent ψ. Both these methods found application in the verification above of the claim ∀x(f (f (x)) = x → ¬P (x)). Its α-instance was verified by falsifying the antecedent; while the β- and γ-instances were verified by verifying their consequents. The general form of the rule for verifying a conditional is accordingly

13

M

(i)

.. . ψ ϕ→ψ

ϕ .. . ⊥ ϕ→ψ

(i)

where the missing dots on the left indicate the presence of a verification of the consequent ψ, while those on the right indicate the presence of a falsification of the antecedent ϕ. Our manipulations of identity statements in the last example were also novel. The step f (β) = γ f (α) = β f (f (α)) = γ is a special case of the rule f (α1 ) = α u1 = α1 f (u1 ) = α where u1 is a saturated D-term and α, α1 are individuals in the domain A moment’s reflection reveals that this is the canonical way to establish a conclusion of the form f (u1 ) = α. How does one work out that f (u1 ) is indeed the object α? One first finds the object α1 denoted by the contained (saturated) term u1 . (See the second premise of the rule just stated.) Then one finds the object (call it α) denoted by the (saturated) term f (α1 ). (See the first premise.) That object α is thereby shown to be (the denotation of the term) f (u1 ). (See the conclusion.) So this rule for identity simply teases out what is involved in computing function-values stage-by-stage. The rule just stated can be generalized to n-place functions as follows: f (α1 , . . . , αn ) = α u1 = α1 ... f (u1 , . . . un ) = α

un = αn

where f is a primitive n-place function sign, u1 , . . . , un are saturated D-terms and α, α1 , . . . , αn are individuals in the domain The computation of the values of the (saturated) terms u1 , . . . , un proceeds in parallel.

14

We turn now to our final example, which is the fifth falsehood: P (f (α)) ∨ P (f (β)). In order to falsify a disjunction ϕ ∨ ψ, one needs to falsify each disjunct: (i)

ϕ∨ψ

(i)

ϕ .. .

ψ .. .

⊥

⊥

(i)

⊥ Applied to the case at hand, we obtain the falsification schema (1)

(1)

P (f (α)) .. .

P (f (β)) .. .

⊥

⊥

P (f (α)) ∨ P (f (β))

(1)

⊥ The missing falsifications to be inserted here are (2)

P (β) M ⊥

M

f (α) = β

P (f (α)) (2)

⊥ and (3)

P (γ) M ⊥

M

f (β) = γ

P (f (β)) (3)

⊥ The final steps of the last two falsifications were applications, for the case n = 1, of the identity rule (i)

A(α1 , . . . , αn ) M ⊥

t1 = α1 . . . tn = αn

A(t1 , . . . , tn )

(i)

⊥ where A is a primitive n-place predicate, t1 , . . . , tn are saturated 15

D-terms, and α1 , . . . , αn are individuals in the domain In order to motivate the formulation of a few more rules governing the construction of verifications and falsifications, let us consider a slight embellishment of our earlier model M . Let M 0 be the following model, identical to M on the vocabulary of M , but with a new, two--place predicate Lxy (to be read as ‘x is to the left of y’). The extension of L is represented by dashed arrows in the diagram for M 0 . α=β α=γ β=γ ⊥ ⊥ ⊥ M0

P (α) P - • • P P α P Pβ P

n - • 1 γ

P (β) P (γ) ⊥ ⊥

f (α) = β

f (β) = γ

f (γ) = γ

L(α, α) L(α, β) L(α, γ) ⊥ L(β, α) L(β, β) L(β, γ) ⊥ ⊥ L(γ, α) L(γ, β) L(γ, γ) ⊥ ⊥ ⊥

Note that all the rules of the model M are rules of the model M 0 . We extend our earlier list of sentences whose truth-values are to be determined by means of the rules framed here. 1. 2. 3. 4. 5. 6. 7.

True-in-M 0 ∃xP (x) ∃x f (x) = x ∃x(P (x) ∧ ¬f (x) = x) ∀x(¬P (x) ∨ ¬f (x) = x) ∀x(f (f (x)) = x → ¬P (x)) f (f (f (α))) = f (f (α)) L(f (α), f (β))

False-in-M 0 ∀xP (x) ∀x ¬f (x) = x ∀x(¬P (x) → f (x) = x) ∀x∃y f (y) = x P (f (α)) ∨ P (f (β)) f (f (f (α))) = α ∃xP (x) ∧ ∀x f (f (x)) = x

16

The seventh sentence on the left, L(f (α), f (β)), is true in M 0 . Its M 0 relative verification is: M

L(β, γ)

M

f (α) = β L(f (α), f (β))

M

f (β) = γ

The final step here is an instance, for the case n = 2, of the following identity rule: A(α1 , . . . , αn ) t1 = α1 . . . A(t1 , . . . , tn )

tn = αn

where A is a primitive n-place predicate, t1 , . . . , tn are saturated D-terms, and α1 , . . . , αn are individuals in the domain. The careful reader will have noticed, from the three substitution rules already stated for identity, that there is a pattern to them that would be completed by having the following rule: (i)

f (α1 , . . . , αn ) = α M ⊥

u1 = α1 . . . un = αn

f (u1 , . . . , un ) = α

(i)

⊥ where f is a primitive n-place function sign, u1 , . . . , un are saturated D-terms, and α, α1 ,. . . ,αn are individuals in the domain. An example in which this rule finds application is the falsification of the sixth sentence on the right in the list above: f (f (f (α))) = α. The rule is applied in the final step of the following falsification, whose rightmost premise is the claim being falsified: (1)

f (γ) = α M ⊥

M

M

f (β) = γ f (α) = β f (f (α)) = γ

f (f (f (α))) = α (1)

⊥ There is only one more ‘model-relative’ rule (or axiom) that needs to be specified in order to have an adequate set of rules for the verification and 17

falsification of claims in a first-order language with identity. It is the rule of reflexivity of identity, stated only for individuals α in the domain of the model in question: where α is an individual in the domain

α=α

This rule enables one to derive the identity claim t = u (where t and u are saturated D-terms of any complexity) when each of those terms has been verified as denoting some same individual α: α=α

t=α u=α t=u

Note that this inference is an application of our earlier substitutivity rule for n-place atomic predicates A—identity being a two-place atomic predicate that can take the place of A in the statement of that rule. In our model M , for example, the claim f (f (f (α))) = f (f (α)) is true. Its verification, whose final step is of the form just displayed, is M

γ=γ

f (β) = γ f (α) = β M f (γ) = γ f (f (α)) = γ f (f (f (α))) = γ f (f (f (α))) = f (f (α))

M M

f (β) = γ f (α) = β f (f (α)) = γ

Consider a conjunction that is false in M , such as ∃xP (x) ∧ ∀x f (f (x)) = x. (This is the seventh sentence on the right in the list above. It is the righthand conjunct that is false.) A falsification of the whole conjunction in such a case must proceed by falsifying the culprit conjunct. The general form for falsification of a conjunction ϕ ∧ ψ via its right conjunct ψ is as follows: (i)

ψ .. . ϕ∧ψ

⊥

(i)

⊥ Of course there must also be a way of falsifying a conjunction when it is only its left conjunct that is false:

18

M

(i)

ϕ .. . ⊥

ϕ∧ψ

(i)

⊥ Even when both conjuncts are false, the most economical route to the falsification of the conjunction is to focus on the conjunct with the easiest falsification. Hence there is no need to add to, or alter, the form of these last two rules for the falsification of conjunctions. We are now in a position to take stock by listing all the rules to which we have had recourse in verifying or falsifying our chosen example sentences above. It is worth stressing that these are rules for determination of the truth-value of an arbitrary sentence, using as synthetic ‘first principles’ the (inferentially coded) basic information in the model in question. The analytic ‘first principles’ are the rules themselves. They characterize the meanings of logical expressions in terms of their roles in determining sentences as true, or as false, in any model. Rules for verification and falsification of primitive saturated formulae.

A(α1 , . . . , αn ) t1 = α1 . . . A(t1 , . . . , tn )

tn = αn

where A is a primitive n-place predicate, t1 , . . . , tn are saturated D-terms, and α1 , . . . , αn are individuals in the domain.

(i)

A(α1 , . . . , αn ) M ⊥

t1 = α1 . . . tn = αn

A(t1 , . . . , tn )

(i)

⊥ where A is a primitive n-place predicate, t1 , . . . , tn are saturated D-terms, and α1 , . . . , αn are individuals in the domain.

19

f (α1 , . . . , αn ) = α u1 = α1 ... f (u1 , . . . un ) = α

un = αn

where f is a primitive n-place function sign, u1 , . . . , un are saturated D-terms and α, α1 , . . . , αn are individuals in the domain

(i)

f (α1 , . . . , αn ) = α M ⊥

u1 = α1 . . . un = αn

f (u1 , . . . , un ) = α

⊥ where f is a primitive n-place function sign, u1 , . . . , un are saturated D-terms, and α, α1 ,. . . ,αn are individuals in the domain.

α=α

where α is an individual in the domain

Rules for verification and falsification of saturated formulae with a connective dominant

(i)

ϕ .. . ⊥ ¬ϕ

.. . ϕ

¬ϕ (i)

⊥

(i)

ϕ ψ ϕ∧ψ

(i)

ϕ .. . ϕ∧ψ ⊥

20

⊥

ψ .. . (i)

ϕ∧ψ ⊥

⊥

(i)

(i)

(i)

ϕ ϕ∨ϕ

ψ ϕ∨ψ

ϕ∨ψ

(i)

ϕ .. .

ψ .. .

⊥

⊥

(i)

⊥

(i)

(i)

.. . ψ ϕ→ψ

ϕ .. . ⊥ ϕ→ψ

(i)

ψ .. .

.. . ϕ

ϕ→ψ

⊥

(i)

⊥

Rules for verification and falsification of saturated formulae with a quantifier dominant

(i)

ψ(α1 ) . . . ψ(αn ) . . . ∀xψ(x)

ψ(α) .. . ⊥

∀xψ(x)

(i)

⊥

(i)

.. . ψ(α) ∃xψ(x)

∃xψ(x)

(i)

ψ(α1 ) ψ(αn ) .. .. ... ... . . ⊥ ⊥ (i)

⊥

We have noted above how quickly one can incur sideways spread in writing down a detailed verification or falsification. This feature militates against the actual construction of these otherwise very illuminating and 21

detailed constructions for sentences (and saturated formulae) relative to a given model M . As soon as one has three or more individuals in the domain of M , along with nested quantifiers (especially when they occasion the use of the two rules that require investigation of all instances of a quantified claim), the blow-up, in the form of sideways spread, is prohibitive. But the resulting construction is only ever as deep as the longest branch within the analysis tree of the sentence (or saturated formula) being evaluated. Moreover, in cases where the domain is infinite, some of these verifications and falsifications will contain steps (for the verification of a universal, or the falsification of an existential) that require infinitely many premises (in the form of instances of the quantified claims in question). In such cases the constructions cannot be written down. Instead, they exist only as infinitary mathematical objects: labelled trees (where the labels are at least finite!) that can have infinite branching, albeit only with branches of finite length. Ultimately, the present ‘inferentialist’ approach to formal semantics via the verifications and falsifications illustrated above requires no more powerful mathematical machinery than is needed in order to vouchsafe the existence of these (rather modest) kinds of infinitary object.

2 2.1

General consequences of the rules of evaluation Identity

Our rules of evaluation, which are framed in the metalanguage, allow for reflexivity of identity only in the form

α=α

where α is an individual in the domain

This is not the same as saying that the rule

t=t

where t is a term of the object-language

is valid. But its validity is not hard to establish, using constructive reasoning in the metalanguage, for models in which all names in the object language denote, and all functions signs represent total functions. Every such model M for any language containing the extra-logical expressions involved in the term t has in its domain some individual α for which there is a verification V , say, of the claim t = α. Applying the first rule for verification of primitive saturated formulae, one obtains the verification

22

V V t=α t=α t=t Thus it follows from our rules of evaluation in the metalanguage that reflexivity of identity holds generally for terms t of the object-language. A similar account can be given of the validity of the rule of substitutivity of identicals for terms t, u in the object-language: ϕ

t=u ψ

where ϕut = ψtu

Reflexivity of identity and substitutivity of identicals therefore become available as general rules of deduction. They preserve truth from their premises to their conclusions. (In the case of reflexivity, of course, the set of premises is empty. Its conclusion is true in every model of any language containing the extra-logical expressions in t .)

2.2

Non-contradiction

It is clear that the basic axioms and rules in the atomic diagram of a model are coherent, in the sense that for no primitive saturated formula does the model contain both the axiom allowing one to infer it from no assumptions, and the rule allowing one to infer ⊥ from it. This point generalizes. Lemma 1 Let M be a model for the extra-logical vocabulary in an M saturated formula ϕ. Then there cannot be both an M -relative verification of ϕ and an M -relative falsification of ϕ. Proof. By induction on the complexity of ϕ. The basis step is obvious. Inductive hypothesis: Assume that the result holds for all M -saturated subformulae of ϕ. Inductive step: By cases, according as ϕ is of the form (i) ¬ψ, (ii) ψ1 ∧ ψ2 , (iii) ψ1 ∨ ψ2 , (iv) ψ1 → ψ2 , (v) ∃xψ, or (vi) ∀xψ. In what follows, λ and µ (with or without numerical subscripts) will be parts of the atomic diagram of M , and V and F (with or without numerical subscripts) will be M -relative verifications and falsifications. We shall also assume that the individuals in the domain of M are α1 , . . . , αn , . . . . Case (i). Any M -relative verification of ¬ψ would take the form

23

(i)

λ, ψ F ⊥ (i) ¬ψ and any M -relative falsification of ψ would take the following form µ V ψ

¬ψ ⊥

But then ψ would have both an M -relative verification and an M -relative falsification, contrary to the inductive hypothesis. Case (ii). Any M -relative verification of ψ1 ∧ ψ2 would take the form µ1 µ2 V1 V2 ψ1 ψ2 ψ1 ∧ ψ2 and any M -relative falsification of ψ1 ∧ ψ2 would take one of the following two forms: (i)

ψ1 ∧ ψ2 ⊥

λ1 , ψ1 F1 ⊥

(i)

(i)

ψ1 ∧ ψ2 ⊥

λ2 , ψ2 F2 ⊥

(i)

Either way, we would have both an M -relative verification and an M -relative falsification of one of the conjuncts, contrary to the inductive hypothesis. Case (iii) and Case (iv) are similar. Case (v). Recall that the individuals in the domain of M are α1 , . . . , αn , . . . . So any M -relative verification of ∀xψ(x) would take the form

24

µ1 V1 ψ(α1 )

µv ... Vn ψ(αn ) ∀xψ(x)

...

and any M -relative falsification of ∀xψ(x) would take the form (i)

∀xψ(x) ⊥

λi , ψ(αi ) Fi ⊥ (i)

The instance ψ(αi ) would therefore have both an M -relative verification and an M -relative falsification, contrary to the inductive hypothesis. Case (vi) is similar.

3

QED

Special features of the rules of evaluation

The foregoing rules of evaluation permit the construction of proof-like objects (verifications and falsifications; or, evaluation proofs and evaluation disproofs). They do, however, have some special limiting characteristics.

3.1

The undischarged assumptions

First, the ‘undischarged assumptions’ of a verification are always either (saturated) primitive formulae, or rules equivalent to negations thereof. When one constructs a verification using the primitive (positive or negative) information in a model (its atomic diagram Λ), there is no complexity in the undischarged assumptions involved (apart from the negation signs in negative literals). The same holds for a falsification, except that the (saturated) formula being falsified may itself be complex. But it will be the only complex formula among the undischarged assumptions of the falsification. So: apart from the complex formula being falsified (when the construction in question is a falsification), construction by means of our rules allows us to ‘reason away from’ at best primitive formulae (and negations thereof). We can emphasize this point by adding mention, within the statement of our rules, of the primitive information upon which the evaluation rests. We shall use λ, λ1 , λ2 as variables ranging over subsets of the atomic diagram Λ.

25

(We use the Greek letter lambda to suggest ‘literals’.) We shall illustrate the point by reference to the rules for the connectives and the quantifiers.

(i)

λ .. .

λ, ϕ .. . ⊥ ¬ϕ

¬ϕ

(i)

λ1 .. .

ϕ ⊥

(i)

λ2 .. .

(i)

λ, ϕ .. .

ϕ ψ ϕ∧ψ

⊥

ϕ∧ψ

λ, ψ .. .

⊥

λ .. .

λ .. .

ϕ ϕ∨ψ

ψ ϕ∨ψ

⊥

(i)

ϕ∨ψ

(i)

λ1 , ϕ .. .

λ2 , ψ .. .

⊥

⊥

λ, ϕ .. .

ψ ϕ→ψ

⊥ ϕ→ψ

(i)

⊥

(i)

(i)

λ .. .

(i)

ϕ→ψ

λ1 .. .

λ2 , ψ .. .

ϕ

⊥

⊥

26

⊥

ϕ∧ψ

(i)

(i)

(i)

λ1 .. . ψ(α1 )

...

λn .. .

ψ(αn ) ∀xψ(x)

3.2

...

⊥

∀xψ(x)

(i)

⊥

(i)

λ .. . ψ(α) ∃xψ(x)

(i)

λ , ψ(α) .. .

λ1 , ψ(α1 ) .. . ∃xψ(x)

(i)

...

λn , ψ(αn ) .. .

⊥

⊥

... (i)

⊥

Conclusions

The second limiting characteristic is that the rules for falsification have ⊥ as their main conclusions, and as conclusions of their subordinate ‘disproofs’. So: the only way to ‘reason away from’ a complex formula (by means of our rules of evaluation) is to reason towards absurdity.

3.3

Domain-dependence of quantifier rules

Thirdly, the rules for verification of universals and for falsification of existentials call for as many subordinate proofs as there are individuals in the domain (one subordinate proof for each individual). And this involves infinite sideways branching when the domain of the model is infinite. (Remember, our verifications and falsifications are model-relative. The are not like deductions in general. The job of a deduction—which is always finitary—is to preserve M -relative truth from its premises to its conclusion, for all models M .)

3.4

The contrast with rules of deduction in general

General rules of deduction allow one in general to reason away from (finite) sets of sentences of any complexity to sentences of any complexity. Of course, primitive sentences (and negations thereof) can stand as assumptions of deductions; and absurdity can stand as a conclusion (in which case the 27

deduction is called a reductio ad absurdum, or refutation, of its set of undischarged assumptions). But deduction in general involves reasoning from a (finite) set of complex sentences as assumptions, to a complex conclusion. We are now in pursuit of rules of inference governing such reasoning, rules in accordance with which more general proofs can be constructed—more general, that is, than our model-relative evaluation proofs (i.e., verifications) and evaluation disproofs (i.e., falsifications). When deducing a sentence ϕ from a set ∆ of sentences, we are no longer working with the atomic diagram of a particular model. Rather, the sentences in the set ∆ (the premises of our sought proof) might all be true ‘simultaneously’ in many different models. The task is to show that in any such model, the sentence ϕ will be true too. That is the job of proof in general. When one has a proof of ϕ whose undischarged assumptions form the set ∆, one must be able to say: any model that verifies every member of ∆ verifies ϕ. This means that we cannot use the present rule for verification of universals when it comes to deductive reasoning towards a universal (trying to establish it as a conclusion); nor can we use the present rule for falsification of existentials when it comes to deductive reasoning away from an existential (trying to use it as a premise). For, both these rules call for a specific number of subordinate deductions, one for each individual in the domain of a specific model (relative to which truth-value determination takes place according to the evaluation rule in question). Deductive reasoning, however, is undertaken without any specific model in mind. What is important is only preservation of truth-value from premises to conclusion—so that every model for the premises is a model for the conclusion.

4

From rules of evaluation to rules for deduction in general

Our task now is to find suitable generalizations or analogues of our rules for verification and falsification that can serve as rules governing deduction towards, and deduction away from, complex sentences. As we survey our rules of verification and falsification, certain of these analogues are immediate. A box subscript on a discharge stroke indicates that the assumption in question must have been used, and therefore be eligible to be discharged. (This was obvious in the case of verifications and falsifications, but now needs to be emphasized, since we are moving towards a statement of rules of inference in general.) 28

We shall take the connectives in turn, as we morph the rules for verification and falsification of sentences with a given connective dominant into more general rules of deduction. These are introduction rules (which tell one how to introduce a dominant occurrence of that connective into the conclusion of an inference), and elimination rules (which tell one how to eliminate a dominant occurrence of the connective from the major premise of an inference).

4.1

Negation

The sought generalizations of the rules for negation are straightforward. 4.1.1

Introduction

The rule for verifying a negation becomes the negation introduction rule upon allowing for more general sets ∆ of side-assumptions in the subordinate reductio:

(i)

∆, ϕ .. . ⊥ ¬ϕ

(i)

Note that the conclusion ¬ϕ depends only on the assumptions in ∆; the assumption ϕ (for reductio ad aburdum) is discharged by applying the rule. Moreover, as indicated by the box subscript on the discharge stroke, the assumption ϕ must have been used, and be undischarged within the subordinate reductio, in order that the rule be applicable. 4.1.2

Elimination

The rule for falsifying a negation becomes the negation elimination rule upon allowing for more general sets ∆ of assumptions in the subordinate proof of the minor premise: ∆ .. . ¬ϕ

ϕ ⊥

29

The conclusion rests both on ¬ϕ and on the assumptions in ∆. Note also that we are not allowing the major premise ¬ϕ to stand, itself, as the conclusion of any proof-work above it. Rather, ¬ϕ stands proud as an undischarged assumption. (It could, however, be discharged by subsequent applications of rules of inference, as the proof-work proceeded further in a downward direction).

4.2

Conjunction

4.2.1

Introduction

The morphing of the rule for model-relative verification of conjunctions into the introduction rule (for inferring conjunctions from arbitrary sets of premises) is straightforward: ∆1 .. .

∆2 .. .

ϕ ψ ϕ∧ψ Note that we allow for ∆1 to be distinct from ∆2 . The conclusion ϕ ∧ ψ depends on their union. 4.2.2

Elimination

Now consider the rule for model-relative falsification of a conjunction: (i)

(i)

∆1 , ϕ .. . ⊥

ϕ∧ψ

∆2 , ψ .. . ⊥

ϕ∧ψ

(i)

⊥

(i)

⊥

We want the elimination rule to allow for the derivation of general conclusions θ in place of ⊥: (i)

(i)

∆1 , ϕ .. . ϕ∧ψ

θ θ

(i)

∆2 , ψ .. . ϕ∧ψ

θ θ 30

(i)

and we may as well economize by allowing for simultaneous discharge of the dischargeable assumptions. At the same time we require that at least one such assumption should have been used, and therefore be eligible to be discharged—this requirement being indicated by a box affixed to the inference strokes: (i)

(i)

∆ ,ϕ, ψ .. . ϕ∧ψ

θ

(i)

θ The conclusion θ depends only on ϕ ∧ ψ and the assumptions in ∆. The major premise ϕ ∧ ψ stands proud.

4.3 4.3.1

Disjunction Introduction

As with conjunction, the introduction rule for disjunction is a straightforward generalization of the rule of model-relative verification:

4.3.2

∆ .. .

∆ .. .

ϕ ϕ∨ψ

ψ ϕ∨ψ

Elimination

The rule for reasoning away from a disjunctive premise ϕ ∨ ψ needs likewise to be generalized so as to permit the deduction of a sentence θ in general, rather than just ⊥. But to this end it would suffice to deduce θ from but one of the cases ϕ and ψ. If the other case closes off with ⊥, then we know the truth does not lie there; hence, lies with the case that leads to θ. So permissible deductive moves would be:

31

(i)

ϕ∨ψ

(i)

∆1 , ϕ .. .

∆2 , ψ .. .

θ

⊥

(i)

ϕ∨ψ

(i)

⊥

(i)

∆1 , ϕ .. .

∆2 , ψ .. .

⊥

θ

(i)

⊥

Naturally also if θ is deducible from each case-assumption, then θ should be deducible overall: (i)

ϕ∨ψ

(i)

∆1 , ϕ .. .

∆2 , ψ .. .

θ

θ

(i)

θ And a special case of θ in this last rule is of course ⊥ itself, as with the rule of falsification with which we began. We can sum up the possibilities just canvassed as follows:

ϕ∨ψ

(i)

∆1 , ϕ .. .

∆2 , ψ .. .

θ/⊥

θ/⊥

(i)

(i)

θ/⊥ The conclusion (θ or ⊥) depends on ϕ ∨ ψ, the members of ∆1 and the members of ∆2 . The sets ∆1 and ∆2 could be distinct. The major premise ϕ ∨ ψ stands proud. The rule of ∨-Elimination is also known as proof by cases. The two subproofs indicated are called the case-proofs. The rule of thumb is: if either one of the case-proofs ends with ⊥, one may bring down the conclusion of the other case-proof as the overall conclusion.

4.4 4.4.1

The conditional Introduction

A refutation of ϕ modulo ∆:

32

∆, ϕ .. . ⊥ guarantees that any model of ∆ falsifies ϕ.2 Thus, by the first half of the model-relative verification rule for the conditional, any model of ∆ verifies ϕ → ψ. So the first half of our model-relative verification rule generalizes into the first half of the sought introduction rule as follows:

(i)

∆, ϕ .. . ⊥ ϕ→ψ

(i)

The second half of our model-relative verification rule needs, however, to be generalized more carefully. We need to allow for the distinct possibility that one might not be in a position to deduce the consequent, given the under-specific information ∆ at hand (as opposed to the highly specific information about a model, which will tell one whether the consequent holds). Since we are now allowing deductions from arbitrary sets of complex sentences, one can imagine a situation in which one has a deduction of the consequent ψ from the antecedent ϕ along with other assumptions forming a set ∆, say: ∆, ϕ .. . ψ Remember that deductions are to be truth-preserving. So, every model of ∆ that verifies ϕ verifies ψ. Therefore one can say of every model M of ∆: if M verifies ϕ, then M verifies ψ—whence, M verifies ϕ → ψ. That justifies the following second half of the introduction rule for the conditional: ∆, ϕ .. .

(i)

ψ (i) ϕ→ψ 2

This innocuous-seeming claim requires, and admits of, proof.

33

The use of the diamond here indicates that the subordinate proof need not have used ϕ as an assumption. But if it did, then that assumption will be discharged by application of the rule. In a case where ϕ is not used as an assumption, the justification of ϕ → ψ is immediate by the verification rule: for every model of ∆ would, ex hypothesi, verify ψ, hence also (by the verification rule) verify ϕ → ψ. 4.4.2

Elimination

The elimination rule for the conditional is obtained by straightforward morphing of the rule for model-relative falsification of conditionals:

ϕ→ψ

(i)

∆1 .. .

∆2 , ψ .. .

ϕ

θ

(i)

θ The conclusion θ depends on ϕ → ψ, all the members of ∆1 and all the members of ∆2 . The sets ∆1 and ∆2 can be distinct. The major premise ϕ → ψ stands proud. The proof of ϕ from ∆1 is called the minor proof; that of θ from ∆2 , ψ is called the major proof (for →-Elimination).

5 5.1

The universal quantifier Introduction

Recall the model-relative rule for verifying a universal claim (without loss of generality here we shall assume that the domain is finite): .. .. . . ... ψ(α1 ) ψ(αn ) ∀xψ(x)

where α1 , . . . , αn are all the individuals in the domain

In the presence of the indicated verifications, one would have a proof of ψ(a) from the assumption a = α1 ∨ . . . ∨ a = αn , to the effect that a is a member of the domain:3 3 In the infinitary case, the disjunctive major premise would have to be infinitary. That would pose no problems in principle, however, since the disjunction in question is being

34

.. . ψ(α1 ) a = α1 ∨ . . . ∨ a = αn

(1)

a = α1 ψ(a)

...

.. . ψ(αn )

(1)

a = αn ψ(a) (1)

ψ(a) The idea behind the introduction rule for ∀ is that any ‘parametric’ proof to the effect a is in the domain .. . ψ(a) should justify one in drawing the conclusion ∀xψ(x): (i)

a is in the domain .. . ψ(a)

(i)

∀xψ(x) Standard logic is the logic of a logically perfect language. The perfection assumption is that every well-formed singular term denotes. In free logic, one gives up this assumption. One allows for empty names, such as ‘Pegasus’. One allows also for partial functions, that is, functions that are ‘not everywhere defined’, such as division (which is not defined when the divisor is 0). And one needs a free logic if one aims to accommodate the definite description operator as a primitive variable-binding term-forming operator. This is because any term of the form ιx(ϕx ∧ ¬ϕx) fails to denote. The same need arises with the set abstraction operator. For, as Russell’s Paradox shows, the term {x | ¬x ∈ x} fails to denote. In free logic, the assumption that a is in the domain is expressed by the formal sentence ∃x x = a, often abbreviated as ∃!a. Thus the rule of ∀-Introduction in free logic is (i)

∃!a .. . ψ(a)

where ∃!a is the only assumption containing a on which ψ(a) depends (i)

∀xψ(x) invoked only by way of motivation of the main idea.

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In standard logic, however, the assumption that a is in the domain does not need to find expression. One can limit oneself to proofs of ψ(a) from assumptions making no mention of a: .. . ψ(a) ∀xψ(x)

5.2

where ψ(a) depends on no assumptions containing a

Elimination

The elimination rule for ∀ can be stated as a rule allowing for multiple discharge (of assumption-instances of the predicate involved). This is on grounds analogous to those on which we allowed the ∧-Elimination rule simultaneously to discharge all assumption-occurrences of either of its conjuncts in the subordinate proof. The rule of ∀-Elimination is accordingly as follows: (i)

(i)

... {z .. .

ψ(t1 ) | ∀xψ(x)

θ

ψ(tn ) } (i)

θ In free logic one would invoke extra premises to the effect that one’s chosen terms t1 , . . . , tn do indeed denote: (i)

(i)

... {z .. .

ψ(t1 ) | ∀xψ(x) ∃!t1 . . . ∃!tn

θ

ψ(tn ) } (i)

θ

6 6.1

The existential quantifier Introduction

The introduction rule for ∃ closely resembles the model-relative rule for verifying existentials. In the latter, it sufficed to verify a single instance: 36

.. . ψ(α) ∃xψ(x) When deducing conclusions from arbitrary sets of sentences, the available instances are no longer saturated formulae using particular individuals α from a domain. Rather, one might prove, for some term t, that ψ(t) holds (conditionally on whatever assumptions are being used). Then—on the background assumption that said term t denotes an individual—one would be able to conclude ∃xψ(x): ψ(t) ∃xψ(x) In free logic one would invoke an extra premise to the effect that one’s chosen term does indeed denote: ψ(t) ∃!t ∃xψ(x)

6.2

Elimination

Recall the model-relative rule for falsifying an existential claim (in the finite case): (i)

∃xψ(x)

(i)

ψ(α1 ) ψ(αn ) .. .. ... . . ⊥ ⊥

where α1 , . . . , αn are all the individuals in the domain

(i)

⊥ Similar considerations apply here as applied in our formulation of the rule of ∀-Introduction above. In the presence of the indicated falsifications, one would have a disproof of ψ(a), using the assumption a = α1 ∨ . . . ∨ a = αn , to the effect that a is a member of the domain:4 4

See footnote 3.

37

(1)

ψ(a)

a = α1 ∨ . . . ∨ a = αn

a = α1 ψ(α1 ) .. .

(1)

ψ(a) ...

⊥

a = α1 ψ(α1 ) .. . ⊥

(1)

⊥ Accordingly, in free logic the rule of ∃-Elimination would be (i)

(i)

∃!a ψ(a) | {z } .. . ∃xψ(x)

θ

(i)

where the parameter a does not occur in any assumption, other than ∃!a and ψ(a), on which the upper occurrence of θ depends; and does not occur in either ∃xψ(x) or θ

θ while in standard logic it would be (i)

ψ(a) .. . ∃xψ(x)

θ

(i)

where the parameter a does not occur in any assumption, other than ψ(a), on which the upper occurrence of θ depends; and does not occur in either ∃xψ(x) or θ

θ

7

Summary of introduction and elimination rules

We collect together the introduction and elimination rules for standard, unfree logic. Where sets of assumptions (other than those being discharged) are permitted within subproofs, we indicate this by means of ∆ (with or without a numerical subscript). In the rule ∧-E, the box indicates that at least one of the conjuncts ϕ, ψ must appear as an (undischarged) assumption in the subordinate proof. Likewise, in the rule ∀-E, the box at the level of the discharge strokes indicates that at least one undischarged assumption of the form ψ(t) must appear in the subordinate proof. Other boxes attached to discharge strokes indicate that an assumption of the form in question must appear undischarged in the subordinate proof. The diamond in the second half of →, however, indicates that no such assumption need appear; if it does, however, it is discharged by the application of the rule.

38

Introduction

¬

(i)

∨

¬ϕ

(i)

∆1 .. .

ϕ ⊥

(i)

∆2 .. . ϕ∧ψ

ϕ ϕ∨ψ

ψ ϕ∨ψ

(i)

∆2 , ψ .. .

θ/⊥

θ/⊥

∀

ψ(a) ∀xψ(x)

(i)

(i)

θ/⊥

(i)

∆2 , ψ .. .

ϕ

θ

ϕ→ψ

ψ (i) ϕ→ψ

(i)

∆1 .. .

(i)

θ ...

(i)

∆ .. .

(i)

∆1 , ϕ .. . ϕ∨ψ

∆, ϕ .. .

∆, ϕ .. .

(i)

∆ .. .

⊥ ϕ→ψ

θ θ

∆ .. .

(i)

(i)

∆ ,ϕ, ψ .. .

ϕ ψ ϕ∧ψ

→

∆ .. .

∆, ϕ .. . ⊥ ¬ϕ

∧

Elimination

∆ |

where a does not occur in any member of ∆

ψ(t1 ) {z .. .

∀xψ(x)

θ θ

39

2... ...

(i)

(i)

ψ(tn ) }

2

∆ .. .

∃

ψ(t) ∃xψ(x)

∃xψ(x)

θ θ

ϕ

=

(i)

∆, ψ(a) | {z } .. .

t=t

t=u ψ

(i)

where a does not occur in ∃xψ(x), θ or any member of ∆

where ϕut = ψtu

Important note: Major premises for eliminations stand proud. They are not drawn as conclusions of any proof-work above them.

8

Conclusion

8.1

Summary

The foregoing rules are those of core logic.5 The aim of this study has been to reveal the natural way in which core logic emerges from reflections on how one establishes sentences as true or as false under interpretations, and how one can generalize those movements in thought so as to deal with complex sentences in general, as starting points and as endpoints of trains of reasoning. Elsewhere, I have argued that core logic is the correct logic according to an anti-realist account of meaning;6 that it suffices for constructive mathematics;7 that it suffices for hypothetico-deductive testing of theories;8 that it enables efficient automated proof-search;9 and that it is the minimal canon invulnerable to revision, every part of which is indispensable for the process of rational belief-revision.10 On this occasion, however, the aim has been to describe a natural conceptual route to core logic, beginning with one’s rudimentary grasp of verification- and falsification-conditions, construed in a suitably inferentialist fashion. 5

Formerly called IR, or intuitionistic relevant logic, as in [4], [5] and [7]. See [4], [7]. 7 See [6]. 8 See [3]. 9 See [5]. 10 See [9]. 6

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8.2

Further developments

Model-relative verifications and falsifications, as introduced here, are rigorously definable as well-understood mathematical objects: they are trees of finite depth, whose nodes are labeled by saturated formulae. When the domain of the model is infinite, the sideways branchings corresponding to verifications of universals and falsifications of existentials will be infinite. But all branches will be of finite length. In a reasonably weak metamathematical theory, one can prove the following. Theorem 1 For any model M , and any saturated formula ϕ, ϕ is true in M (in Tarski’s sense) ⇔ there is an M -relative verification of ϕ and ϕ is false in M (in Tarski’s sense) ⇔ there is an M -relative falsification of ϕ The theory of model-relative verifications and falsifications is a structuretheory for truth-makers and falsity-makers. The current philosophical literature on truth-makers appears to be bereft of such a structure-theory. In future work I intend to demonstrate some advantages in conceiving of philosophers’ ‘truth-makers’ as these (appropriately structured) model-relative verifications. Other results, to be presented in detail elsewhere,11 are the following. Theorem 2 Given any two proofs in core logic, where the conclusion of the first proof is a premise of the second proof (call it the ‘cut-sentence’), one can effectively find a core proof of the second proof ’s conclusion, or of absurdity, from premises of the given proofs other than the cut sentence. Theorem 3 Proofs in core logic provide an ‘effective’ means of transforming (for any model M ), M -relative verifications of their undischarged assumptions into M -relative verifications of their conclusions.

11

These results were stated in [8].

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References [1] Alfred Tarski. The Concept of Truth in Formalized Languages. In J. H. Woodger, editor, Logic, Semantics, Metamathematics, pages 152–278. Clarendon Press, Oxford, 1956. [2] Neil Tennant. Natural Logic. Edinburgh University Press, 1978. [3] Neil Tennant. Minimal logic is adequate for Popperian science. British Journal for Philosophy of Science, 36:325–329, 1985. [4] Neil Tennant. Anti-Realism and Logic: Truth as Eternal. Clarendon Library of Logic and Philosophy, Oxford University Press, 1987. [5] Neil Tennant. Autologic. Edinburgh University Press, 1992. [6] Neil Tennant. Intuitionistic mathematics does not need ex falso quodlibet. Topoi, pages 127–133, 1994. [7] Neil Tennant. The Taming of The True. Oxford University Press, 1997. [8] Neil Tennant. Cut for Core Logic. Paper presented to the Sixth Quadrennial Fellows Conference of the Pittsburgh Center for Philosophy of Science, July 2008, Ohio University, Athens and to the NYU Conference in Philosophy of Mathematics, April 2009, Unpublished typescript. [9] Neil Tennant. Rational Belief Revision. Unpublished typescript.

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