Lewis Carroll, Gentzen and Entailment Relations. Thierry Coquand. Preliminary version at September 9, 1999. 1 Introduction. This note summarizes some ...
Lewis Carroll, Gentzen and Entailment Relations Thierry Coquand Preliminary version at September 9, 1999
1 Introduction This note summarizes some remarks about entailment relations. Basically these are elementary facts about propositional logic, but I nd it suggestive to formulate it in the simple framework of inference rules (no logical connective is needed!).
2 Entailment Relation Let S be a set, we think of its elements as abstract \statements" or propositions. We denote by X; Y; Z; : : : arbitrary nite subsets of S . We write X; Y for X [ Y and X; s for X [ fsg.
De nition 2.1 An entailment relation ! on the set S is a relation satisfying the following conditions of re exivity, monotonicity and transitivity:
X ! Y if X \ Y is inhabited X !Y X; X 0 ! Y; Y 0 X ! s; Z X; s ! Z X !Z
(R) (M ) (T )
Notice that this de nition is \symmetric": the converse of an entailment relation is also an entailment relation. In presence of (M ) the rule (T ) is actually equivalent to the following rule:
X ! s; Z X 0 ; s ! Z 0 X; X 0 ! Z; Z 0
As emphasised by Scott [Sco74], this notion of entailment relation can be seen as an abstract generalisation of Gentzen's multi-conclusion sequent calculus.
3 Completeness Scott proves a completeness theorem that can be seen as the purest form of completeness theorem for a logic (recall that S is any set, not necessarily a set of syntactical formulae). Theorem: An entailment is a semantical consequence of a given set of rules i it can be deduced by the three rules above. 1
This is actually equivalent to the completeness of Quine's method for enumerating all prime implicants of a given formule in conjunctive normal forms. It is quite remarkable that we can enumerate in this way all interesting consequences of a given set of axioms. This is quite similar to Knuth-Bendix algorithm, that enumerates interesting equational consequences of a given set of equations. Let us give Quine's proof: we suppose that we have only nitely many atoms. Suppose then that A ! B is not derived. Then A and B are disjoint by the rule (R). If all atoms are in A; B then we have a valuation (all As set to 1 and all B s set to 0) that shows that A ! B is not a consequence of the set of rules. Otherwise, let x be an atom not in A; B . It follows from the rule (C) that either A ! B; x or A; x ! B is not derived. Eventually we get back to the cases where all atoms are in A; B:
4 Gentzen The rst published work of Gentzen who was the rst to formulate the rule (T ) was concerned with the problem of nding independent axiomatisations of a given set of formula. Gentzen was working in the restricted system where there is at most one atom in the right side of an entailment. In this framework he gives an example of an in nite system that has no independent axiomatisation. The set S is the set of atoms b; c; a0 ; a1 ; : : : and the axioms are
a0 ; b ! c a 1 ; b ! c a 2 ; b ! c : : : and ai ! aj for i < j: This in nite set of axioms is not independent: for instance a0 ; b ! c is a consequence of a0 ! a2 and a2 ; b ! c: Theorem: This system has no independent axiomatisation. Notice that we can characterise the theorems that are consequence of this theory: either X ! c with X containing either c or b and one of the ai , or X ! aj with X containing ai ; i j or X ! b with X containing b: The theorem is a simple consequence of this remark.
5 Problems of Lewis Carroll There are two contributions: one is a quick method to guess what may be an interesting consequence of a set of rules, the second is the method of trees to nd if a given set of rules is inconsistent. Most problems analysed by Lewis Carroll (in Symbolic Logic Part I and II) can be presented in the following way. We have a nite sets of entailment, and we are asked to nd an interesting consequence. An example is the following: 1. 2. 3. 4.
h; m; k ! ! d; e; c h ! k; a b; l ! h 2
5. c; k ! m 6. h; e ! c 7. b; a ! k Like Gentzen, Lewis Carroll spends some time at looking at the linear case, where all rules are of the form a ! b and the binary case, where the rules may also be of the form a; b ! or ! a; b: The case of rules that are ternary or more is clearly much more dicult and constitutes the part II of his book of Symbolic logic. Lewis Carroll usually did not formulate his rules with letters. Here is a typical (simple) example: what can you deduce from the clauses 1. Babies are illogical; 2. Nobody is despised who can manage a crocodile; 3. Illogical persons are despised? However in this note, I will limit myself to examples with letters. Lewis Carroll makes an interesting remark: among the letters we can distinguish the retinends that occur only on one side of the entailment: here b; l occuring only at the left and d only at the right, and the eliminands tha occur both in the left and right side. We thus expect that b; l ! k should be a consequence. In order to prove this we rewrite the clauses by taking away all occurences of the eliminands: 1. 2. 3. 4. 5. 6. 7.
h; m; k ! ! e; c h ! k; a !h c; k ! m h; e ! c a!k
We see then that h has to hold and we can take away all the occurence of h : 1. 2. 3. 4. 5. 6.
m; k ! ! e; c ! k; a c; k ! m e!c a!k 3
The problem is then to test if this set of rules is inconsistent. The method consists in chosing a rule with a minimum number of atoms and to make a case analysis. For instance from the second rule we either have e = 1 or e = 0; c = 1: The rst case gives 1. m; k ! 2. ! k; a 3. c; k ! 4. ! c 5. a ! k which is contradictory. The second case gives 1. m; k ! 2. ! k; a 3. k ! m 4. a ! k which is also contradictory.
6 Improvement and Ternary Case We can recognize consequence of binary clauses: if we have ! a; b and a ! b then ! b is a consequence. Similarly b ! is a consequence of b ! a and b; a ! : Also, it is standard that the general case can be coded in the ternary case. For instance D A _ B _ C can be represented as d ! u, u ! d and a ! v, b ! v, v ! a; b, expressing v a _ b and v ! u, c ! u and u ! c; v expressing u c _ v:
7 Genealogical Tree Beside the observation about retinends and eliminands, the method that is used to test if a given set of rules is inconsistent is quite remarkable. In this approach we build a tree of possibilities, similar to Beth trees. Carroll calls this method the method of trees or the \Genealogical Method" and was quite proud of it. He had also a justi cation of the fact that the trees should grow head-downwards. \A Genealogical tree always grows downwards: then why may not a Logical tree do likewise?" If we have a rule with only one atom, we immediatly simplify our set of rules by taking away all occurences of this atom. So we are left with rules with two or more atoms. If we have a proof of ! this proof should start by using a rule with no atom on the left or no atom on the right. So we know that if the set of rules is inconsistent we have at least one rule with no atom on the left or symmetrically with no atom on the right. The method consists in chosing one such rule and doing a case analysis on this rule. We know that all proofs of ! if there are any, will be captured by this method. So, like in the examples above, at each node of the tree is associated a set of rules such that: 4
all rules have more than one atom, there is at least one rule with no atoms on the left, and one rule with no atom on the right.
8 Summary of the method and variation The problem to solve is: we have a set of rules, or sequents, is a given sequent a consequence of this set? We introduce the notion of simpli cation of a set of rules: if a is an atom the right simpli cation of a set of rules by a consists in taking away all rules that have a in the right handside and taking away all occurence of a in the left handside of the remaining rules. We de ne dually the left simpli cation by a. To test if a given sequent, say a; b ! c; d is a consequence of a set of sequent we do a left simpli cation by a; b and a right simpli cation by c; d and we test if the remaining set is inconsistent. To test if a set is inconsistent, we choose one rule, say e; f ! g and we do a case analysis: we have three cases, left simpli cation by e; by f and right simpli cation by g: It is understood that if we have a rule with one atom, we do the corresponding simpli cation. A variation of this method is that, instead of taking one rule, we can choose one atom a and do a case analysis: left simpli cation by a and right simpli cation by a:
9 Application Given a xed number of variables, what is the most complicated set of rules which is contradictory?? Here is a simple argument proving that Carroll's method is exponential when applied to the pigeon-hole problems which is the set of clauses, for n > m : ! pi1 ; : : : ; pim for i n; pi1 j ; pi2 j ! for i1 < i2 and all j m I think that the method of trees is quite suggestive here. Consider a possible logical tree for the pigeon-hole problem. We make a distinction between right rules like like ! pi1 ; : : : ; pim and left rules like p11 ; p21 ! : We look at what happens when one moves one step lower in the tree of research: either we make a case using a right rule or a left rule. If we make a case using a left rule pi1 j ; pi2 j !, the tree splits in two parts and by moving for instance in the case pi1 j ! it is like taking away all occurences of pi1 j that we know is false. If we make a case using a right rule ! pij the tree splits in more than two parts and in each case pij we take away one right rule and all occurences of pij . If we follow a branch of length l < m there are at least n ? l remaining right rules, each of one has at least n ? l atoms. It follows that the tree has at least 2m?1 nodes. Intuitively it seems that this method of tree is optimal for solving this kind of problem, and I wonder how to formalise this intuition. A rst step will be to show that any proof by using the rules (T ); (M ); (R) has to be bigger than a proof that comes from a tree. This should hold, but we can also argument as follows l
l
l
5
to show that any resolution proof has to be exponential, by considering the modi cation of Carroll's method where we do a case analysis on one atom. We show by induction on the deduction that any deduction of a sequent, and not only the empty sequent, has to be bigger than one we get by this modi cation of Carroll's method. This is direct if the last step in (M ) and it is by induction if the last step is (T ): for instance if the sequent is a; b ! c; d from a; b; e ! c; d and a; b ! c; d; e we apply the method by doing a case analysis on e:
References [Carroll96] L. Carroll. Symbolic Logic, Part I and II edited by W.W. Bartley, Harvester Press. [Sco74] D. Scott. Completeness and axiomatizability. Proceedings of the Tarski Symposium, pages 411{435, 1974.
6
