An Automated Derivation of Łukasiewicz's CN Sentential Calculus from Church's P 2

Transcription

1 An Automated Derivation of Łukasiewicz's CN Sentential Calculus from Church's P 2 Jack K. Horner P. O. Box 266 Los Alamos, New Mexico USA Abstract Two logics are implicationally equivalent if the axioms and inference rules of each imply the axioms of the other. Characterizing the implicational equivalences of various formulations of the sentential calculi is foundational to the study of logic. Using an automated deduction system, I show that Łukasiewicz's CN sentential calculus can be derived from Church's P 2 ; the proof appears to be novel. Keywords: propositional logic, automated deduction, sentential calculus 1.0 Introduction Two logics are implicationally equivalent if the axioms and inference rules of each imply the axioms of the other. Characterizing the implicational equivalences of various formulations of the sentential calculi is foundational to the study of logic ([1],[3]- [7],[9]-[10],[12]-[15]). "CN", the formulation of the sentential calculus in [1], is among the most austere: its vocabulary contains only two logical connectives (C, and N) and sentence variables (p, q, r,...). It has two inference rules (condensed detachment and substitution), and three axioms. In CN, any expression of the form Cxy or Nz, where x, y, and z are sentences, is a sentence. Cpq is interpreted as "sentence p implies sentence q"; Np is interpreted as "not-p". C and N are right-associative; N has higher associative precedence than C. For example, CCqrCpNr translates to the more common "arrow-andparenthesis" notation as (q r) (p ~r) where " " designates "implies" and "~" designates "not". The axioms of CN in [1] are: CN1. CCpqCCqrCpr CN2. CCNppp CN3. CpCNpq Cast in CN notation, the axioms of the Church's P 2 sentential calculus (P2, [12]) are P CpCqp P CCsCpqCCspCsq P CCNpNqCqp The main result of this paper is that [12] implies [1].

2 2.0 Method To show that [12] implies [1], the prover9 ([2]) script shown in Figure 1 was executed under on a Dell Inspiron 545 with an Intel Core2 Quad CPU 2.33 GHz and 8.00 GB RAM, running under the Windows Vista Home Premium (SP2)/Cygwin operating environment. set(hyper_resolution). formulas(usable). P ( i(x, i(y,x)) ) # label("p2-202"). P ( i(i(z, i(x,y)), i(i(z,x), i(z,y))) ) # label("p2-203"). P ( i(i(-x,-y), i(y,x)) ) # label("p2-204"). formulas(sos). -P(i(x,y)) -P(x) P(y) # label("infcondet"). formulas(goals). P( i(i(x,y), i(i(y,z), i(x,z))) ) # label("axcn1"). P( i(i(-x,x), x) ) # label("axcn2"). P( i(x, i(-x,y)) ) # label("axcn3"). Figure 1. The prover9 script used to show that P2 implies CN. The implementation of condensed detachment is the formula in the "sos" list; substitution is derived from prover9's hyperresolution rule (introduced in the "set" command at the top of the script). Details of prover9's syntax and semantics can be found in [2]. 3.0 Results Figure 2 shows that P2 implies CN.

3 % Proof 1 at 0.11 (+ 0.03) seconds: "AxCN2". % Length of proof is 18. % Level of proof is 6. % Maximum clause weight is 20. % Given clauses P(i(i(-x,x),x)) # label("axcn2") # label(non_clause) # label(goal). [goal]. 6 P(i(i(-x,-y),i(y,x))) # label("p2-204"). [assumption]. 9 -P(i(i(-c4,c4),c4)) # label("axcn2") # answer("axcn2"). [deny(2)]. 11 P(i(i(i(-x,-y),y),i(i(-x,-y),x))). [hyper(7,a,5,a,b,6,a)]. 14 P(i(x,i(i(-y,-z),i(z,y)))). [hyper(7,a,4,a,b,6,a)]. 18 P(i(x,i(i(i(-y,-z),z),i(i(-y,-z),y)))). [hyper(7,a,4,a,b,11,a)]. 152 P(i(x,i(i(-y,-x),y))). [hyper(7,a,33,a,b,18,a)]. 154 P(i(-x,i(x,y))). [hyper(7,a,33,a,b,14,a)]. 158 P(i(i(-x,x),i(-x,y))). [hyper(7,a,5,a,b,154,a)]. 185 P(i(i(x,i(-y,-x)),i(x,y))). [hyper(7,a,5,a,b,152,a)] P(i(i(-x,x),x)). [hyper(7,a,185,a,b,158,a)] $F # answer("axcn2"). [resolve(1009,a,9,a)]. % Proof 2 at 0.11 (+ 0.03) seconds: "AxCN3". % Length of proof is 25. % Level of proof is 9. % Maximum clause weight is 20. % Given clauses P(i(x,i(-x,y))) # label("axcn3") # label(non_clause) # label(goal). [goal]. 6 P(i(i(-x,-y),i(y,x))) # label("p2-204"). [assumption]. 10 -P(i(c5,i(-c5,c6))) # label("axcn3") # answer("axcn3"). [deny(3)]. 11 P(i(i(i(-x,-y),y),i(i(-x,-y),x))). [hyper(7,a,5,a,b,6,a)]. 13 P(i(i(x,y),i(x,x))). [hyper(7,a,5,a,b,4,a)]. 14 P(i(x,i(i(-y,-z),i(z,y)))). [hyper(7,a,4,a,b,6,a)]. 18 P(i(x,i(i(i(-y,-z),z),i(i(-y,-z),y)))). [hyper(7,a,4,a,b,11,a)].

4 22 P(i(x,x)). [hyper(7,a,13,a,b,4,a)]. 23 P(i(i(i(x,y),x),i(i(x,y),y))). [hyper(7,a,5,a,b,22,a)]. 152 P(i(x,i(i(-y,-x),y))). [hyper(7,a,33,a,b,18,a)]. 154 P(i(-x,i(x,y))). [hyper(7,a,33,a,b,14,a)]. 159 P(i(x,i(-y,i(y,z)))). [hyper(7,a,4,a,b,154,a)]. 165 P(i(i(i(-x,i(x,y)),z),z)). [hyper(7,a,23,a,b,159,a)]. 168 P(i(i(x,-y),i(x,i(y,z)))). [hyper(7,a,5,a,b,159,a)]. 185 P(i(i(x,i(-y,-x)),i(x,y))). [hyper(7,a,5,a,b,152,a)]. 996 P(i(--x,x)). [hyper(7,a,165,a,b,185,a)] P(i(x,--x)). [hyper(7,a,6,a,b,996,a)] P(i(x,i(-x,y))). [hyper(7,a,168,a,b,1029,a)] $F # answer("axcn3"). [resolve(1050,a,10,a)]. % Proof 3 at 4.93 (+ 0.53) seconds: "AxCN1". % Length of proof is 18. % Level of proof is 8. % Maximum clause weight is 24. % Given clauses P(i(i(x,y),i(i(y,z),i(x,z)))) # label("axcn1") # label(non_clause) # label(goal). [goal]. 8 -P(i(i(c1,c2),i(i(c2,c3),i(c1,c3)))) # label("axcn1") # answer("axcn1"). [deny(1)]. 15 P(i(x,i(i(y,i(z,u)),i(i(y,z),i(y,u))))). [hyper(7,a,4,a,b,5,a)]. 29 P(i(x,i(y,i(z,i(u,z))))). [hyper(7,a,4,a,b,16,a)]. 65 P(i(i(x,i(i(y,i(z,y)),u)),i(x,u))). [hyper(7,a,12,a,b,29,a)]. 89 P(i(i(i(x,i(y,z)),i(i(i(x,y),i(x,z)),u)),i(i(x,i(y,z)),u))). [hyper(7,a,12,a,b,15,a)]. 153 P(i(i(x,y),i(i(z,x),i(z,y)))). [hyper(7,a,33,a,b,15,a)] P(i(i(i(x,y),z),i(y,z))). [hyper(7,a,65,a,b,153,a)] P(i(x,i(i(i(y,z),u),i(z,u)))). [hyper(7,a,4,a,b,2531,a)] P(i(i(x,i(y,z)),i(y,i(x,z)))). [hyper(7,a,89,a,b,4557,a)] P(i(i(x,y),i(i(y,z),i(x,z)))). [hyper(7,a,20399,a,b,153,a)] $F # answer("axcn1"). [resolve(23067,a,8,a)]. Figure 2. Summary of a prover9 ([2]) proof showing that P2 ([12]) implies CN ([1]).

5 The total time to complete the proofs shown in Figure 2 was ~6 seconds on the platform described in Section Conclusions and discussion Section 3 demonstrates that P 2 implies CN. A companion paper ([16]) proves CN implies P 2. The proof in Figure 2 appears to be novel. 5.0 References [1] Łukasiewicz J. Elements of Mathematical Logic. Second Edition (1958). Trans. by Wojtasiewicz O. Pergamon Press [2] McCune WW. prover9 and mace [3] Aristotle. Prior Analytics. Trans. by A. J. Jenkinson. In Aristotle. The Basic Works of Aristotle. Ed. by R. McKeon. Random House pp [4] Aristotle. Posterior Analytics. Trans. by G. R. G. Mure. In Aristotle. The Basic Works of Aristotle. Ed. by R. McKeon. Random House pp [5] Tarski A. Introduction to Logic. Trans. by O. Helmer. Dover [6] Hempel C. Studies in the logic of explanation. In Hempel C. Aspects of Scientific Explanation and Other Essays in the Philosophy of Science. Free Press pp [9] Russell B and Whitehead AN. Principia Mathematica. Volume I (1910). Merchant Books [10] Frege G. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle Translated in van Heijenoort J. Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought. From Frege to Gödel: A Source Book in Mathematical Logic, Harvard pp [11] Horn A. On sentences which are true of direct unions of algebras. Journal of Symbolic Logic 16 (1951), [12] Church A. Introduction to Mathematical Logic. Volume I. Princeton [13] Birkhoff G and von Neumann J. The logic of quantum mechanics. Annals of Mathematics 37 (1936), [14] Kant I. Kant's Introduction to Logic (1800). Trans. by Abbott TK. Greenwood Press [15] Cohen MR and Nagel E. An Introduction to Logic and Scientific Method. Harcourt, Brace, and Company [16] Horner JK. An automated derivation of Church's P 2 sentential calculus from Łukasiewicz's CN. Proceedings of the 2011 International Conference on Artificial Intelligence. CSREA Press. Forthcoming. [7] Quine WVO. Philosophy of Logic. Second Edition. Harvard [8] Chang CC and Keisler HJ. Model Theory. North-Holland