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Distributivity in Lℵ0 and Other Sentential Logics

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Distributivity in Lℵ0 and Other Sentential Logics 2 Harris & Fitelson Distributivity inL ℵ0 and Other Sentential Logics We choose Polish notation not just because of its historical dominance in this area, but also because of its syntactic economy. Some of the proofs we will present are quite complex and would be difficult to parse Kenneth Harris and Branden Fitelson and/or typeset in either the infix or the prefix notation used by Otter. University of Wisconsin–Madison The proofs presented in this paper have been translated directly 2 November 10, 2000 from Otter output files into our Polish notation. Hyperresolution is used, throughout, to implement all logical rules of inference in the Abstract. Certain distributivity results forLukasiewicz’s infinite-valued logicL ℵ0 are proved axiomatically (for the first time) with the help of the automated reasoning systems presented. The notation ‘[i, j, k]’written to the left of each program Otter [16]. In addition, non-distributivity results are established for a formula in our proofs is shorthand for ‘the present formula was obtained wide variety of positive substructural logics by the use of logical matrices discovered by applying the rule at line i (via a single hyperresolution step), with with the automated model findingprograms Mace [15] and MaGIC [25]. the formula at line j as major premise and the formula at line k as minor premise.’Rules of inference are stated using commas to separate the Keywords: distributivity, sentential, logic,Lukasiewicz, detachment, substructural premises of the rule, and a double arrow ‘⇒’to indicate its conclusion. For instance, modus ponens (or detachment) looks like: ‘Cpq,p ⇒ q’. 1. Motivation 3.Lukasiewicz’s Infinite-Valued LogicL ℵ This research was originally motivated by the following uncharacteristic 0 remark made by Rose and Rosser in their classic paper onLukasiewicz’s The Semantics ofL 3.1. ℵ0 infinite-valued logicL ℵ0 [23, pp. 11-12]: A B K L With both and servingas disjunctions and both and servingas The semantics of the infinite-valued logicL ℵ0 were first presented by conjunctions, one can write a number of possible distributive laws. Some are Lukasiewicz [13, pp. 129-130]. His original motivation was to provide a not valid, and of the valid ones we have been able to prove only two from the “numerical interpretation” of the sentential variables which appear in axiom schemes A1-A4. Whitehead and Russell’s Principia Mathematica. This numerical inter- We took up the challenge to prove the validL ℵ distributivity results 0 pretation was intended to (i) provide a way to show that “some logical that Rose and Rosser had been unable to establish axiomatically.1 Our laws are independent of the others,” and (ii)toextendLukasiewicz’s success lead us to a much deeper understanding of distributivity, not three-valued logic [13, pp. 87-109] to a logic with infinitely many values only inL ℵ , but in a wide variety of positive sentential logics, both 0 or “degrees of probability corresponding to various possibilities.”3 classical and non-classical. Much of what we learned is reported here. We findLukasiewicz’s original presentation of the semantics some- what cumbersome to apply. What follows is Wajsberg’s [28, p. 123, note 8] simplification of the semantics forL ℵ . We introduce a valuation 2. Polish Notation and Other Conventions 0 function v which assigns rational numbers on the closed unit interval We will adopt the well-known Polish notation ofLukasiewicz. In par- [0, 1] to each atomic sentence in the language ofL ℵ0 .Thevaluesv ticular, ‘C’stands for implication, ‘ N’for negation, ‘ A’for disjunction, assigns to complexL ℵ0 sentences are determined recursively, as follows: and ‘K’for conjunction. Lower case letters ‘ p’,‘q’, etc., are used as 0ifp ≥ q, sentential or propositional variables. To illustrate how Polish notation −˙ v(Cpq)=q p = v(Kpq)=max(p, q) differs from infix and prefix notation (both of which are supported by q − p if p<q. Otter), we show how a distributive law looks in each of the three: Polish: CKpAqrAKpqKpr v(Apq)=min(p, q) v(Np)=1− p ∨ → ∨ Infix: (p &(q r)) ((p & q) (p & r)) 2 See www.mcs.anl.gov/˜fitelson/distrib/ for relevant input files. Prefix: C(K(p, A(q, r)), A(K(p, q), K(p, r))) 3 See [11, pp. 569–572] for the historical development of the thoughts of Lukasiewicz and his contemporaries on many-valued logics. See [26] for a recent 1 Note added in proof: Beavers [2] has independently taken up this challenge. survey of interpretations and applications of many-valued logics, includingL ℵ0 . Distributivity inL ℵ0 and Other Sentential Logics 3 4 Harris & Fitelson In other words, v assigns a conditional of the form Cpq the value sections that follow, we will provide various axiomatic proofs of (1). q −˙ p,where−˙ is cutoff subtraction, v assigns a conjunction Kpq the An axiomatic proof of (1) eluded Rose and Rosser [23]. maximum of the values of p and q, v assigns a disjunction Apq the minimum of the values of p and q,andv assigns a negation Np the The Axiomatics ofL 3.2. ℵ0 value 1 − p.4 Thinking in terms of cutoff subtraction, max, and min, 6 makes calculating the values of complexL ℵ0 sentences relatively easy. Lukasiewicz [13, pp. 143-144] conjectured that the following set of five There is just one designated value in this semantics: zero. For any axiom schemes (A1–A5) and the single rule (CD) of detachment7 is sentence p ofL ℵ , p is said to be semantically valid inL ℵ just in case 0 0 complete with respect to the semantics ofL ℵ0 described above. v(p) = 0, for all valuations v. As an illustration, we now present a proof (CD) Cpq,p ⇒ q that the following distributive law is semantically valid inL ℵ0 . (A1) CpCqp (1) CKpAqrAKpqKpr (A2) CCpqCCqrCpr The proof that (1) is valid inL ℵ0 involves calculating v[(1)] in six cases, corresponding to all of the different ways in which p, q,andr can be (A3) CCCpqqCCqpp 5 partially ordered, according to any possible valuation v. To this end, (A4) CCNpNqCqp we present the following “Lℵ truth-table” for the formula (1). 0 (A5) CCCpqCqpCqp Six Cases v(Aqr) v(KpAqr) v(Kpq) v(Kpr) v(AKpqKpr) v[(1)] Lukasiewicz’s conjecture is rumored to have been proven first by his p ≤ q ≤ r q q q r q 0 student Mordchaj Wajsberg [13, p. 144] [28, p. 105]. However, Wajs- berg’s proof was never made public. Rose and Rosser [23] were the first p ≤ r ≤ q r r q r r 0 to publish a proof of the completeness of A1–A5. Since then, others q ≤ p ≤ r q p p r p 0 have provided completeness proofs with respect to more general classes of algebraic structures [6, 7]. q ≤ r ≤ p q p p p p 0 InL ℵ0 , A and K are defined in terms of C and N, as follows: r ≤ p ≤ q r p q p p 0 Apq =df CCpqq r ≤ q ≤ p r p p p p 0 Kpq =df NCCNpNqNq In the next section, we will report an Otter proof of the distribu- Provingthat the Distributive Law (1) is Valid in Lℵ 0 tivity of K over A (expressedintermsofC and N), from A1–A4. Inspection of the above table reveals that every possible valuation v is 6 Axiom A5 was shown to be dependent by both Meredith [17] and Chang[5]. such that v[(1)] = 0. Therefore, (1) is semantically valid inL ℵ .Inthe 0 Recently, Wos [31, §11.4.3] has used Otter to find much shorter and more elegant 4 Strictly speaking, we should always write ‘v(p)’ when we talk about the value v proofs of the dependence of A5 (see footnote 10). Since A5 is dependent, we will ℵ assigns to the sentence p. But, we will, for simplicity, often write things like ‘p ≤ q’, restrict ourselves to axioms A1–A4 when workingin the C-N fragment ofL 0 . where it is understood in such cases that ‘p’and‘q’denotethenumerical values Although A5 is dependent, it plays a crucial role in our proofs of distributivity. 7 that v assigns to the sentences p and q, respectively (not the sentences themselves). Lukasiewicz and his contemporaries allowed themselves both the rule of de- 5 tachment and the rule of arbitrary substitution (or instantiation) into theorems. In Wajsberg[28, p. 125, Theorem 17] proves that the set of valid Lℵ0 sentences is decidable. The validity of (1) is equivalent to a conjunction of six statements in the the late 1950’s, Meredith (as explained by Prior [19, Appendix II]) eliminated the first-order theory of the rationals. For example, the first case can be rephrased as: arbitrariness of the rule of substitution by insistingthat all instantiations be most general (much like what we would now call most general unification). This was the ∀ ∀ ∀ ≤ ≤ ⇒ −˙ ( p)( q)( r)[p q & q r max(p, min(q, r)) min(max(p, q), max(p, r)) = 0] birth of condensed detachment (CD), which is easily implemented in Otter using Generally, theL ℵ0 -validity of a sentence p is equivalent to a conjunction of univer- hyperresolution. (CD) is the main rule of inference used throughout this paper, and sally quantified Boolean combinations of linear equalities and inequalities. For this it is what makes the automation of logical calculi feasible [31, page 11]. See [10] class of first-order formulas, Wajsbergproves a quantifier elimination theorem, along for a detailed discussion of the history and automated implementation of condensed the lines of Langford’s [12] decidability proof for the theory of dense linear orders.
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