Improved Decision Procedures for Lewis's Calculus S4 and von Wright's Calculus M by AlanRoss Anderson; Correction to a paper on Modal Logic by Alan Ross Anderson; On AlternativeFormulations of a Modal System of Feysvon Wright by Alan Ross AndersonReview by: Naoto YonemitsuThe Journal of Symbolic Logic, Vol. 20, No. 3 (Sep., 1955), pp. 302-303Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2268272 .

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302 REVIEWS

The litigation which is proposed by Ryle can be appreciated by a consideration of the clever analogies which he draws between the function of the formal logician and that of the philosopher. It is argued that formal logic is to philosophy as parade-ground tactics are to that of the battlefield or as pure geometry is to cartography or what the role of accounting is to that of the merchant whose problems are not arithmetical but who is in constant need of the "back-room check of the properly balanced ledger."

Conseqgpjitly, philosophy is a kind of informal or applied logic which is such that there si arl essential dependence relationship between its function and that of formal logic.

When viewed in such a light, it is absurd for either the logician or philosopher to encourage the recent domestic issue which constitutes the dilemma under study. Its horns are a hoax which only frustrates a rich harvest which can be the result of joint effort. Even if Ryle's analogies should prove defective at certain points, the moral which he draws from them is of such a quality that it ouldl appear unfortunate if it were not taken seriously by both logicians and philosophers. A. R. TuiQUzTTi

MARY PRIOR and ARTHUR IPRIOR. Erotetic logic. The philosophical review, vol. 64 (1955), pp. 43-59.

This article is taxonomic in character: It draws and discusses several useful dis- tinctions which arise in the logical analysis of questions. Include among these are: (1) The distinction between what-questions andl whether-questions, i.e., between ques- tions whose answer consists in supplying a predicate and questions whose answer consists in the selection of the suitable member of a series of stated alternatives. (2) The distinction between questions which are or are not "genuinely conditional" in the sense of not presupposing a statement whose falsity would make the question improper. (3) The distinction between questions which are or are not mnodal in the sense of permitting "perhaps" or "possibly" as an adequate answer, rather than a mere evasion.

Although this is not made explicit, the present article confines discussion to questions serving as requests for information. Questions can, of course, serve in other capacities: to determine an opinion, attitude or intention, to secure obedience, to reproach, to reassure, and sundry other purposes as well.

Use is made (pp. 54-55) of an interesting necessary condition for the equivalence of questions - the condition of admitting the same answers. This has the consequence that the informal intuitive requirement for the equivalence of questions, that of requesting the same item of information, is not a sufficient condition. In consonance with the authors' criterion, the interrogatives "Are you wearing your green hat tonight?" and "If you wear a hat tonight, will it be your green one ?" do not ask the same question, since "Yes, but I may go hatless" is a proper answer to the second, but not the first.

The authors summarize as the "net result" of their inquiry that the logic of in- terrogatives cannot be developed by means of a symbolic calculus. In the reviewer's opinion, however, their discussion does no more than show that the logic of questions has special and peculiar features which cannot readily be accomodated within existing symbolical systems of (non-interrogative) logic. NICHOLAS RESCHER

ALAN Ross ANDERSON. Improved decision procedures for Lewis's calculus S4 and von Wright's calculus M. The journal of symbolic logic, vol. 19 (1954), pp. 201- 214.

ALAN Ross ANDERSON. Correction to a paper on modal logic. Ibid., vol. 20 (1955), P. 150.

ALAN Ross ANDERSON. On alternative formulations of a modal system of Feys- von Wright. The journal of computing systems, vol. 1 no. 4 (1954), pp. 211-212.

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REVIEWS 303

The contents of the first paper were already outlined, in a slightly different form, in an abstract of this paper in this JOURNAL, vol. 18(1953), pp. 187-188. (The second paper is a correction to the first, indicating that the definition of tautology for S4 should be altered to be the same as that contained in the abstract.)

In section 2 and 3 (and in the Correction), the author describes in detail a new decision procedure for Lewis's calculus S4 (4561, Appendix II) with the proof; and in section 4 a decision procedure for von Wright's calculus M (XVIII 174, Appendix II) without proof, because the proof that the method for M is adequate can be con- structed along the lines of the proof for S4. Decision methods for both systems are already known (McKinsey VII 118, von Wright XVIII 174); however, the author's methods, which constitute simplifications of methods by von Wright, are by far the simplest yet discovered, and they may be regarded as "practical" for modal functions of degree two or less, with three or fewer distinct propositional variables. In section 5, it is remarked that (i) while an analogous development is possible for Lewis's calculus S5 (4561), the method appears to have no advantage over other known methods, (ii) the method of decision for S4 provides new decision procedures for Heyting's propositional calculus (McKinsey-Tarski XIII 171) and for topological equations (McKinsey VII 118).

On page 212 of the first paper the author presents, without proof, a new formulation of von Wright's M, or the modal system T of Feys (Sobocifiski XIX 293), which consists of infinitely many axiom schemata, and the rule of detachment for material implication as the only primitive rule of inference. In the third paper, he gives a proof of equivalence of his formulation with von Wright's Al, notes that his formulation contains an infinite number of non-independent axiom schemata, and conjectures that his infinite set of axiom schemata for M cannot be replaced by an equivalent finite set of axiom schemata if his rule is kept as the only primitive rule.

Erratum. On page 212 of the first paper, in line 13 from the bottom (of text), for f read -P. NAOTO YONEMITSU

TAKEO SUGIHARA. Strict implication free from implicational paradoxes. Memoirs of the Faculty of Liberal Arts, Fukui University, ser. 1 no. 4 (1965), pp. 55-59.

Halld~n (XIV69) has shown that (1) 0p.D.p-3q and (2) Op.D.fqp are theorems of Lewis's SI (4561, p. 500). (They are also theorems of Vredenduin's N (IV 124), which avoids corresponding theorems with -3 replacing D.) In XIV 199 Halld6n shows that (p Up) -9 q is not a theorem of his system SO, obtained from SI by taking -g rather than ( as primitive, and dropping the definition p -3 q . = . - 0 (Peq) . Halld6n's technique can also be used to show that (1) and (2) are not theorems of SO, where <)p is short for .(p -3 -p).

Sugihara, without reference to N or SO, considers SA (a subsystem of N) which consists of SO with the additional axiom Up -3 q . - .q - p. An infinite non-charac- teristic matrix shows that for no k is Jk(P) D . p -3 q a theorem of SA (where Jk(P) has a designated value if and only if p has the value k, and where p 0 q has an undesignated value if and only if p has a designated value and q has an undesignated value). It is shown also (1) that SA is equivalent to no finite-valued logic, (2) that addition of pq r p r q to SA yields S1, and (3) that addition of pq..r: -: p - . q -3 r to SA turns -3 into material implication.

The results are purely formal; reasons for wishing to avoid the allegedly paradoxical theorems of SI are not discussed. ALAN Ross ANDERSON

BOLESLAW SOBOCI*SKI. Axiomatization of a conjunctive-negative calculus of prop- ositions. The journal of computing systems, vol. 1 no. 4 (1954), pp. 229-242.

The conjunctive-negative system of the.classical propositional calculus was used in Gbdel's salient result 41811, to interpret the classical propositional logic as part of the

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