A New System of Modal Logic by G. H. von WrightReview by: Naoto YonemitsuThe Journal of Symbolic Logic, Vol. 19, No. 1 (Mar., 1954), pp. 66-67Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2267672 .

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66 REVIEWS

symbols. The general concept of derivability of an expression in a given calculus is supposedly "introduced" by giving examples of actual derivations rather than through an explicitly formulated recursive definition in terms of the basic rules. The protological study of a given calculus may show that by virtue of the basic rules, certain additional rules are valid for the calculus in the sense that any step performed by reference to one of them can always be replaced by steps directly governed by the basic rules. Thus, there is obtained a system of meta-rules each of which is to the effect that (for the calculus in question) a specified additional rule R can be "eliminated" by virtue of antecedently established rules R1, R2, ... , Rn . Each of these meta-rules of elimination is construed as an operational rule for a meta-calculus of rules; the latter, in turn, gives rise to a meta-meta-calculus, and so forth. Lorenzen asserts that this operational conception of rules, meta-rules, etc., leads "exactly to those expressions which are known as theorems of the positive, or intuitionist, logic of implication." At this point, protologic goes over into logic in the more customary sense of a theory of the logical constants, which can then be constructed according to operational principles.

The brevity with which the author had to sketch his ideas precludes the possibility of a fair appraisal of their formal potentialities. As for its philosophical justification, however, it would seem that the operational conception avoids the infinite regress attributed to the ontological view only through the problematic and unclear assumption that protologic can be developed, at least in part, in a not explicitly verbalized or symbolized manner, through concrete use of certain rules of construction, and through a kind of intuitive, not explicitly formulated grasp of their scope. CARL G. HEMPEL

G. H. VON WRIGHT. A new system of modal logic. Ibid., pp. 59-63. The paper gives an outline of a new modal system which, instead of the usual

possibility, has relative possibility M(p/q) as a primitive: M(p/q) may be read, "p is possible, given q." The author first states, after a slightly vague argument, that Lewis's )(p&q) (4561) is incapable of expressing a notion of compatibility which is the same as his notion of relative possibility. The system is formalized as follows: We deal only with homogeneous M-expressions of the first order, i.e., (i) expressions of the form M( / ) in which the blanks are filled by propositional variables or molecular complexes of propositional variables, or (ii) molecular complexes of such expressions. Where -, v, &, -*, i-+ are the usual connectives of classical propositional logic PL, and t is an arbitrary tautology of PL, there are four axioms: M(t/t), -M(-t/t), M(p/t) -- M(q/p) v M(-q/p), M(p&q/t) - M(p/t) & M(q/p). Also a rule of substitution: If a +-. b is a theorem of PL, then a and b are intersubstitutable in M-expressions. After two theorems are proved as examples in the calculus, it is asserted without any argument that (valid) theorems of the system will be obtained if we replace all con- stituents of the form 0 (c) by M(c/t) in those theorems of Lewis's SI1-S5, or of Becker's system (XVIII 327), or of the system M of the author's XVIII 174, which are homo- geneous modal expressions of the first order.

Under the title of the concept of entailment, the author, taking up the so-called 'paradoxes" of strict implication, makes ~M(.q/p) denote that p entails q, and states without proof that the two corresponding paradoxes for the notion of relative necessity, -M(p/t) ->. M(-q/p) and -M(-p/t) -+ -M(-p/q), are unprovable in the present system.

Finally, a certain structural similarity is shown between the system and the calculus of probability, and the paper concludes with the suggestion that the system of relative modality can, in various ways, be extended to contain properties of higher order modalities, and these extensions may differ from one another in the same way that various classical systems of absolute modality differ.

Reviewer's note: (i) It must be understood that, in the above formalization of the

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REVIEWS 67

system, PL is implicitly presupposed. (ii) Unprovability of the paradoxes can be shown by the following independence example: M(1/1) = M(O/O) = 1, M(1/0) = M(0/1) = 0, truth-tables of the other connectives being the usual ones and 1 the designated value. (iii) Concerning relative modality, refer also to Vredenduin's system (IV 124(1), XVI 278(2)). Von Wright's system may be regarded as weaker than Vredenduin's in the sense that if, in any theorem of the former system, we replace the constituents of the form M(p/q) by (q -q ip), the resulting formula is a theorem of the latter, but the latter system, in opposition to the former, has further an absolute possibility as a primitive. (iv) In the author's system, some usual properties of entailment can be shown to be unprovable, for instance (a) -M(.q/p) -*M(~ (p/q) (law of contraposition), (b) -M(-r/p&q) -- -M(,fq/p&,), and (c) ~M(.q/p) ->~ iM((q&rY)/P&Y). (For consider an independence example which is like that of (ii) above except that M(I1/0) = 1.) In view of this it would seem to the reviewer natural to add, as an axiom, at least M(p/q) ->~ M(q/p), or M(q v -*p&.y M(q/p), in order to obtain such properties of entailment. Then it is noticed that the first extension is the minimal one which has the law of contraposition as a theorem, and the second, the minimal one which has (c) as a theorem, and that the paradoxes are left unprovable in both extensions (as shown by considering again the independence example of (ii)). On the other hand, in the first extension, which causes us to have the property of commutativity of the author's so-called "compatibility," the structural similarity seems to be no longer preserved between modality and probability. And if we consider a minimal extension (of the original system) which preserves the rules of procedure and in which (b) and (c) are theorems, we find that (a), and the paradoxes, and then M(p&q/t) * M(q/p) can no longer be unprovable. NAOTO YONEMITSU

ANDREA GALIMBERTI. L'analyse linguistique de la representation. Ibid., pp. 139 -145.

The author states that language is "objectified intelligence" and that, because of the development of mathematical logic, intelligence can now be studied "objectively" rather than subjectively and psychologically. The first of these theses is supported by remarks about language, memory, and imagination. RODERICK M. CHISHOLM

PAUL F. LINKE. Die Implikation als echte Wenn-so-Beziehung. Bemerkungen zu den "Fundamental-Paradoxien" der Logistik. Ibid., pp. 146-150.

The author holds that, despite the so-called "paradoxes" of material implication, the ordinary concept of implication can be expressed by means merely of truth-functional connectives. He notes that implications, or conditionals, in their ordinary use are "indifferent" to the truth-value of their antecedents, i.e., that ordinarily, when one asserts an implication, one means to convey that the implication is true no matter whether the antecedent is true or false. He proposes a method of translating ordinary implications into statements which (i) express this "indifference" and which (ii) contain no logical connectives other than those expressing conjunction and negation. But the method he proposes does not seem to satisfy the second of these conditions. Applied to "If Jena is on the Saale, then Jena is in Germany," his method yields "No matter whether the objective, Jena's being on the Saale, is true or false [ob dies objectiv wahr oder falsch ist], it is false that both Jena is on the Saale and Jena is not in Germany." The author makes no note of the fact that, in the formulas thus derived, the expression "no matter whether" ["ob"], or ,,no matter whether ... or functions as a logical connective. RODERICK M. CHISHOLM

J. C. C. MCKINSEY. Systems of modal logic which are not unreasonable in the sense of Hallddn. The journal of symbolic logic, vol. 18 (1953), pp. 109-113.

The paper is based on Halld6n's XVI 273(1), and treats various of the Lewis systems

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