Vague Terms by Tadeusz KubińskiReview by: Czeslaw LejewskiThe Journal of Symbolic Logic, Vol. 24, No. 3 (Sep., 1959), pp. 270-271Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2963884 .

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270 REVIEWS

foundation of mathematics. Ontological commitments are said, in accordance with Quine XV 152, to be tied to what ranges of bound variables are accepted.

The article also contains a section about logical truth. The idea of vacuous variants developed by Quine in I 42 is taken as a starting point.

Reviewer's comments: It seems better to translate "there are at least two different things" (p. 70) as "(3x)(3y)(x # y)" than as "(x)(3y)(x # y)". The partial reformu- lation of the sentence, "The discoverer of the North Pole was a Norwegian" (p. 74), as "Some one discovered the North Pole and no one else discovered the North Pole," given as an example of the analysis of definite descriptions by Russell, may mislead the reader, as it gives the impression that the definite-description part of the original sentence is translatable to an expression taking truth-values. ERIK G6TLIND

TADEUSZ KUBIN'SKI. Nazwy nieostre (Vague terms). Polish, with English and Russian summaries. Studia logica, vol. 7 (1958), pp. 115-179.

Two-noun-expressions are contradictory in a universe U if and only if the sum of their respective denotations, which are disjoint, is included in and exhausts U. Two noun-expressions are contrary in a universe U if and only if the sum of their respective denotations, which are disjoint, is included in U. It is generally accepted that the functor of negation for noun-expressions forms the contradictory of a noun-expression. Now, the author attaches a weaker sense to the functor of negation by postulating that it should form the contrary of a given noun-expression. Thus his negation aims at accommodating not only the meaning of the usual negation for noun-expressions but also the meaning of the prefix in such expressions as 'unjust' or 'immoral.' Given a noun-expression a and a universe U we can distinguish within U: (i) the positive denotation of a, i.e., the class of objects designated by a, (ii) the negative denotation of a, i.e., the class of objects designated by the contrary of a, and (iii) the fringe of a, i.e., the class of objects designated neither by a nor by its contrary. A noun-expression a is vague in a language J if and only if the fringe of a is not empty. The author assumes that the problem of determining the fringe for every noun-expression be- longing to a given vocabulary has been solved, and on this assumption he proceeds to characterise deductive systems which allow for the use of vague noun-expressions. He calls these systems quasi-ontologies because each of them contains System Q., which is a fragment of Le'niewski's Ontology. System fl is based on the following axioms and definitions, which are given here in the author's symbolism:

Al. x9[EXy _z(ezx) A v W(SvxA eWx -ew) A il(EUX -UY)].

A2. xA[exy - (exNy)].

Dl. x9z(exAyz exy v exz).

D2. xz(ESxKyz _Xy A EXZ).

A3. x'z[(exNAyz exNy A exNz) A (exNKyz _ exNy v exNz) A (exNNy _= xy)].

If we added to the system thesis A2. 1, which says that x[Sxx A (exNy)' -Ey], then A2 and A2. 1 between them would define 'N' as the ordinary negation for noun- expressions, A3 would be deducible as a theorem, and, given appropriate rules of inference, System n would coincide with Lesniewski's Ontology. As it is, the axioms of Q turn out to be independent. A quasi-ontology is obtained by subjoining to D a number of "axioms of individual existence" and a number of "empirical theses." A considerable part of the paper is devoted to the analysis of quasi-ontologies and to the classification of vague noun-expressions.

As far as the reviewer can judge, the most interesting thing about System Q is the fact that without contradiction we can extend it by adding to it theses of the

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REVIEWS 271

form 'eaa A (eab)' A (eaNb)", which in the author's opinion have to be admitted if vague noun-expressions are to be part of our vocabulary.

CZESLAW LEJEWSKI

ZBIGNIEW CZERWUISKI. 0 paradoksie implikacii (On the paradox of implication). Ibid., vol. 7 (1958), pp. 265-271.

The author takes up the topics discussed in Adjukiewicz XXII 407. Contrary to

Ajdukiewicz's thesis that 'p D q' states the same as 'if p then q' although the two

need not express the same things, CzerwiAski argues that in ordinary usage false

implications of the form 'if p then q' can be found whose counterparts of the form

'p D q' are considered to be true. The implication 'If Copernicus had a son then Coper- nicus was not a father' is offered as an example. It is contended that in harmony

with the sense ordinarily attached to the connective 'if ... then' the above implication

is false whereas, in view of the fact that Copernicus had no sons, it turns out to be

true provided we interpret the connective in accordance with the usual truth-table.

The author goes on to suggest that the connective 'if ... then' as used in ordinary

language gives rise to the following rule: a proposition of the form 'if /(a) then g(a)'

is false if the corresponding proposition of the form 'llx(/(x) j -g(x))' is true. He

further suggests that an implication 'if p then q' is true if and only if it is the counter-

part of the result of a substitution in a true formal implication of the form

'llx(/(x) D g(x))'. Implications which, in this sense, are not particular cases of formal

implications, can, in the author's view, be interpreted as material implications.

The reviewer, who rejects the dogma of the infallibility of ordinary usage, is tempted

to remark that indeed the true implication 'If Copernicus had a son then Copernicus

was not a father' is sometimes mistakenly regarded to be false because mistakenly it is thought to be the contradictory of the true implication 'If Copernicus had a son

then Copernicus was a father,' whose correct contradictory is the false conjunction

'Copernicus had a son and Copernicus was a father.' CZEESLAW LEJEWSKI

J. J. C. SMART. Incompatible colors. Philosophical studies (Minneapolis), vol. 10 (1959), pp. 39-42.

Smart accepts, at first, the argument of H. Putnam (XXII 318), and claims that

Goodman's objection to it (XXII 318) is unfounded. According to Smart, Goodman's objection rests on the suggestion that the relation "exactly the same colour as" is

not transitive; but, in fact, Goodman makes no use of this suggestion in his criticism.

Smart assumes, with Putnam, that the question whether an object can be both red

and green all over is settled by the question whether the relation "exactly the same

colour as" is transitive. This is a dubious assumption: the former question depends upon the latter only if all objects which are red are exactly the same colour, and all

objects which are green are exactly the same colour. So far from this being so, it is

not clear that all objects which are green are even the same colour. Grass and gorgon-

zola are both green but they are not the same colour. And this is what is to be expected in comparing a green-and-red object with a green-but-not-red object: they would

both be green but they would not be the same colour. Putnam argues in fact only that

all objects of a specific shade of green must be exactly the same colour, and that they cannot therefore be exactly the same colour as an object of a specific shade of blue.

But this begs the question. at issue, for an object might be green and blue all over just

because there existed a shade of green that was also a shade of blue. Smart goes on to question Putnam's position by giving an example of a type of

colour vision rather different from that of a human being. He supposes a robot that

can separately distinguish each wavelength of a beam of light of mixed wavelength.

Suppose the robot can distinguish three wavelengths, L1, L2, and L3. A surface reflecting L1 could also reflect L2, and a surface reflecting L2 could also reflect L3,

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