For, whether or not a consistent calculus of modal functions—to be distinguished from modal operators, prefixed to names of propositions—be possible, none of the mentioned paradoxes would arise if the Leibnizian definition were restricted to first-order functions in the sense of the ramified theory of types. But, A universe which is just like ours except that it does not contain language, and thus contains no sentences, is surely logically possible. Positivism pursued two closely related projects. This definition, however, breaks down under the weight of three objections: a an expression which occurs essentially in one argument may occur inessentially in another argument. The interpretation that I have been pursuing identifies attributes with quantifiers over individuals. Physicalism as an Analytic Thesis.
Reduction Sentences and Open Concepts 1953. With regard to a contingent proposition it cannot be logically impossible that a given sentence expresses it in a given language and yet that sentence is false; in fact, if p is false, then any sentence designating it in L will be false in L. Logic and the Synthetic A Priori 1949. The Analytic-Synthetic Distinction: Dispositional and Open Concepts. But, as Pap formulates the problem, given the definition of numbers as classes of similar sets, if only n individuals existed, then the number n + 1, being defined as the class of all classes that have n + 1 members or that would have exactly n members if exactly one element were withdrawn from them , would be equal to the null class; for in that case no classes with n+1 members would exist. One cannot deduce consequences from one singular postulate or hypothesis; there is, for the possibility of deduction, required a set of postulates or hypotheses.
To my mind, this is true as far as it goes, but its emphasis is not quite right, nor does it go far enough. Thus we have proved that no unicorns dislike cake. The cardinality operator is governed by axioms based on the intuitive criterion of identity for numbers discussed above: the number of Fs is identical to the number of Gs just in case there is a 1-1 correspondence between the Fs and the Gs. Classes, Attributes, and the Logical Analysis of Mathematics. Pap's work played an important role in the development of the analytic tradition.
But I cannot, then, assuming the absolute necessity of a, b, c go on to ask whether D is a fact or even a possibility: for it is only on the hypothesis that D is a fact that a, b, c were declared as necessities. He gave definitions of the numeral 0, the relation of immediate successor, and the predicate of being a number, all in terms of classes and predicates and relations of classes. Disposition Concepts and Extensional Logic 1958. It follows, then, that the contingency of the length of S can be expressed using S itself. The Linguistic Hierarchy and the Vicious-Circle Principle 1954. It may be described as a confusion of material and logical equivalence.
Why should that definition have existential import? At least, one might suggest, the universal conditional of the semantic meta-language has inductive character if the proposition of which truth is predicated is contingent. But then whatever knowledge one might express by the statement is not arrived at by reflection on meanings alone. This is true enough; but it is by no means conflicting to call a judgment both functionally and materially a priori. The axioms of a system of pure geometry, for example, implicitly define the geometrical primitives in the sense of specifying the structural meanings they have within the deductive system—not to be confused with their material meanings in empirical applications of the system. Subscripted variables, xn , yn , zn range over entities of type n.
In particular, the principle does not require that verification is by sensory perception. Universals in Re and the Resemblance Theory. Mathematics, Abstract Entities, and Modern Semantics 1957. Semantic Examination of Realism 1947. First, in the 1950s, W. To be sure, as long as. They are the following: Necessity: sections 2-7 The Analytic-Synthetic Distinction: sections 4 and 5 Dispositional and Open Concepts in Science: section 5 Logical Consequence and Material Entailment: section 7 Logicism: section 10 The Semantic Concept of Truth: section 8.
Instead, propositional identity should be relativized to specified kinds of substitution-contexts. In general, there is no contradiction in conceiving that, by using a sentence, one could make a true description of conditions in Introduction 29 which there are no sentences or users of them. This quantity collects essentially the most major papers of Arthur Pap. Surely, if p is contingent, then it is logically possible that a sentence designating p in some language should be false in that language. Logical Consequence and Material Entailment The laws of logic, for Pap, are not simply a set of statements of determinate syntactic forms. His mistake lies only in this, that he regarded Euclidean geometry as the only interpretation of formal geometry that is of any use to physics. This difficulty cannot be avoided either by stipulating that the formalization should be as abstract or generic as possible.
If we define, then, the formal a priori as characteristic of implications rather than of categorical statements, we must explicitly rule out material, extensionally defined implications. Indeed, if we appeal to any evidence at all for laws of logic, we appeal only to intuitive evidence: we try, and fail, to conceive of any object not being identical to itself. And if we argued that what makes these principles necessary is the fact that they are demonstrable with the help of adequate definitions of the logical constants involved, our argument would evidently be circular. If the basic postulates of physics have methodological value and can be profitably taken as a priori in the functional sense, it is because they have fundamentum in re. In that case the definiens will be true of a given proposition, provided there are no sentences designating that proposition in some language. Or, we might say that such terms are not logical constants, just because they are not definable in terms of our primitive logical constants, and that therefore statements which contain them essentially are synthetic. Perhaps the most well-known thesis of positivism, the verificationist theory of meaning, subserved both of these projects.
Whereas the sciences seem to display steady if not uninterrupted progress and continued general agreement on results, philosophy seems mired in endless irresolvable disputes. Suppose that P1 and Q1 are the properties for which this existential statement holds. Quine has shown that unrestricted substitutivity of identity also leads to paradox if modal functions are admitted: it used to be thought that the morning star is identical with the evening star, but this was to overlook that the morning star is necessarily identical with itself whereas it is not necessarily identical with the evening star. The principles of logic themselves, which we just saw to be essentially involved in the definition of the formal a priori, are materially a priori. Are Statements about Other Minds Conclusively Verifiable? But what guarantee, now, do we have that any new logical constant which we might discover in the process of analyzing arguments would be definable in terms of our logical primitives? Catalogue record for this book is available from the Library of Congress.