Quantifiers, substitutional and objectual

Understood substitutionally, ‘Something is F’ is true provided one of its substitution instances (a sentence of the form ‘a is F’) is true. This contrasts with the objectual understanding, on which it is true provided ‘is F’ is true of some object in the domain of the quantifier. Substitutional quantifications have quite different truth-conditions from objectual ones. For instance, ‘Something is a mythological animal’ is true if understood substitutionally, since the substitution instance ‘Pegasus is a mythological animal’ is true. But understood objectually, the sentence is not true, since there are no mythological creatures to make up a domain for the quantifier. Since substitutional quantifiers do not need domains over which they range, it is easy to introduce substitutional quantifiers which bind predicate or sentential variables, even variables within quotation marks. One reason for interest in substitutional quantification is the hope that it may provide a way to understand discourse which appears to be about numbers, properties, propositions and other ‘troublesome’ sorts of entities as being free of exceptional ontological commitments. Whether natural language quantification is sometimes plausibly construed as substitutional is not, however, clear.

Out-Gunning Skepticism

Bredo C. Johnsen1 misconceives my strictures concerning acceptance of the following principle (where ‘p’ stands for any empirical proposition):(1) If A both knows that p and knows that p entails q, then A can come to know that q.Johnsen seems unaware that my criticism was intended to apply only after (1) is made to appear in its most plausible light; that is, only after its consequent is interpreted as: ’It is logically possible for A to know that q.’ Without this interpretation (1) might be dismissed simply on the grounds that A suffers from some physical or psychological disability that prevents him from drawing inferences from what he knows.Properly interpreted, (1) remains acceptable as long as the propositions substituted for p and q are such that it is at least logically possible for A to get evidence enough to make them known. Agreement on this point is itself enough to render Johnsen's own examples irrelevant. For instance, even though it may be physically impossible for A to get adequate evidence that in the constellation Andromeda there is a planet intermediate in size between Venus and Earth, the foregoing is still a fit substitution instance for q; but since such a q does not suffice to falsify the consequent of (1), it does nothing to generate any skeptical argument, either.

Semantic analysis of tense logics

Although we believe the results reported below to have direct philosophical import, we shall for the most part confine our remarks to the realm of mathematics. The reader is referred to [4] for a philosophically oriented discussion, comprehensible to mathematicians, of tense logic.The “minimal” tense logic T0 is the system having connectives ∼, →, F (“at some future time”), and P (“at some past time”); the following axioms:(where G and H abbreviate ∼F∼ and ∼P∼ respectively); and the following rules:(8) from α and α → β, infer β,(9) from α, infer any substitution instance of α,(10) from α, infer Gα,(11) from α, infer Hα.A tense logic is a system T whose language is that of T0 and whose axioms and rules include (1)–(11). The axioms and rules of T other than (1)–(11) are called proper axioms and rules.We shall investigate three systems of semantics for tense logics, i.e. three notions of structure and three relations ⊧ which stand between structures and formulas. One reads ⊧ α as “α is valid in .” A structure is a model of a tense logic T if every formula provable in T is valid in . A semantics is adequate for T if the set of models of T in the semantics is characteristic for T, i.e. if whenever T ∀ α then there is a model of T in the semantics such that ∀ α. Two structures and , possibly from different semantics, are called equivalent ( ∼ ) if exactly the same formulas are valid in as in .

Banishing the rule of substitution for functional variables

Let and be (well-formed) formulas of the functional calculus, let c be an n-adic functional variable, and let a1, …, an be distinct individual variables. Church has defined the metalogical notation to indicate the formula resulting from when each part of of the form c{1, …, n) (such that the occurrence of c is free in ) is replaced by the formula which arises from by replacing every free occurrence of ai by i, i, = 1, …, n. (Here 1, …, n may be any individual variables or constants, not necessarily all distinct.) The notation is not defined for all , , c, a1, …, an, however, but only for those cases (specified in detail by Church) when the resulting formula constitutes a valid substitution instance of the formula according to the standard interpretation of the functional calculus.The full and correct syntactical statement of the conditions (under which this type of substitution is permissible) has proved so arduous, that it seems to have been rendered in error more often than not. Unfortunately the functional calculi are often set up as deductive systems in which this type of substitution occurs as one of the primitive rules of inference, or in one of the axiom schemata. Thus the beginning student who is introduced to the calculi through such a formulation is forced to cope from the outset with details which have proved treacherous even to the initiate. For this reason it is desirable to seek alternative formulations of the functional calculi in which this type of substitution is not mentioned.