Hugues Leblanc

Alternatives to Standard first-order Semantics

Alternatives to standard semantics are legion, some even antedating standard semantics. I shall study several here, among them: substitutional semantics, truth-value semantics, and probabilistic semantics. All three interpret the quantifiers substitutionally, i.e. all three rate a universal (an existential) quantification true if, and only if, every one (at least one) of its substitution instances is true.2 As a result, the first, which retains models, retains only those which are to be called Henkin models. The other two dispense with models entirely, truth-value semantics using instead truth-value assignments (or equivalents thereof to be called truth-value functions) and probabilistic semantics using probability functions. So reference, central to standard semantics, is no concern at all of truth-value and probabilistic semantics; and truth, also central to standard semantics, is but a marginal concern of probabilistic semantics.

Probability functions and their assumption sets — the binary case

It is argued in Leblanc 1983b that statements are accorded probabilities in light of assumptions — or, as mathematicians often put it, under conditions.1 It is further argued that each singulary probability function in Kolmogorov 1933 or, equivalently, Popper 1955 comes with (a set of) assumptions, to wit, those statements evaluating to 1 under P. With P a singulary probability function of the Kolmogorov-Popper sort and A a statement from a certain language L, Leblanc thus takes P(A) to be the probability that P accords to A in light of the assumptions in {A : P(A) = 1}. His rationale for interpreting P(A) in this manner is two-fold. He first contends that any assumption set in light of which a rational agent would accord probabilities must be deductively closed and, for convenience’s sake, may be presumed consistent as well. He then establishes that a set S of statements of L is consistent and deductively closed if and only if there is a singulary probability function P for L such that S = {A : P(A) = 1}. The second result is called by Leblanc The Fundamental Theorem on Assumption Sets, Case One.

Truth-Functionality and the Ramified Theory of Types

Publisher Summary This chapter discusses the truth-functionality and the ramified theory of types. Calculus C has a truth-functional interpretation if there is a family ∑ of functions from the wffs (non-atomic as well as atomic) of C to {T, F}. The sentential calculus (SC) has a strictly truth-functional interpretation; so does the first-order quantificational calculus (QC 1 ). By contrast, the principal interpretation of QC 2 (the second-order quantificational calculus), though truth-functional, is not strictly truth-functional.

On Meyer and Lambert's Quantificational Calculus FQ

The semantical account that Meyer and Lambert give in [7] of their quantificational calculus FQ can be considerably simplified, and—supposing as the authors do at the close of their paper that ‘ = ’ counts as a primitive sign—so can their axiom system for FQ. (1) As the authors remark, axiom schema 102 can be simplified to read : A ⊃ (∀Χ)A , where Χ does not occur free in A . Following Tarskis [9], it can be further simplified: A ⊃ (∀Χ)A , where Χ does not occur in A .

The Autonomy of Probability Theory (Notes on Kolmogorov, Rényi, and Popper)

Kolmogorovs account in his [1933] of an absolute probability space presupposes given a Boolean algebra, and so does Rényis account in his [1955] and [1964] of a relative probability space. Anxious to prove probability theory ‘autonomous’. Popper supplied in his [1955] and [1957] accounts of probability spaces of which Boolean algebras are not and [1957] accounts of probability spaces of which fields are not prerequisites but byproducts instead.1 I review the accounts in question, showing how Poppers issue from and how they differ from Kolmogorovs and Rényis, and I examine on closing Poppers notion of ‘autonomous independence’. So as not to interrupt the exposition, I allow myself in the main text but a few proofs, relegating others to the Appendix and indicating as I go along where in the literature the rest can be found.

Free Intuitionistic Logic: A Formal Sketch

The inference rule If ⊢

Subformula Theorems for

If \vdash A(t),{\text{ then }} \vdash (\exists x)A(x)

-Sequents

(1) and its dual