Peter Milne

Of his numerous investigations ... Tarski was most proud of two: his work on truth and his design of an algorithm in 1930 to decide the truth or falsity of any sentence of the elementary theory of the high school Euclidean geometry. [...] His mathematical treatment of the semantics of languages and the concept of truth has had revolutionary consequences for mathematics, linguistics, and philosophy, and Tarski is widely thought of as the man who "defined truth". The seeming simplicity of his famous example that the sentence "Snow is white" is true just in case snow is white belies the depth and complexity of the consequences which can be drawn from the possibility of giving a general treatment of the concept of truth in formal mathematical languages in a rigorous mathematical way. (J.W. Addison).

While there is now considerable experimental evidence that, on the one hand, participants assign to the indicative conditional as probability the conditional probability of consequent given antecedent and, on the other, they assign to the indicative conditional the “defective truth-table” in which a conditional with false antecedent is deemed neither true nor false, these findings do not in themselves establish which multi-premise inferences involving conditionals participants endorse. A natural extension of the truth-table semantics pronounces as valid numerous inference patterns that do seem to be part of ordinary usage. However, coupled with something the probability account gives us—namely that when conditional-free ϕ entails conditional-free ψ, “if ϕ then ψ” is a trivial, uninformative truth—we have enough logic to derive the paradoxes of material implication. It thus becomes a matter of some urgency to determine which inference patterns involving indicative conditionals participants do endorse. Only thus will we be able to arrive at a realistic, systematic semantics for the indicative conditional.

Various natural deduction formulations of classical, minimal, intuitionist, and intermediate propositional and first-order logics are presented and investigated with respect to satisfaction of the separation and subformula properties. The technique employed is, for the most part, semantic, based on general versions of the Lindenbaum and Lindenbaum–Henkin constructions. Careful attention is paid to which properties of theories result in the presence of which rules of inference, and to restrictions on the sets of formulas to which the rules may be employed, restrictions determined by the formulas occurring as premises and conclusion of the invalid inference for which a counterexample is to be constructed. We obtain an elegant formulation of classical propositional logic with the subformula property and a singularly inelegant formulation of classical first-order logic with the subformula property, the latter, unfortunately, not a product of the strategy otherwise used throughout the article. Along the way, we arrive at an optimal strengthening of the subformula results for classical first-order logic obtained as consequences of normalization theorems by Dag Prawitz and Gunnar Stålmarck.

hese notes don’t reach any conclusions. Their purpose is to point to issues one needs to think through seriously when thinking about logic teaching. They indicate some of the relevant literature where some of these issues are addressed, but they also raise points that seem to have been overlooked. They aim to promote informed discussion. That indeed was their origin: they are descended from an internal discussion document prepared a few years ago when the then Department of Philosophy at the University of Edinburgh was reviewing its logic teaching.

As Wilfrid Hodges has observed, there is no mention of the notion truth-in-a-model in Tarski's article 'The Concept of Truth in Formalized Languages'; nor does truth make many appearances in his papers on model theory from the early 1950s. In later papers from the same decade, however, this reticence is cast aside. Why should Tarski, who defined truth for formalized languages and pretty much founded model theory, have been so reluctant to speak of truth in a model? What might explain the change in his practice? The answers, I believe, lie in Tarski's views on truth simpliciter

Starting from John MacFarlane's recent survey of answers to the question ‘What is assertion?’, I defend an account of assertion that draws on elements of MacFarlane's and Robert Brandom's commitment accounts, Timothy Williamson's knowledge norm account, and my own previous work on the normative status of logic. I defend the knowledge norm from recent attacks. Indicative conditionals, however, pose a problem when read along the lines of Ernest Adams' account, an account supported by much work in the psychology of reasoning. Furthermore, there seems to be no place for degrees of belief in the accounts of belief and assertion given here. Degrees of belief do have a role in decision‐making, but, again, there is much evidence that the orthodox theory of subjective utility maximization is not a good description of what we do in decision‐making and, arguably, neither is it a good normative guide to how we ought to make decisions

From the point of view of proof-theoretic semantics, we examine the logical background invoked by Neil Tennant's abstractionist realist account of mathematical existence. To prepare the way, we must first look closely at the rule of existential elimination familiar from classical and intuitionist logics and at rules governing identity. We then examine how well free logics meet the harmony and uniqueness constraints familiar from the proof-theoretic semantics project. Tennant assigns a special role to atomic formulas containing singular terms. This, we find, secures harmony and uniqueness but militates against the putative realism.

Intervals in boolean algebras enter into the study of conditional assertions (or events) in two ways: directly, either from intuitive arguments or from Goodman, Nguyen and Walker's representation theorem, as suitable mathematical entities to bear conditional probabilities, or indirectly, via a representation theorem for the family of algebras associated with de Finetti's three-valued logic of conditional assertions/events. Further representation theorems forge a connection with rough sets. The representation theorems and an equivalent of the boolean prime ideal theorem yield an algebraic completeness theorem for the three-valued logic. This in turn leads to a Henkin-style completeness theorem. Adequacy with respect to a family of Kripke models for de Finetti's logic, Łukasiewicz's three-valued logic and Priest's Logic of Paradox is demonstrated. The extension to first-order yields a short proof of adequacy for Körner's logic of inexact predicates

Our starting point is Michael Luntley's falsificationist semantics for the logical connectives and quantifiers: the details of his account are criticised but we provide an alternative falsificationist semantics that yields intuitionist logic, as Luntley surmises such a semantics ought. Next an account of the logical connectives and quantifiers that combines verificationist and falsificationist perspectives is proposed and evaluated. While the logic is again intuitionist there is, somewhat surprisingly, an unavoidable asymmetry between the verification and falsification conditions for negation, the conditional, and the universal quantifier. Lastly we are lead to a novel characterization of realism.