Peter Milne

Taking as starting point two familiar interpretations of probability, we develop these in a perhaps unfamiliar way to arrive ultimately at an improbable claim concerning the proper axiomatization of probability theory: the domain of definition of a point-valued probability distribution is an orthomodular partially ordered set. Similar claims have been made in the light of quantum mechanics but here the motivation is intrinsically probabilistic. This being so the main task is to investigate what light, if any, this sheds on quantum mechanics. In particular it is important to know under what conditions these point-valued distributions can be thought of as derived from distribution-pairs of upper and lower probabilities on boolean algebras. Generalising known results this investigation unsurprisingly proves unrewarding. In the light of this failure the next topic investigated is how these generalized probability distributions are to be interpreted.

This article begins by outlining some of the history—beginning with brief remarks of Quine's—of work on conditional assertions and conditional events. The upshot of the historical narrative is that diverse works from various starting points have circled around a nexus of ideas without convincingly tying them together. Section 3 shows how ideas contained in a neglected article of de Finetti's lead to a unified treatment of the topics based on the identification of conditional events as the objects of conditional bets. The penultimate section explores some of the consequences of the resulting logic of conditional events while the last defends it.

Some propositions add more information to bodies of propositions than do others. We start with intuitive considerations on qualitative comparisons of information added . Central to these are considerations bearing on conjunctions and on negations. We find that we can discern two distinct, incompatible, notions of information added. From the comparative notions we pass to quantitative measurement of information added. In this we borrow heavily from the literature on quantitative representations of qualitative, comparative conditional probability. We look at two ways to obtain a quantitative conception of information added. One, the most direct, mirrors Bernard Koopman’s construction of conditional probability: by making a strong structural assumption, it leads to a measure that is, transparently, some function of a function P which is, formally, an assignment of conditional probability (in fact, a Popper function). P reverses the information added order and mislocates the natural zero of the scale so some transformation of this scale is needed but the derivation of P falls out so readily that no particular transformation suggests itself. The Cox–Good–Aczél method assumes the existence of a quantitative measure matching the qualitative relation, and builds on the structural constraints to obtain a measure of information that can be rescaled as, formally, an assignment of conditional probability. A classical result of Cantor’s, subsequently strengthened by Debreu, goes some way towards justifying the assumption of the existence of a quantitative scale. What the two approaches give us is a pointer towards a novel interpretation of probability as a rescaling of a measure of information added

In making assertions one takes on commitments to the consistency of what one asserts and to the logical consequences of what one asserts. Although there is no quick link between belief and assertion, the dialectical requirements on assertion feed back into normative constraints on those beliefs that constitute one's evidence. But if we are not certain of many of our beliefs and that uncertainty is modelled in terms of probabilities, then there is at least prima facie incoherence between the normative constraints on belief and the probability-like structure of degrees of belief. I suggest that the norm-governed practice relating to degrees of belief is the evaluation of betting odds.

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

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.

Plausibly, when we adopt a probabilistic standpoint any measure Cb of the degree to which evidence e confirms hypothesis h relative to background knowledge b should meet these five desiderata: Cb > 0 when P > P < 0 when P < P; Cb = 0 when P = P. Cb is some function of the values P and P assume on the at most sixteen truth-functional combinations of e and h. If P < P and P = P then Cb ≤ Cb; if P = P and P < P then Cb ≥ Cb. Cb – Cb is fully determined by Cb and Cbe – Cbe; if Cb = 0 then Cb + Cbe = 0. If P = P then Cb = Cb.

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).

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.

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

A conception of probability as an irreducible feature of the physical world is outlined. Propensity analyses of probability are examined and rejected as both formally and conceptually inadequate. It is argued that probability is a non-dispositional property of trial-types; probabilities are attributed to outcomes as event-types. Brier's Rule in an objectivist guise is used to forge a connection between physical and subjective probabilities. In the light of this connection there are grounds for supposing physical probability to obey some standard set of axioms. However, there is no a priori reason why this should be the case.

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.