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Ian Rumfitt has proposed systems of bilateral logic for primitive speech acts of assertion and denial, with the purpose of ‘exploring the possibility of specifying the classically intended senses for the connectives in terms of their deductive use’ (Rumfitt Mind109, 781–823 (4): 810f). Rumfitt formalises two systems of bilateral logic and gives two arguments for their classical nature. I assess both arguments and conclude that only one system satisfies the meaning-theoretical requirements Rumfitt imposes in his arguments. I then formalise an intuitionist system of bilateral logic which also meets those requirements. Thus Rumfitt cannot claim that only classical bilateral rules of inference succeed in imparting a coherent sense onto the connectives. My system can be extended to classical logic by adding the intuitionistically unacceptable half of a structural rule Rumfitt uses to codify the relation between assertion and denial. Thus there is a clear sense in which, in the bilateral framework, the difference between classicism and intuitionism is not one of the rules of inference governing negation, but rather one of the relation between assertion and denial.

We show that the implicational fragment of intuitionism is the weakest logic with a non-trivial probabilistic semantics which satisfies the thesis that the probabilities of conditionals are conditional probabilities. We also show that several logics between intuitionism and classical logic also admit non-trivial probability functions which satisfy that thesis. On the other hand, we also prove that very weak assumptions concerning negation added to the core probability conditions with the restriction that probabilities of conditionals are conditional probabilities are sufficient to trivialize the semantics.

We distinguish three different readings of the intuitionistic notions of validity, soundness, and completeness with respect to the quantification occurring in the notion of validity, and we establish certain relations between the different readings. For each of the meta-logical notions considered we suggest that the “most natural” reading (which is not the same for all cases) is precisely the one which is required by the recent intuitionistic completeness theorems for IPC.

A version of intuitionistic type theory is presented here in which all logical symbols are defined in terms of equality. This language is used to construct the so-called free topos with natural number object. It is argued that the free topos may be regarded as the universe of mathematics from an intuitionist's point of view.

At the joint APA-ASL meeting in December 1981, there was a symposium on the foundations of intuitionism and this volume is based on the papers presented there. Dana Scott, William Tait and Scott Weinstein were the co-symn-posiasts; Solomon Feferman acted as chair. Because of their bearing on the issues raised at the symposium, two other papers, one by A.S. Troelstra and the other by Giiran Sundholm, have also been included.

Les AA. soutiennent que les mathematiques intuitionnistes ne donnent pas lieu au paradoxe de Skolem. Ils soutiennent de plus que les preuves connues du theoreme de Lowenheim-Skolem sont inacceptables d'un point de vue constructiviste. Ils en tirent les consequences pour une philosophie des mathematiques intuitionniste

The system C of Pottinger (1974) has as primitive logical constants = (entailment), -9 (S4 strict implication), -, (relevant implication), D (intuitionist implication), & (intuitionist conjunction), v (intuitionist disjunction), and A (absurdity). In Section 8 of Pottinger (1974) it was pointed out that the addition of either 2-A A or 2B 2A .A B2 to C collapses -* into material implication and that the addition of either 21A * A or