Beyond first-order logic: the historical interplay between mathematical logic and axiomatic set theory

What has been the historical relationship between set theory and logic? On the óne hand, Zermelo and other mathematicians developed set theory as a Hilbert-style axiomatic system. On the other hand, set theory influenced logic by suggesting to Schröder, Löwenheim and others the use of infinitely long expressions. The question of which logic was appropriate for set theory — first-order logic, second-order logic, or an infinitary logic — culminated in a vigorous exchange between Zermelo and Gödel around 1930.     

Bertrand Russell and the introduction of mathematical logic in China

Bertrand Russell was one of the most prominent figures in the formation and development of mathematical logic. It is widely acknowledged that his work in this field exerted tremendous influence in the West, especially in the first three decades of the twentieth century. The important role he played in inspiring Chinese interest in this subject, however, is virtually unknown. This paper describes Russell's contributions to the introduction of mathematical logic in China through a discussion of his lectures in Beijing in March of 1921, and the subsequent Chinese translations of his Introduction to Mathematical Philosophy and Principia Mathematica.     

The Fregean Axiom and Polish mathematical logic in the 1920s

Summary of the talk given to the 22^nd Conference on the History of Logic, Cracow (Poland), July 5–9, 1976.Man, so far as I know, is the only animal capable of lying to himself. ROBERT ARDREYAllatum est 1 Junii 1976     

A logic for deontic dilemmas

The possibility of deontic dilemmas poses a significant problem for deontic logic. Here I review some proposals to resolve this problem, and then offer a new account. This is a simple modification of standard deontic logic that enables the system to accommodate deontic dilemmas without inconsistency and without deontic explosion, while at the same time accounting for the range of genuinely valid inferences.     

A conditional logic for abduction

We propose a logic of abduction that (i) provides an appropriate formalization of the explanatory conditional, and that (ii) captures the defeasible nature of abductive inference. For (i), we argue that explanatory conditionals are non-classical, and rely on Brian Chellas’s work on conditional logics for providing an alternative formalization of the explanatory conditional. For (ii), we make use of the adaptive logics framework for modeling defeasible reasoning. We show how our proposal allows for a more natural reading of explanatory relations, and how it overcomes problems faced by other systems in the literature.     

A tense logic for boethius

An interpretation in modal and tense logic is proposed for Boethius's reconciliation of God's foreknowledge with human freedom from The consolation of philosophy, Book V. The interpretation incorporates a suggestion by Paul Spade that God's special status in time be explained as a restriction of God's knowledge to eternal sentences. The argument proves valid, and the seeming restriction on omnipotence is mitigated by the very strong expressive power of eternal sentences.     

A topological logic of action

We consider a quantifier-free language in which there are terms as well as formulas. The proposition-forming propositional operators are the usual ones, and the term-making term operators are the usual lattice theoretical ones. In addition there is a formula-making term operator, does. We study a new logic in which does is claimed to approximate some features of the informal concept the agent performs the action .     

A deontic logic of action

The formal language studied in this paper contains two categories of expressions, terms and formulas. Terms express events, formulas propositions. There are infinitely many atomic terms and complex terms are made up by Boolean operations. Where and are terms the atomic formulas have the form = ( is the same as ), Forb ( is forbidden) and Perm ( is permitted). The formulae are truth functional combinations of these. An algebraic and a model theoretic account of validity are given and an axiomatic system is provided for which they are characteristic.The closure principle, that what is not forbidden is permitted is shown to hold at the level of outcomes but not at the level of events. In the two final sections some other operators are considered and a semantics in terms of action games.