Ahti Pietarinen

Peirce's linear versions of the language of his Existential Graphs (EGs), presented in 1902, are examined. Differences between linear and non-linear languages are explained by permutational invariance and type- vs. occurrence-referentiality: Standard EGs are permutationally invariant with respect to linear EGs, while the Beta part of the system, which corresponds to first-order quantificational theory with identity, is occurrence-referential. However, occurrence-referentiality of Beta graphs constitutes a defect of expressivity: Since the meaning of a quantifier is inextricably connected to the meaning of the sign of identity, certain complex assertions cannot be expressed in the language of Beta graphs without a new extension of its standard notation. © 2021 Elsevier B.V., All rights reserved.

The problem of intentional identity, as originally offered by Peter Geach, says that there can be an anaphoric link between an indefinite term and a pronoun across a sentential boundary and across propositional attitude contexts, where the actual existence of an individual for the indefinite term is not presupposed. In this paper, a semantic resolution to this elusive puzzle is suggested, based on a new quantified intensional logic and game-theoretic semantics (GTS) of imperfect information. This constellation leads to an expressive intensional language with a property of informational independence, argued to produce a purely semantic explication to intentional identity statements. One consequence is that various extra-logical and pragmatic factors become of secondary concern; it is possible to solve the puzzle by logico-semantic methods, albeit somewhat radically renewed ones.

Peirce and Frege both distinguished between the propositional content of an assertion and the assertion of a propositional content, but with different notational means. We present a modification of Peirce’s graphical method of logic that can be used to reason about assertions in a manner similar to Peirce’s original method. We propose a new system of Assertive Graphs, which unlike the tradition that follows Frege involves no ad hoc sign of assertion. We show that axioms of intuitionistic logic can be derived from AGs, and argue that AGs analyse and represent assertions and illocutionary content in a way which is motivated both by its logical properties and its historical connection with the ideas that led to the development of the graphical method.

A graphical approach to intuitionistic propositional logic is presented. The system GrIn is a deep inference system and it is formulated in terms of Peirce’s existential graphs. GrIn is shown to be sound and complete with respect to the class of all Heyting algebras. Moreover, the system GrIn is shown to be equivalent to the Gentzen sequent calculus G3ip via translations between intuitionistic graphs and intuitionistic formulas. We also discuss the intuitionistic interpretation of logical constants proposed by this graphical approach. © 2019 Elsevier B.V., All rights reserved.

We study partiality in propositional logics containing formulas with either undefined or over-defined truth-values. Undefined values are created by adding a four-place connective W termed transjunction to complete models which, together with the usual Boolean connectives is shown to be functionally complete for all partial functions. Transjunction is seen to be motivated from a game-theoretic perspective, emerging from a two-stage extensive form semantic game of imperfect information between two players. This game-theoretic approach yields an interpretation where partiality is generated as a property of non-determinacy of games. Over-defined values are produced by adding a weak, contradictory negation or, alternatively, by relaxing the assumption that games are strictly competitive. In general, particular forms of extensive imperfect information games give rise to a generalised propositional logic where various forms of informational dependencies and independencies of connectives can be studied

“Intuitive time,” Gödel says, “is what everyone understood by time before relativity theory.” Such an understanding includes the perception or experience of objective lapse of time, presupposing a layer of successive “nows,” and a change in the existing. The notion of temporality that Gödel came up with in his work on the general theory of relativity has, in contrast, often been taken to demonstrate the impossibility of such a theory to account for the intuitive notion of time, and even be inconsistent with it. So instead, what Gödel suggested is temporal idealism, that intuitive time is an illusion.

This book contains three Henriette Hertz Lectures delivered as the annual Philosophical Lectures for the British Academy, covering three genres of philosophical logic. In the first paper, James Higginbotham explores arguments for and against second- and higher-order logic as taken to be motivated by the properties of natural language. R. M. Sainsbury investigates the consequences of treating constraints on reporting speech as guides to meaning, with special reference to indexicals in natural language. In the third paper, Timothy Williamson analyzes an aspect of intensionality, namely that sometimes two coextensive sentential operators do not satisfy the same set of principles, a question important to the iteration of attitudes. Each of these papers is followed by an extensive commentary by a distinguished scholar.