Aboutness Paradox

The present work outlines a logical and philosophical conception of propositions in relation to a group of puzzles that arise by quantifying over them: the Russell-Myhill paradox, the Prior-Kaplan paradox, and Prior's Theorem. I begin by motivating an interpretation of Russell-Myhill as depending on aboutness, which constrains the notion of propositional identity. I discuss two formalizations of of the paradox, showing that it does not depend on the syntax of propositional variables. I then extend to propositions a modal predicative response to the paradoxes articulated by an abstraction principle for propositions. On this conception, propositions are “shadows” of the sentences that express them. Modal operators are used to uncover the implicit relation of dependence that characterizes propositions that are about propositions. The benefits of this approach are shown by application to other intensional puzzles. The resulting view is an alternative to the plenitudinous metaphysics of impredicative comprehension principles.

Grounding and propositional identity

I show that standard grounding conditions contradict standard conditions for the identities of propositions.

An Investigation into Intuitionistic Logic with Identity

We define Kripke semantics for propositional intuitionistic logic with Suszko’s identity (ISCI). We propose sequent calculus for ISCI along with cut-elimination theorem. We sketch a constructive interpretation of Suszko’s propositional identity connective.

PURE LOGIC OF ITERATED FULL GROUND

This article develops the Pure Logic of Iterated Full Ground (plifg), a logic of ground that can deal with claims of the form “ϕ grounds that (ψ grounds θ)”—what we call iterated grounding claims. The core idea is that some truths Γ ground a truth ϕ when there is an explanatory argument (of a certain sort) from premisses Γ to conclusion ϕ. By developing a deductive system that distinguishes between explanatory and nonexplanatory arguments we can give introduction rules for operators for factive and nonfactive full ground, as well as for a propositional “identity” connective. Elimination rules are then found by using a proof-theoretic inversion principle.

Propositional Identity and Logical Necessity

In two early papers, Max Cresswell constructed two formal logics of propositional identity, PCR and FCR, which he observed to be respectively deductively equivalent to modal logics S4 and S5. Cresswell argued informally that these equivalences respectively “give … evidence” for the correctness of S4 and S5 as logics of broadly logical necessity. In this paper, I describe weaker propositional identity logics than PCR that accommodate core intuitions about identity and I argue that Cresswell’s informal arguments do not firmly and without epistemic circularity justify accepting S4 or S5. I also describe how to formulate standard modal logics (K, S2, and their extensions) with strict equivalence as the only modal primitive.

Functions of Propositions

In [1] (p. 131 et seq) Professor A. N. Prior suggests a calculus of functions of propositions in which the range of the function variables is not restricted to truth functions.1 If f, g, … etc. represent such variables and we have quantification over all variables we can introduce propositional identity by definition as (ƒ)(ƒp ⊃ ƒq) Alternatively identity may be primitive with the usual axioms and schemata (v.e.g. [4] p. 190 et seq). We shall refer to such systems as ‘functorial calculi’2 (abbreviated as FC).