Free Logic

This dissertation develops an inferentialist theory of meaning. It takes as a starting point that the sense of a sentence is determined by the rules governing its use. In particular, there are two features of the use of a sentence that jointly determine its sense, the conditions under which it is coherent to assert that sentence and the conditions under which it is coherent to deny that sentence. From this starting point the dissertation develops a theory of quantification as marking (...) coherent ways a language can be expanded and modality as the means by which we can reflect on the norms governing the assertion and denial conditions of our language. If the view of quantification that is argued for is correct, then there is no tension between second-order quantification and nominalism. In particular, the ontological commitments one can incur through the use of a quantifier depend wholly on the ontological commitments one can incur through the use of atomic sentences. The dissertation concludes by applying the developed theory of meaning to the metaphysical issue of necessitism and contingentism. Two objections to a logic of contingentism are raised and addressed. The resulting logic is shown to meet all the requirement that the dissertation lays out for a theory of meaning for quantifiers and modal operators. (shrink)

This paper presents a way of formalising definite descriptions with a binary quantifier ι, where ιx[F, G] is read as ‘The F is G’. Introduction and elimination rules for ι in a system of intuitionist negative free logic are formulated. Procedures for removing maximal formulas of the form ιx[F, G] are given, and it is shown that deductions in the system can be brought into normal form.

Starting from a generalization of the standard axioms for a monoid we present a stepwise development of various, mutually equivalent foundational axiom systems for category theory. Our axiom sets have been formalized in the Isabelle/HOL interactive proof assistant, and this formalization utilizes a semantically correct embedding of free logic in classical higher-order logic. The modeling and formal analysis of our axiom sets has been significantly supported by series of experiments with automated reasoning tools integrated with Isabelle/HOL. We also address the (...) relation of our axiom systems to alternative proposals from the literature, including an axiom set proposed by Freyd and Scedrov for which we reveal a technical issue (when encoded in free logic where free variables range over defined and undefined objects): either all operations, e.g. morphism composition, are total or their axiom system is inconsistent. The repair for this problem is quite straightforward, however. (shrink)

Paul Oppenheimer and Edward Zalta's formalisation of Anselm's ontological argument for the existence of God is automated by embedding a free logic for definite descriptions within Isabelle/HOL.

I argue that neutral free logic and existence hedging are incompatible. Primarily, I respond to proposals by James Pryor intended to reconcile the two. Consideration of those proposals will reveal that on any neutral free logic either some existence hedges will entail some undesired existence claims, or they will not entail some desired existence claims.

Logic: the Basics is an accessible introduction to the core philosophy topic of standard logic. Focussing on traditional Classical Logic the book deals with topics such as mathematical preliminaries, propositional logic, monadic quantified logic, polyadic quantified logic, and English and standard ‘symbolic transitions’. With exercises and sample answers throughout this thoroughly revised new edition not only comprehensively covers the core topics at introductory level but also gives the reader an idea of how they can take their knowledge further and the (...) philosophical questions around logic. -/- Logic: the Basics is essential reading for first-year undergraduate philosophy students on standard introductory logic courses. (shrink)

By a free logic is generally meant a variant of classical first-order logic in which constant terms may, under interpretation, fail to refer to individuals in the domain D over which the bound variables range, either because they do not refer at all or because they refer to individuals outside D. If D is identified with what is assumed by the given interpretation to exist, in accord with Quine’s dictum that “to be is to be the value of a [bound] (...) variable,” then a free variation on classical semantics does not require that all constant terms refer to existents, and in this sense such terms lack existential import. (shrink)

Fregean theories of descriptions as terms have to deal with improper descriptions. To save bivalence various proposals have been made that involve assigning referents to improper descriptions. While bivalence is indeed saved, there is a price to be paid. Instantiations of the same general scheme, viz. the one and only individual that is F and G is G, are not only allowed but even required to have different truth values.

Two subject-predicate calculi with equality,SP = and its extensionUSP =, are presented as systems of natural deduction. Both the calculi are systems of free logic. Their presentation is preceded by an intuitive motivation.It is shown that Aristotle's syllogistics without the laws of identitySaP andSiP is definable withinSP =, and that the first-order predicate logic is definable withinUSP =.

This paper asks which free logic a Fregean should adopt. It examines options within the tradition including Carnap’s (1956) chosen object theory, Lehmann’s (1994, 2002) strict Fregean free logic, Woodruff’s (1970) strong table about Boolean operators and Bencivenga’s (1986, 1991) supervaluational semantics. It argues for a neutral free logic in view of its proximity towards explaining natural languages. However, disagreeing with Lehmann, it claims a Fregean should adopt the strong table based on Frege’s discussion on generality. Supervaluation uses strong table (...) and aims to give it a semantic justification. However, supervaluation is in turn justified by convention or mental experiments, which Lehmann argues as inadequate. The paper proposes a new justification of supervaluation based on sense and two-dimensional semantics. The resulting model, coined Supervaluational Neutral Free Logic (SNFL), resolves many conflicts between Lehmann and Bencivenga while staying close with Frege’s discussions about non-denotation. It also provides new insights into the relations among truth, logical truth, and supervaluated truth (or supertruth, for short). (shrink)

The partial structures approach has two major components: a broad notion of structure (partial structure) and a weak notion of truth (quasi-truth). In this paper, we discuss the relationship between this approach and free logic. We also compare the model-theoretic analysis supplied by partial structures with the method of supervaluations, which was initially introduced as a technique to provide a semantic analysis of free logic. We then combine the three formal frameworks (partial structures, free logic and supervaluations), and apply the (...) resulting approach to accommodate semantic paradoxes. (shrink)

The result of combining classical quantificational logic with modal logic proves necessitism – the claim that necessarily everything is necessarily identical to something. This problem is reflected in the purely quantificational theory by theorems such as ∃x t=x; it is a theorem, for example, that something is identical to Timothy Williamson. The standard way to avoid these consequences is to weaken the theory of quantification to a certain kind of free logic. However, it has often been noted that in order (...) to specify the truth conditions of certain sentences involving constants or variables that don’t denote, one has to apparently quantify over things that are not identical to anything. In this paper I defend a contingentist, non-Meinongian metaphysics within a positive free logic. I argue that although certain names and free variables do not actually refer to anything, in each case there might have been something they actually refer to, allowing one to interpret the contingentist claims without quantifying over mere possibilia. (shrink)

IPC, the intuitionistic predicate calculus, has the property(i) Vc(A c /x) xA.Furthermore, for certain important , IPC has the converse property (ii) xA Vc(A c /x). (i) may be given up in various ways, corresponding to different philosophic intuitions and yielding different systems of intuitionistic free logic. The present paper proves the strong completeness of several of these with respect to Kripke style semantics. It also shows that giving up (i) need not force us to abandon the analogue of (ii).