Renewed worries about the unity of the proposition have been taken as a crucial stumbling block for any traditional conception of propositions. These worries are often framed in terms of how entities independent of mind and language can have truth conditions: why is the proposition that Desdemona loves Cassio true if and only if she loves him? I argue that the best understanding of these worries shows that they should be solved by our theory of truth and not our theory (...) of content. Specifically, I propose a version of the redundancy theory according to which ‘it is true that Desdemona loves Cassio’ expresses the same proposition as ‘Desdemona loves Cassio’. Surprisingly, this variant of the redundancy theory treats ‘is true’ as an ordinary predicate of the language, thereby defusing many standard criticisms of the redundancy theory. (shrink)

Intensional contexts are typically characterised by an apparent failure of either (A) the principle of the inter-substitution of co-referring terms salva veritate, or (B) existential generalisation. The difficulties which are seen to occur do so in contexts involving either modality or the propositional attitudes. In this paper attempts are made to determine whether or not Scheffler’s inscriptional analysis can provide a viable means of accounting for the problems which are thought to occur in intensional contexts. Somewhat unexpectedly, little effort has (...) been made in the past to address this issue. In this paper it is shown that Scheffler’s theory may be employed to account for the difficulties mentioned above, though further work needs to be done to show precisely how his analysis may be adapted so as to handle modal statements. Popular objections to Scheffler’s inscriptionalism are also addressed, particularly in the light of his theory being used to account for the problems of intensionality. It is found that, with certain qualifications, the aforesaid objections do not show Scheffler’s theory to be an unviable means of accounting for the intensionality problems. (shrink)

Leon Henkin (1921–2006) was not only an extraordinary logician, but also an excellent teacher, a dedicated professor and an exceptional person. The first two sections of this paper are biographical, discussing both his personal and academic life. In the last section we present three aspects of Henkin’s work. First we comment part of his work fruit of his emphasis on teaching. In a personal communication he affirms that On mathematical induction, published in 1969, was the favourite among his articles with (...) a somewhat panoramic nature and not meant exclusively to specialists. This subject is covered in the first subsection. Needless to say that we also analyse Henkin’s better known contribution: his completeness method. His renowned results on completeness for both type theory and first order logic were part of his thesis, The Completeness of Formal Systems, presented at Princeton in 1947 under the advise of Alonzo Church. It is interesting to note that he obtained the proof of completeness for first order logic readapting the argument for the theory of types. The last subsection is devoted to philosophy. The work most directly related to philosophy is an article entitled: Some Notes on Nominalism which appeared in the Journal of Symbolic Logic in 1953. Unfortunately, we are not covering his contribution to the field of cylindric algebras. As a matter of fact, Henkin spent many years investigating algebraic structures with Alfred Tarski and Donald Monk, among others. (shrink)

Kripke’s Fregean quantification logic FQ fails to formalize the usual first-order logic with identity due to the interpretation of the conditional operator. Motivated by Kripke’s syntax and semantics, the three-valued Fregean quantification logic FQ3 is proposed. This three valued logic differs from Kleene and Łukasiewicz’s three-valued logics. The logic FQ3 is decidable. A sound and complete Hilbert-style axiomatic system for the logic FQ3 is presented.

This paper is dedicated to Alonzo Church, who died in August 1995 after a long life devoted to logic. To Church we owe lambda calculus, the thesis bearing his name and the solution to the Entscheidungsproblem.His well-known book Introduction to Mathematical LogicI, defined the subject matter of mathematical logic, the approach to be taken and the basic topics addressed. Church was the creator of the Journal of Symbolic Logicthe best-known journal of the area, which he edited for several decades This (...) paper is in three sections. The first is written in journalistic style:the story of the life of AlonzoChurch is told, including some of the many anecdotes I have collected from different sources. The secondpart is devoted to his work, but is far from being exhaustive. The last part is more original; in it I attempto show that Church’s great discovery was lambda calculus and that his remaining contributions weremainly inspired afterthoughts in the sense that most of his contributions as well as some of his pupils derivefrom that initial achievement. Included are Kleene’s Recursion Theory and the completeness proof ofHenkin. I have added an appendix in which is presented the typed lambda calculus and a proof of theundecidability of first-order logic. (shrink)

This paper is about the meaning and function of identity statements involving proper names. There are two prominent views on this topic, according to which identity statements ascribe a relation: the object-view, on which identity statements ascribe a relation borne by all objects to themselves, and the name-view, on which an identity statement 'a is b' says that the names 'a' and 'b' codesignate. The object- and name-views may seem to exhaust the field. I make a case for treating identity (...) statements as sui generis instead of attempting to explain them by means of the idea that they ascribe a relation. My contention is that once we do this, no analysis is required. -/- I do not wish to insist that we stop saying that identity statements ascribe a relation. The point is that there is a fundamental disanalogy between identity statements and other two-termed statements which we overlook to our peril. This will be seen to parallel the more recognized disanalogy between existence statements and other one-termed statements. One way of registering the fundamental disanalogy is to say that identity statements are not relational, but this is not essential. Following my negative arguments in section 2, I employ some simple diagrammatical models in section 3 to exhibit the fundamental disanalogy. In a final section I respond to some possible objections which may be raised against this kind of approach. (shrink)

The identity predicate can be defined using second-order quantification: a=b =df ∀F(Fa↔Fb). Less familiarly, a dyadic sentential operator analogous to the identity predicate can be defined using third-order quantification: ϕ≡ψ =df ∀X(Xϕ↔Xψ), where X is a variable of the same syntactic type as a monadic sentential operator. With this notion in view, it is natural to ask after general principles governing its application. More grandiosely, how fine-grained is reality? -/- I will argue that reality is not structured in anything like (...) the way that the sentences we use to talk about it are structured. I do so by formulating a higher-order analogue of Russell’s paradox of structured propositions. I then relate this argument to the Frege-Russell correspondence. When confronted with the alleged paradox, Frege agreed that reality was not structured, but maintained that propositions (i.e. thoughts) were structured all the same. Russell replied that his paradox showed Frege’s theory of structured thoughts to be inconsistent, to which Frege replied that Russell’s argument failed to heed the distinction between sense and reference. Most recent commentators have sided with Russell. In defense of Frege, I establish the consistency of one version of his rejoinder. I then consider and reject some ways of resisting the argument against a structured conception of reality. I conclude that, if propositions are structured, this is because they correspond not to distinctions in reality, but rather to ways in which those distinctions can be represented. (shrink)

Paraconsistent quantum logics are weak forms of quantum logic, where the noncontradiction and the excluded-middle laws are violated. These logics find interesting applications in the operational approach to quantum mechanics. In this paper, we present an axiomatization, a Kripke-style, and an algebraic semantical characterization for two forms of paraconsistent quantum logic. Further developments are contained in Giuntini and Greuling's paper in this issue.

Arguments are given against the thesis that properties and propositional functions are identical. The first shows that the familiar extensional treatment of propositional functions -- that, for all x, if f(x) = g(x), then f = g -- must be abandoned. Second, given the usual assumptions of propositional-function semantics, various propositional functions (e.g., constant functions) are shown not to be properties. Third, novel examples are given to show that, if properties were identified with propositional functions, crucial fine-grained intensional distinctions would (...) be lost. (shrink)

This paper provides a new approach to a family of outstanding logical and semantical puzzles, the most famous being Frege's puzzle. The three main reductionist theories of propositions (the possible-worlds theory, the propositional-function theory, the propositional-complex theory) are shown to be vulnerable to Benacerraf-style problems, difficulties involving modality, and other problems. The nonreductionist algebraic theory avoids these problems and allows us to identify the elusive nondescriptive, non-metalinguistic, necessary propositions responsible for the indicated family of puzzles. The algebraic approach is also (...) used to defend antiexistentialism against existentialist prejudices. The paper closes with a suggestion about how this theory of content might enable us to give purely semantic (as opposed to pragmatic) solutions to the puzzles based on a novel formulation of the principle of compositionality. (shrink)