Online research in philosophy

We give a syntactic translation from first-order intuitionistic predicate logic into second-order intuitionistic propositional logic IPC2. The translation covers the full set of logical connectives ∧, ∨, →, ⊥, ∀, and ∃, extending our previous work, which studied the significantly simpler case of the universal-implicational fragment of predicate logic. As corollaries of our approach, we obtain simple proofs of nondefinability of ∃ from the propositional connectives and nondefinability of ∀ from ∃ in the second-order intuitionistic propositional logic. We also show that the ∀-free fragment of IPC2 is undecidable.

Provided here is an account, both syntactic and semantic, of first-order and monadic second-order quantification theory for domains that may be non-atomic. Although the rules of inference largely parallel those of classical logic, there are important differences in connection with the identification of argument places and the significance of the identity relation.

In this paper, I shall provide a defence of second-order logic in the context of its use in the philosophy of mathematics. This shall be done by considering three problems that have been recently posed against this logic: (1) According to Resnik [1988], by adopting second-order quantifiers, we become ontologically committed to classes. (2) As opposed to what is claimed by defenders of second-order logic (such as Shapiro [1985]), the existence of non-standard models of first-order theories does not establish the inadequacy of first—order axiomatisations (Melia [1995]). (3) In contrast with Shapiro’s suggestion (in his [1985]), second-order logic does not help us to establish referential access to mathematical objects (Azzouni [1994]). As I shall argue, each of these problems can be neatly solved by the second-order theorist. As a result, a case for second-order logic can be made. The first two problems will beconsidered rather briefly in the next section. The rest of the paper is dedicate to a discussion of the third.

Because of its capacity to characterize mathematical concepts and structures?a capacity which first-order languages clearly lack?second-order languages recommend themselves as a convenient framework for much of mathematics, including set theory. This paper is about the credentials of second-order logic:the reasons for it to be considered logic, its relations with set theory, and especially the efficacy with which it performs its role of the underlying logic of set theory.

particular alternative logic could be relevant to another one? The most important part of a response to this question is to remind the reader of the fact that independence friendly (IF) logic is not an alternative or “nonclassical” logic. (See here especially Hintikka, “There is only one logic”, forthcoming.) It is not calculated to capture some particular kind of reasoning that cannot be handled in the “classical” logic that should rather be called the received or conventional logic. No particular epithet should be applied to it. IF logic is not an alternative to our generally used basic logic, the received first-order logic, aka quantification theory or predicate calculus. It replaces this basic logic in that it is identical with this “classical” first-order logic except that certain important flaws of the received first-order logic have been corrected. But what are those flaws and how can they be corrected? To answer these questions is to explain the basic ideas of IF logic. Since this logic is not as well known as it should be, such explanation is needed in any case. I will provide three different but not unrelated motivations for IF logic.

Symbolic logic is a marvelous thing. It allows for an explicit expression of existence, viz. by means of the existential quantifier, and by it only. This is the true gist in Quine’s slogan “to be is to be a value of a bound variable.” Accordingly, one can also formulate explicitly the thesis of nominalism in terms of such logic. What this thesis says is that all the values of existential quantifiers we need in our language are particular objects, not higher-order objects such as properties, relations, functions and sets. This requirement is satisfied by the first-order languages using the received first-order logic. The commonly used basic logic is therefore nominalistic. But this result does not tell anything, for the received first-order logic is far too weak to capture all we need in mathematics or science. According to conventional wisdom, we need for this purpose either higher-order logic or set theory. Now both of them deal with higher-order entities and hence violate the canons of nominalism. This does not refute nominalism, however. For I will show that both set theory and higher-order logic can be made dispensable by developing a more powerful first-order logic that can do the same job as they do. Moreover, there are very serious problems connected with both of them. This constitutes an additional reason for dispensing with them in the foundations of mathematics. I will show how we can do just that. But we obviously need a better first-order logic for the purpose. Hence my first task is to develop one. But is this a viable construal of the problem of nominalism? The very distinction between particular and higher-order entities might perhaps seem to be hard to capture in logical terms — harder than has been indicated so far. Logicians like Jouko Väänänen (2001) have emphasized the complexities involved in trying to distinguish first-order logic from higher-order logic..

A propositional system of modal logic is second-order if it contains quantifiers ∀p and ∃p, which, in the standard interpretation, are construed as ranging over sets of possible worlds (propositions). Most second-order systems of modal logic are highly intractable; for instance, when augmented with propositional quantifiers, K, B, T, K4 and S4 all become effectively equivalent to full second-order logic. An exception is S5, which, being interpretable in monadic second-order logic, is decidable.

This text is a short introduction to logic that was primarily used for accompanying an introductory course in Logic for Linguists held at the New University of Lisbon (UNL) in fall 2010. The main idea of this course was to give students the formal background and skills in order to later assess literature in logic, semantics, and related fields and perhaps even use logic on their own for the purpose of doing truth-conditional semantics. This course in logic does not replace a proper introduction to semantics and is not intended as such, although parts of Chapter 1 and 4 could be used to supplement an introductory course in semantics. In contrast to other introductions it has a certain focus on ‘writing things down correctly.’ Proofs of metatheorems are omitted, though.

This is work in progress. Please send suggestions and corrigenda to erich@snafu.de. Have fun!

This is a free book, 165 pages. It is for anyone who has had a solid introductory logic course and wants more. Topics covered include soundness and completeness for first-order logic, Tarski's theorem on the undefinability of truth, Gödel's incompleteness theorems, the undecidability of first-order logic, a smattering of second-order logic, and modal logic (both propositional and quantificational). I wrote it for use in my own course, because I thought I could present the most important results and concepts more clearly than the available textbooks.

Pure second-order logic is second-order logic without functional or first-order variables. In "Pure Second-Order Logic," Denyer shows that pure second-order logic is compact and that its notion of logical truth is decidable. However, his argument does not extend to pure second-order logic with second-order identity. We give a more general argument, based on elimination of quantifiers, which shows that any formula of pure second-order logic with second-order identity is equivalent to a member of a circumscribed class of formulas. As a corollary, pure second-order logic with second-order identity is compact, its notion of logical truth is decidable, and it satisfies a pure second-order analogue of model completeness. We end by mentioning an extension to n th-order pure logics.