Second-Order Logic and Foundations of Mathematics

Philosophy Compass, 2015

Both second order logic and set theory can be used as a foundation for mathematics, that is, as a formal language in which propositions of mathematics can be expressed and proved. We take it upon ourselves in this paper to compare the two approaches, second order logic on one hand and set theory on the other hand, evaluating their merits and weaknesses. We argue that we should think of first order set theory as a very high order logic.

Epistemology versus Ontology, 2012

The question, whether second order logic is a better foundation for mathematics than set theory, is addressed. The main difference between second order logic and set theory is that set theory builds up a transfinite cumulative hierarchy while second order logic stays within one application of the power sets. It is argued that in many ways this difference is illusory. More importantly, it is argued that the often stated difference, that second order logic has categorical characterizations of relevant mathematical structures, while set theory has non-standard models, amounts to no difference at all. Second order logic and set theory permit quite similar categoricity results on one hand, and similar non-standard models on the other hand.

The question of the semantic interpretation of higher-order logics has long been a matter of contention. Even though second-order quantification is quite natural, entangled interpretations have famously caused philosophers of logic such as Quine to reject second-order logic completely. In this paper I take a liberal attitude, open to maximizing the scope of logic, but careful to avoid conflation with other disciplines – and to avoid epistemological confusion. Higher-order logic (HOL) is perfectly acceptable, but one should be careful as to which semantics deserves to be called " standard ".

In the paper we consider the classical logicism program restricted to first-order logic. The main result of this paper is the proof of the theorem, which contains the necessary and sufficient conditions for a mathematical theory to be reducible to logic. Those and only those theories, which don't impose restrictions on the size of their domains, can be reduced to pure logic. Among such theories we can mention the elementary theory of groups, the theory of combinators (combinatory logic), the elementary theory of topoi and many others. It is interesting to note that the initial formulation of the problem of reduction of mathematics to logic is principally insoluble. As we know all theorems of logic are true in the models with any number of elements. At the same time, many mathematical theories impose restrictions on size of their models. For example, all models of arithmetic have an infinite number of elements. If arithmetic was reducible to logic, it would had finite models, including an one-element model. But this is impossible in view of the axiom 0 ̸ = x ′ .

Journal of Philosophical Logic

Philosophia Mathematica, 1999

I once heard a story about a museum that claimed to have the skull of Christopher Columbus. In fact, they claimed to have two such skulls, one of Columbus when he was a small boy and one when he was a grown man. Whether there was such a museum or not, the clear moral is that one should not claim too much. The purpose of this paper is to apply the moral to the contrast between first-order logic and second-order logic, as articulated in my Foundations without foundationalism: A case for second-order logic (Shapiro [1991]; see also Shapiro [1985]). Important philosophical issues concerning the nature of logic and logical theory lie in the vicinity. In a review of my book, John Burgess [1993] wrote: ... there is a tendency, signaled by the use of the word 'case' in the subtitle and the phrase 'the competition' as the title for the last chapter, for the author to step into the role of a lawyer or salesman for Second-Order, Inc., and this approach leads to some exaggerated and tendentious formulations. Thus Burgess thinks that in my enthusiastic defense of second-order logic, I claim too much. So do a few other commentators. There is little point to an exegetical study of my own book, to see whether it contains the exaggerated claims in question, but a study of the critical remarks will reveal what should and should not be claimed for second-order logic. The focus in my book, and here, is on second-order languages with standard model-theoretic semantics. In each interpretation, the property or set variables range over the entire powerset of the domain d, the binary relation variables range over the powerset of d 2 , etc. I do not insist on extensionality. If one takes the higher-order variables to range over intensional items, like concepts, then the issue of standard semantics is whether, for each subset s of d, there is a concept whose extension is s, and similarly for relation and function variables. Let AR be the conjunction of the standard Peano axioms, including the

Journal of Philosophical Logic, 2000

Henkin quantifiers have been introduced in Henkin (1961). Walkoe (1970) studied basic model-theoretical properties of an extension L 1 * (H) of ordinary first-order languages in which every sentence is a first-order sentence prefixed with a Henkin quantifier. In this paper we consider a generalization of Walkoe's languages: we close L 1 * (H) with respect to Boolean operations, and obtain the language L 1 (H). At the next level, we consider an extension L 2 * (H) of L 1 (H) in which every sentence is an L 1 (H)-sentence prefixed with a Henkin quantifier. We repeat this construction to infinity. Using the (un)definability of truth-inN for these languages, we show that this hierarchy does not collapse. In addition, we compare some of the present results to the ones obtained by Kripke (1975), McGee (1991), and Hintikka (1996).

Synthese, 2013

In a first part, I defend that formal semantics can be used as a guide to ontological commitment. Thus, if one endorses an ontological view O and wants to interpret a formal language L, a thorough understanding of the relation between semantics and ontology will help us to construct a semantics for L in such a way that its ontological commitment will be in perfect accordance with O. Basically, that is what I call constructing formal semantics from an ontological perspective. In the rest of the paper, I develop rigorously and put into practice such a method, especially concerning the interpretation of second-order quantification. I will define the notion of ontological framework: it is a set-theoretical structure from which one can construct semantics whose ontological commitments correspond exactly to a given ontological view. I will define five ontological frameworks corresponding respectively to: (i) predicate nominalism, (ii) resemblance nominalism, (iii) armstrongian realism, (iv) platonic realism, and (v) tropism. In those different frameworks, I will construct different semantics for first-order and second-order languages. Notably I will present different kinds of nominalist semantics for second-order languages, thus showing how we can quantify over properties and relations while being ontologically committed only to individuals. More generally I will show in what extent those semantics differ from each other; it will make clear how the disagreements between the ontological views extend from ontology to logic, and how metaphysical questions can be correctly treated, in those semantics, as simple questions of logic.

Journal of Logic Language and Information, 1998

Price: $59.95/£45.00, xxii + 388 pages, ISBN 0-521-35435-8.