Some reasons to regard the cumulative hierarchy of sets as potential rather than actual are discussed. Motivated by this, a modal set theory is developed which encapsulates this potentialist conception. The resulting theory is equi-interpretable with Zermelo Fraenkel set theory but sheds new light on the set-theoretic paradoxes and the foundations of set theory.

Proof-theoretic reflection principles are schemas which attempt to express the soundness of arithmetical theories within their own language, e.g., ${\mathtt{{Prov}_{\mathsf {PA}} \rightarrow \varphi }}$ can be understood to assert that any statement provable in Peano arithmetic is true. It has been repeatedly suggested that justification for such principles follows directly from acceptance of an arithmetical theory $\mathsf {T}$ or indirectly in virtue of their derivability in certain truth-theoretic extensions thereof. This paper challenges this consensus by exploring relationships between reflection principles (...) and principles of mathematical and transfinite induction as well as the status of the latter with respect to various foundational characterizations of number theory. (shrink)

Hartry Field distinguished two concepts of type-free truth: scientific truth and disquotational truth. We argue that scientific type-free truth cannot do justificatory work in the foundations of mathematics. We also present an argument, based on Crispin Wright's theory of cognitive projects and entitlement, that disquotational truth can do justificatory work in the foundations of mathematics. The price to pay for this is that the concept of disquotational truth requires non-classical logical treatment.

This article introduces, studies, and applies a new system of logic which is called ‘HYPE’. In HYPE, formulas are evaluated at states that may exhibit truth value gaps and truth value gluts. Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and the (...) predicate logic of the system. The propositional logic of HYPE is shown to contain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionistic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE’s logical connectives. Furthermore, HYPE’s first-order logic is a conservative extension of intuitionistic logic with the Constant Domain Axiom, when intuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler-Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a general logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of relevance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. Finally, HYPE may be used as a background system for modal operators that create hyperintensional contexts, though the details of this application need to be left to follow-up work. (shrink)

Philosophy and model theory frequently meet one another. Philosophy and Model Theory aims to understand their interactions -/- Model theory is used in every ‘theoretical’ branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics. But these wide-ranging appeals to model theory have created a highly fragmented literature. On the one hand, many philosophically significant mathematical results are found only in mathematics textbooks: these are aimed squarely at mathematicians; (...) they typically presuppose that the reader has a significant background in mathematics; and little clue is given as to their philosophical significance. On the other hand, the philosophical applications of these results are then scattered across disconnected pockets of papers, separated by decades or more. -/- The first aim of Philosophy and Model Theory, then, is to consider the philosophical uses of model theory. On a technical level, we try to show how philosophically significant results connect to one another, and also to state the best version of a result for philosophical purposes. On a philosophical level, we show how similar dialectical situations arise repeatedly, across fragmented debates in varied philosophical areas. -/- The second aim of Philosophy and Model Theory, though, is to consider the philosophy of model theory. Model theory itself is rarely taken as the subject matter of philosophising (contrast this, say, with the philosophy of biology, or the philosophy of set theory). But model theory is a beautiful part of pure mathematics, and worthy of philosophical study in its own right. (shrink)

According to structuralism in philosophy of mathematics, arithmetic is about a single structure. First-order theories are satisfied by models that do not instantiate this structure. Proponents of structuralism have put forward various accounts of how we succeed in fixing one single structure as the intended interpretation of our arithmetical language. We shall look at a proposal that involves Tennenbaum's theorem, which says that any model with addition and multiplication as recursive operations is isomorphic to the standard model of arithmetic. On (...) this account, the intended models of arithmetic are the notation systems with recursive operations on them satisfying the Peano axioms. [A]m Anfang […] ist das Zeichen. (shrink)

We show that the categoricity of second-order Peano axioms can be proved from the comprehension axioms. We also show that the categoricity of second-order Zermelo–Fraenkel axioms, given the order type of the ordinals, can be proved from the comprehension axioms. Thus these well-known categoricity results do not need the so-called “full” second-order logic, the Henkin second-order logic is enough. We also address the question of “consistency” of these axiom systems in the second-order sense, that is, the question of existence of (...) models for these systems. In both cases we give a consistency proof, but naturally we have to assume more than the mere comprehension axioms. (shrink)

The essay addresses the well‐known idea that there has to be a place for intuition, thought of as a kind of non‐inferential rational insight, in the epistemology of basic logic if our knowledge of its principles is non‐empirical and is to allow of any finite, non‐circular reconstruction. It is argued that the error in this idea consists in its overlooking the possibility that there is, properly speaking, no knowledge of the validity of principles of basic logic. When certain important distinctions (...) are observed, for instance that between recognising that Modus Ponenes is sound and recognising that it is proof against the competent discovery of basic counterexamples, the case for thinking that there is indeed no space for genuine recognition of the validity of Modus Ponens becomes increasingly impressive. It is argued however that, the impossibility of knowledge notwithstanding, we are, in an important sense, entitled to take it that Modus Ponens is sound and that this notion of entitlement can help break the trichotomy ‐ intuition, inference, experience ‐ which imprisons our ordinary thinking about logical knowledge. (shrink)

In the first part of this paper we discuss some aspects of Detlefsen's attempt to save Hilbert's Program from the consequences of Godel's Second Incompleteness Theorem. His arguments are based on his interpretation of the long standing and well-known controversy on what, exactly, finitistic means are. In his paper [1] Detlefsen takes the position that there is a form of the ω-rule which is a finitistically valid means of proof, sufficient to prove the consistency of elementary number theory Z. On (...) the other hand, he claims that Z with its first-order logic is not strong enough to allow a formalization of such an ω-rule. This would explain why the unprovability of $\operatorname{Con}(Z)$ in Z does not imply that the consistency of Z cannot be proved by finitistic means. We show that Detlefsen's proposal is unacceptable as originally formulated in [1], but that a reasonable modification of the rule he suggest leads to a partial program already studied for many years. We investigate the scope of such a program in terms of proof-theoretic reducibilities. We also show that this partial program encompasses mathematically important theories studied in the "Reverse Mathematics" program. In order to investigate the provability with such a modified rule, we define new consistency and provability predicates which are weaker than the usual ones. We then investigate their properties, including a few that have no apparent philosophical significance but compare interestingly with the properties of the program based on the iteration of our ω-rule. We determine some of the limitations of such programs, pointing out that these limitations partly explain why partial programs that have been successfully carried out use quite different and substantially more radical extensions of finitistic methods with more general forms of restricted reasoning. (shrink)

Deflationsism about truth is a pot-pourri, variously claiming that truth is redundant, or is constituted by the totality of 'T-sentences', or is a purely logical device (required solely for disquotational purposes or for re-expressing finitarily infinite conjunctions and/or disjunctions). In 1980, Hartry Field proposed what might be called a 'deflationary theory of mathematics', in which it is alleged that all uses of mathematics within science are dispensable. Field's criterion for the dispensability of mathematics turns on a property of theories, called (...) conservativeness. I present some technical results, some of which may be found in Tarski (1936), concerning the logical properties of truth theories; in particular, concerning the conservativeness of adding a truth theory for an object level language to any theory expressed in it. It transpires that various deflationary truth theories behave somewhat differently from the standard Tarskian truth theory. These results suggest that Tarskian theories of truth are not redundant or dispensable. Finally, I hint at an analogy between the behaviour of mathematical theories and of standard (Tarskian) theories of truth with respect to their indispensability to, as Quine would put, our 'scientific world-view'. (shrink)

This article gives an epistemological analysis of the reflection process by means of which you can come to know the consistency of a mathematical theory that you already accept. It is argued that this process can result in warranted belief in new mathematical principles without justifying them.

The uniform reflection principle for the theory of uniform T-sentences is added to PA. The resulting system is justified on the basis of a disquotationalist theory of truth where the provability predicate is conceived as a special kind of analyticity. The system is equivalent to the system ACA of arithmetical comprehension. If the truth predicate is also allowed to occur in the sentences that are inserted in the T-sentences, yet not in the scope of negation, the system with the reflection (...) schema for these T-sentences assumes the strength of the Kripke-Feferman theory KF, and thus of ramified analysis up to go. (shrink)

Hartry Field distinguished two concepts of type‐free truth: scientific truth and disquotational truth. We argue that scientific type‐free truth cannot do justificatory work in the foundations of mathematics. We also present an argument, based on Crispin Wright's theory of cognitive projects and entitlement, that disquotational truth can do justificatory work in the foundations of mathematics. The price to pay for this is that the concept of disquotational truth requires non‐classical logical treatment.