This paper presents a uniform semantic treatment of nonmonotonic inference operations that allow for inferences from infinite sets of premisses. The semantics is formulated in terms of selection functions and is a generalisation of the preferential semantics of Shoham, Kraus et al., and Makinson. A selection function picks out from a given set of possible states (worlds, situations, models) a subset consisting of those states that are, in some sense, the most preferred ones. A proposition alpha$$ \alpha $$ is a nonmonotonic consequence of a set of propositions Gamma$$ \Gamma $$ iff alpha$$ \alpha $$ holds in all the most preferred Gamma$$ \Gamma $$-states. In the literature on revealed preference theory, there are a number of well-known theorems concerning the representability of selection functions, satisfying certain properties, in terms of underlying preference relations. Such theorems are utilised here to give corresponding representation theorems for nonmonotonic inference operations. At the end of the paper, the connection between nonmonotonic inference and belief revision, in the sense of Alchourron, Gardenfors, and Makinson, is explored. In this connection, infinitary belief revision operations, that allow for the revision of a theory with a possibly infinite set of propositions, are introduced and characterised axiomatically. Several semantic representation theorems are proved for operations of this kind.

Vann McGee has proposed a counterexample to the Ramsey Test. In the counterexample, a seemingly trustworthy source has testified that p and that if not-p, then q. If one subsequently learns not- p (and so learns that the source is wrong about p), then one has reason to doubt the trustworthiness of the source (perhaps even the identity of the source) and so, the argument goes, one has reason to doubt the conditional asserted by the source. Since what one learns is that the antecedent of the conditional holds, these doubts are contrary to the Ramsey Test. We argue that the counterexample fails. It rests on a principle of testimonial dependence that is not applicable when a source hedges his or her claims.

There is a traditional picture of logic that may be spelled out roughly as follows: Logic is concerned with the principles for correct reasoning and valid arguments; its principles are universal, necessary, a priori and formal; logically valid argument are necessarily truth-preserving and have a fundamental epistemic significance; and finally, logic is in some sense a normative discipline.This traditional picture might appear appealing but it is at the same time deeply problematic and gives rise to many questions.The notions of universality, logical necessity, aprioricity and formality are notoriously difficult to analyse.In what sense, if any, is logic normative?Is there a principled way of distinguishing between logical and non-logical concepts?Is there a way of understanding logical consequence and logical inference that is compatible with the traditional picture?While continuing to face these foundational questions, logic has developed into an advanced mathematical discipline-mathematical logic-where the informal notions of logical proof, validity and logical consequence are given rigorous mathematical explications.In mathematical logic, there are two major kinds of approaches to these notions: model-theoretic and proof-theoretic ones.

This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice (ZFC). This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued today is predicativistic constructivism based on Martin-Lof type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Lof. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Lof and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics?

Abstract This chapter discusses a paradox that, in Kaplan's view, threatens the use of possible worlds semantics as a model‐theoretic framework for intensional logic. Kaplan's paradox starts out from an intuitively reasonable principle referred to as the principle of plenitude. From this principle he derives a contradiction in what he calls naive possible world theory. Kaplan's metatheoretic argument can be restated in the modal object language as an intensional version of the Liar paradox.

Peter Aczel: Predicate logic over a type setup There are a variety of closely related notions aimed at capturing the abstract structure of type dependency in the syntax and semantics of dependent type theories. Examples of such notions are category with attributes, category with families, category with display maps, contextual category, comprehension category, and there are more. The notion of a type setup is yet one more such notion, which differs from the others in taking a more syntactic approach in its explicit use of contexts, as finite lists of typed variable declarations. This makes it closer to the syntax of dependent type theories, while still abstracting away from the usual inductive structure of syntax and the corresponding recursive definition of substitution. Predicate logic over a type setup is simply defined as a sorted predicate logic where the sorts are the types of the type setup and the sorted terms and substitution are also given by the type setup. Logic over a type setup generalises logic-enriched type theory which, in turn generalises dependently sorted logic, a dependent generalisation of many-sorted logic. Many results of predicate logic generalise to predicate logic over a type setup. I will consider the disjunction and existence properties of intuitionistic predicate logic and end with a characterisation of the logic of the propositions-as-types interpretation of intuitionistic predicate logic. Mark van Atten: Different times: Kant and Brouwer on real numbers Kant held that under the concept of the square root of 2 falls only a geometrical magnitude, but not a number. In particular, he explicitly distinguished the square root of 2 from infinite converging sequences of rationals. Like Kant, Brouwer based his foundations of mathematics on the a priori intuition of time, and indeed he presented his position as fundamentally Kantian. Yet, unlike Kant, Brouwer did identify the square root of 2 with an infinite sequence. The question arises where this difference comes from. I will suggest that it has its origin in the difference in their views on the relation of time to intuition. Steve Awodey: Type theory and homotopy theory In recent research it has become clear that there are deep and fascinating connections between the intensional type theory of Per Martin-Löf and homotopy theory, via the modern approach to the latter in terms of Quillen model categories, as well as the theory of higher dimensional categories. This talk will survey some of these developments. Thierry Coquand: Forcing and type theory Forcing is an important tool in constructive mathematics since it is a general technique to give constructive meaning to some ideal elements. A typical example, that I will recall, is Joyal’s constructive explanation of the algebraic closure of a field. (Even the construction of a splitting field of a polynomial requires this technique.) I will then explain how to adapt this technique to type theory, giving for instance a way to extend type theory with a decidable algebraic closure of a (decidable) field. Peter Dybjer: Program testing and constructive validity In this talk I will discuss the connection between program testing and Martin-Löf's meaning explanations for intuitionistic type theory. First I give a short overview of the historical development of the ideas behind the meaning explanations. Then I explain the connection with program testing. Finally, I will mention the possibility of pursuing the testing point of view for some other logical systems including impredicative ones. Juliet Floyd: Wittgenstein, Gödel and Turing In 1946, recalling his discussions with Turing in Cambridge before the war, Wittgenstein stressed that Turing’s ‘machines’ are really “humans who calculate” (RPP I 1096). Was this intended to embrace or to reject Turing’s model of human calculative activity? What form of anthropomorphism was (and is) at stake in regarding humans as machines, and in playing imitation games? The question becomes even more intriguing when we reflect that while Gödel held that it was Turing’s “precise and unquestionably adequate” definition of the notion of a formal system that allowed his own incompleteness theorems to be proved rigorously for the first time, Gödel also held that Turing made a “philosophical error” in holding that human mental procedures cannot go beyond mechanical procedures. We shall contrast the viewpoints of Wittgenstein, Gödel and Turing, emphasizing the evolution of the logical systems of notation that each one of them provided, and discussing how each viewed the philosophical significance of logic and mathematics. Jean-Yves Girard: Towards non-commutative foundations Quantum physics, operator algebra and the non-commutative geometry of Connes deeply challenge old style foundations. Roughly speaking, the object is entangled, since non-commutative, whereas the subject appears as a commutative window, hence a settheoretic reduction. Issues, partial results, working hypotheses, will be discussed in the talk. Sten Lindström, Church-Fitch’s knowability paradox revisited According to a non-realist conception, the notion of truth is epistemically constrained: the anti-realist accepts one version or another of Dummett’s Knowability Principle: (K) If a statement is true, then it must in principle be possible to know that it is true. There is, however, a well-known argument, due to Alonzo Church and Frederic Fitch, which seems to threaten the anti-realist position. Starting out from seemingly innocuous assumptions, Fitch (JSL 1963) claims to prove: if there is some true proposition which nobody knows to be true, then there is a true proposition which nobody can know to be true. The Church-Fitch argument is simple. Suppose that q is a true proposition that is not known to be true. Consider then the proposition (p): q and it is not known that q. This is obviously a true proposition. And it cannot be known. For suppose that p were known to be true. Then the following proposition would be true: It is known that (q and it is not known that q). Since knowledge distributes over conjunction, it would then also be true that: it is known that q and it is known that it is not known that q. Since knowledge implies truth, it would then follow that it is known that q and it is not known that q. That is, the proposition (p) could not be known to be true. Roughly speaking, we can envisage the following reactions to this argument: • The argument is valid and constitutes a refutation of the anti-realist position. • The argument is valid, but it does not constitute a threat to the anti-realist position. • A detailed analysis of the argument shows it to be invalid. In the talk I plan to discuss these three kinds of reactions to the Knowability argument of Church and Fitch. Per Martin-Löf: Logic: epistemological or ontological? What is logic? Is it the study of the process of inference or reasoning, called demonstration in mathematics, by means of which we justify our judgements? Or is it the study of the logical and set-theoretical concepts, like proposition, truth and consequence on the one hand, and set, element and function on the other, that make their appearance in the contents of our judgements? This is the fundamental question whether logic is in its essence epistemological or ontological. The answer is presumably that it is both, which is to say that, within logic, one can distinguish between two parts, or two layers, the one epistemological and the other ontological. But there remains the question of the order of priority between these two layers: Which comes first? Is epistemology prior to ontology, or is it the other way round? Bolzano, whose logic in four volumes, called Wissenschaftslehre, has the most clear architectonic structure of all logics that have so far been written, treated of the ontological notions of proposition, truth and logical consequence (Ableitbarkeit) in the first two volumes of his Wissenschaftslehre, relegating the epistemology to the third volume. Thus he let ontology take priority over epistemology. Although the line of demarcation between the two was drawn in exactly the right place by Bolzano, my own work on constructive type theory has forced me to the conclusion that the order of priority between ontology and epistemology is nevertheless the reverse of the order in which they are treated in the Wissenschaftslehre. The epistemological notions of judgement and inference have to be in place already when you begin to deal with propositions, truth and consequence, as well as with other purely ontological notions, like the set-theoretical ones. Colin McLarty: What are the things of mathematics? –Identity and existence in categorical foundations Philosophical treatments of identity and existence in mathematics most often take the Zermelo-Frankel conception of extensionality as the norm for individuating objects, which a structuralist account must elude in some way. We will look at the issues with an axiomatic foundation in the category of categories where that kind of individuation is the exception from the start. Peter Pagin: Assertion, truth, and judgment There is an interesting connection between Martin-Löf’s proposition/judgment distinction and a certain puzzle about speech acts. When a speaker asserts (1) The moon reflects light from the sun her assertion is in a sense about its own possible world, even though the proposition she asserts can be evaluated at many worlds. If we treat the world of utterance as an index, we get the content of (2) In w, the moon reflects light from the sun where w gets the world of utterance as value. As a result, the content is either the necessary proposition, if true, or the impossible proposition, if false. This reduces content to truth value. An alternative is to separate the world parameter from the proposition asserted, so that the assertoric content is a distinct entity: (3) w: The moon reflects light from the sun Th

Is it possible to give a justification of our own practice of deductive inference? The purpose of this paper is to explain what such a justification might consist in and what its purpose could be. On the conception that we are going to pursue, to give a justification for a deductive practice means to explain in terms of an intuitively satisfactory notion of validity why the inferences that conform to the practice coincide with the valid ones. That is, a justification should provide an analysis of the notion of validity and show that the inferences that conform to the practice are just the ones that are valid. Moreover, a complete justification should also explain the purpose, or point, of our inferential practice. We are first going to discuss the objection that any justification of our deductive practice must use deduction and therefore be circular. Then we will consider a particular model of justificatory explanation, building on Kreisel’s concept of informal rigour. Finally, in the main part of the paper, we will discuss three ideas for defining the notion of validity: (i) the classical conception according to which the notion of (bivalent) truth is taken as basic and validity is defined in terms of the preservation of truth; (ii) the constructivist idea of starting instead with the notion of (a canonical) proof (or verification) and define validity in terms of this notion; (iii) the idea of taking the notions of rational acceptance and rejection as given and define an argument to be valid just in case it is irrational to simultaneously accept its premises and reject its conclusion (or conclusion s, if we allow for multiple conclusions). Building on work by Dana Scott, we show that the last conception may be viewed as being, in a certain sense, equivalent to the first one. Finally, we discuss the so-called paradox of inference and the informativeness of deductive arguments.

Modal logic was born in philosophy, and has travelled widely; it retains important links with the discipline. This chapter discusses the historical heartland of philosophical modal logic—namely, the scope and limitations of modal logic as an account of necessity and possibility. It also examines modal logic and the logic of belief change, and modal logic as logic of action. The relationship between the logical and metaphysical interpretation of the alethic modalities is discussed. The epistemic logic and deontic logic are meant to illustrate two different uses that modal logic or indeed any logic can have: it may be applied to already existing (non-logical) theory, or it can be used to develop new theory. Modal logic is brought to bear on an area that has already reached a degree of maturity and that is formulated with little or no regard to modal logic. There is a strong connection between the theory of belief change and the logic of conditionals.