In this dissertation we study two-cardinal phenomena—both of the admitting cardinals variety and of the Chang’s Conjecture variety—under the assumption that all our models have stable theories. All our results involve two, relatively widely accepted generalizations of the traditional definitions in this area. First, we allow the relevant subsets of our models to be picked out by (perhaps infinitary) partial types; second we consider δ-cardinal problems as well as two-cardinal problems.

We examine two-cardinal problems for the class of O-minimal theories. We prove that an O-minimal theory which admits some (κ, λ) must admit every (κ , λ ). We also prove that every “reasonable” variant of Chang’s Conjecture is true for O-minimal structures. Finally, we generalize these results from the two-cardinal case to the δ-cardinal case for arbitrary ordinals δ.

We examine two-cardinal problems for the class of O-minimal theories. We prove that an O-minimal theory which admits some must admit every. We also prove that every "reasonable" variant of Chang's Conjecture is true for O-minimal structures. Finally, we generalize these results from the two-cardinal case to the $\delta$-cardinal case for arbitrary ordinals $\delta$.

While working on various medieval philosophers, I have noticed an affinity between their remarks on the reasonableness of accepting propositions that are not matters of proof and strict deduction and St. John Henry Newman’s remarks that we accept unconditionally and rightly everyday ordinary propositions without calibrating them to demonstrable arguments. In particular, Cardinal Matthew of Aquasparta and Blessed John Duns Scotus both claim there is a sense in which assent to everyday propositions is tantamount to knowledge, even though there (...) is no adequate argumentation or demonstrative reasoning compelling us to assent to such propositions. Newman’s distinction between notional and real apprehension of propositions, notional and real assents, and his insistence on the existence of real assents to propositions that are not necessarily proved, or in some cases provable, seem, at first glance, a case parallel to that of the medieval philosophers we have mentioned. (shrink)

[Ian Rumfitt] Frege's logicism in the philosophy of arithmetic consisted, au fond, in the claim that in justifying basic arithmetical axioms a thinker need appeal only to methods and principles which he already needs to appeal in order to justify paradigmatically logical truths and paradigmatically logical forms of inference. Using ideas of Gentzen to spell out what these methods and principles might include, I sketch a strategy for vindicating this logicist claim for the special case of the arithmetic of the (...) finite cardinals. /// [Timothy Williamson]The paper defends the intelligibility of unrestricted quantification. For any natural number n, 'There are at least n individuals' is logically true, when the quantifier is unrestricted. In response to the objection that such sentences should not count as logically true because existence is contingent, it is argued by consideration of cross-world counting principles that in the relevant sense of 'exist' existence is not contingent. A tentative extension of the upward Löwenheim-Skolem theorem to proper classes is used to argue that a sound and complete axiomatization of the logic of unrestricted universal quantification results from adding all sentences of the form 'There are at least n individuals' as axioms to a standard axiomatization of the first-order predicate calculus. (shrink)

In this paper the ultrafilter properties of canonical frames are used to produce a non-standard map between canonical frames of different cardinalities. While this map is not a p-morphism, it is presented as a step towards the full understanding of canonical structures.

This chapter contains sections titled: This chapter generalizes the technique to other cases, including the identity of species and persons, and makes the appropriate concept of approximation precise. What need to be found are generalizations with worthwhile instances, beyond those already considered. The first section abstracts a formal schema: when a supposed criterion of identity M for objects of some kind turns out to be non‐transitive, the best approximation to the original criterion is a maximal M‐relation. The second section applies (...) the schema to operationally defined quantities, species, and non‐Cantorian cardinal numbers. The third section applies it in more detail to persons. The fourth section considers the case in which M is held to be a sufficient condition rather than a necessary one. The final section analyses the relevant concept of a best approximation. (shrink)

[Ian Rumfitt] Frege's logicism in the philosophy of arithmetic consisted, au fond, in the claim that in justifying basic arithmetical axioms a thinker need appeal only to methods and principles which he already needs to appeal in order to justify paradigmatically logical truths and paradigmatically logical forms of inference. Using ideas of Gentzen to spell out what these methods and principles might include, I sketch a strategy for vindicating this logicist claim for the special case of the arithmetic of the (...) finite cardinals. /// [Timothy Williamson]The paper defends the intelligibility of unrestricted quantification. For any natural number n, 'There are at least n individuals' is logically true, when the quantifier is unrestricted. In response to the objection that such sentences should not count as logically true because existence is contingent, it is argued by consideration of cross-world counting principles that in the relevant sense of 'exist' existence is not contingent. A tentative extension of the upward Löwenheim-Skolem theorem to proper classes is used to argue that a sound and complete axiomatization of the logic of unrestricted universal quantification results from adding all sentences of the form 'There are at least n individuals' as axioms to a standard axiomatization of the first-order predicate calculus. (shrink)

Almost all set theorists pay at least lip service to Cantor’s definition of a set as a collection of many things into one whole; but empty and singleton sets do not fit with it. Adapting Dana Scott’s axiomatization of the cumulative theory of types, we present a ‘Cantorian’ system which excludes these anomalous sets. We investigate the consequences of their omission, examining their claim to a place on grounds of convenience, and asking whether their absence is an obstacle to the (...) theory’s ability to represent ordered pairs or to support the arithmetization of analysis or the development of the theory of cardinals and ordinals. (shrink)

We live with the legacy of injustice, political as well as personal. Even if our governments are now democratically elected and governed, our societies are scarred by forms of power and privilege accrued from a time in which people’s race, sex, class and religion were grounds for denying them a role in government, or in the selection of those who governed them. What does that past imply for the treatment of religion in democratic states? The problem is particularly pressing once (...) one accepts that religious freedom is not just a matter of individuals’ freedom of conscience and worship, but of people’s claims to associate with others through institutions whose powers, status and commitment to equality are very different (Laborde, 2015). If this means that churches pose some of the same philosophical and practical problems as families, from a democratic perspective, the fact that churches have no obvious point or justification, beyond being the repository of the claims to conscience of their members, appears to distinguish them from the former. In principle, this should make it easier to think about the claims of government, as compared to those of churches. In practice, however, it may simply bring into sharper focus philosophical and political challenges to equality that contemporary democracies now face. (shrink)

I consider the first-order modal logic which counts as valid those sentences which are true on every interpretation of the non-logical constants. Based on the assumptions that it is necessary what individuals there are and that it is necessary which propositions are necessary, Timothy Williamson has tentatively suggested an argument for the claim that this logic is determined by a possible world structure consisting of an infinite set of individuals and an infinite set of worlds. He notes that only (...) the cardinalities of these sets matters, and that not all pairs of infinite sets determine the same logic. I use so-called two-cardinal theorems from model theory to investigate the space of logics and consequence relations determined by pairs of infinite sets, and show how to eliminate the assumption that worlds are individuals from Williamson’s argument. (shrink)

In Fixing Language, Herman Cappelen defends the project of conceptual engineering from a family of objections that he calls “the Strawsonian challenges.” Those objections are all versions of this: “If I ask you a question about the F’s, and you give me an answer that’s not about the F’s but rather about the G’s, then you haven’t answered my question. You have changed the subject.” I argue that Cappelen’s response succeeds in reply to one understanding of the Strawsonian challenge—on which (...) it is motivated by ordinary judgments of samesaying and continuity of topic—but that it fails as a response to another version—on which a parallel objection is motivated by philosophical considerations and is stated in a theoretical register. (shrink)

This paper explains and defends the idea that metaphysical necessity is the strongest kind of objective necessity. Plausible closure conditions on the family of objective modalities are shown to entail that the logic of metaphysical necessity is S5. Evidence is provided that some objective modalities are studied in the natural sciences. In particular, the modal assumptions implicit in physical applications of dynamical systems theory are made explicit by using such systems to define models of a modal temporal logic. Those assumptions (...) arguably include some necessitist principles. -/- Too often, philosophers have discussed ‘metaphysical’ modality — possibility, contingency, necessity — in isolation. Yet metaphysical modality is just a special case of a broad range of modalities, which we may call ‘objective’ by contrast with epistemic and doxastic modalities, and indeed deontic and teleological ones (compare the distinction between objective probabilities and epistemic or subjective probabilities). Thus metaphysical possibility, physical possibility and immediate practical possibility are all types of objective possibility. We should study the metaphysics and epistemology of metaphysical modality as part of a broader study of the metaphysics and epistemology of the objective modalities, on pain of radical misunderstanding. Since objective modalities are in general open to, and receive, natural scientific investigation, we should not treat the metaphysics and epistemology of metaphysical modality in isolation from the metaphysics and epistemology of the natural sciences. -/- In what follows, Section 1 gives a preliminary sketch of metaphysical modality and its place in the general category of objective modality. Section 2 reviews some familiar forms of scepticism about metaphysical modality in that light. Later sections explore a few of the many ways in which natural science deals with questions of objective modality, including questions of quantified modal logic. (shrink)

Disorders of volition are often accompanied by, and may even be caused by, disruptions in the phenomenology of agency. Yet the phenomenology of agency is at present little explored. In this paper we attempt to describe the experience of normal agency, in order to uncover its representational content.

This work presents a historically informed, systematic exposition of the Christology of the first seven Ecumenical Councils of undivided Christendom, from the First Council of Nicaea in 325 AD to the Second Council of Nicaea in 787 AD. Assuming the truth of Conciliar Christology for the sake of argument, Timothy Pawl considers whether there are good philosophical arguments that show a contradiction or incoherence in that doctrine. He presents the definitions of important terms in the debate and a helpful (...) metaphysics for understanding the incarnation. -/- In Defense of Conciliar Christology discusses three types of philosophical objections to Conciliar Christology. Firstly, it highlights the fundamental philosophical problem facing Christology: how can one thing be both God and man, when anything deserving to be called "God" must have certain attributes, and yet it seems that nothing that can aptly be called "man" can have those same attributes? It then considers the argument that if the Second Person of the Holy Trinity were immutable or atemporal, as Conciliar Christology requires, then that Person could not become anything, and thus could not become man. Finally, Pawl addresses the objection that if there is a single Christ then there is a single nature or will in Christ. However, if that conditional is true, then Conciliar Christology is false, since it affirms the antecedent of the conditional to be true, but denies the truth of the consequent. Pawl defends Conciliar Christology against these charges, arguing that all three philosophical objections fail to show Conciliar Christology inconsistent or incoherent. (shrink)

Isn't probability 1 certainty? If the probability is objective, so is the certainty: whatever has chance 1 of occurring is certain to occur. Equivalently, whatever has chance 0 of occurring is certain not to occur. If the probability is subjective, so is the certainty: if you give credence 1 to an event, you are certain that it will occur. Equivalently, if you give credence 0 to an event, you are certain that it will not occur. And so on for other (...) kinds of probability, such as evidential probability. The formal analogue of this picture is the regularity constraint: a probability distribution over sets of possibilities is regular just in case it assigns probability 0 only to the null set, and therefore probability 1 only to the set of all possibilities. (shrink)

The possibility of justified true belief without knowledge is normally motivated by informally classified examples. This paper shows that it can also be motivated more formally, by a natural class of epistemic models in which both knowledge and justified belief are represented. The models involve a distinction between appearance and reality. Gettier cases arise because the agent's ignorance increases as the gap between appearance and reality widens. The models also exhibit an epistemic asymmetry between good and bad cases that sceptics (...) seem to ignore or deny. (shrink)

The paper challenges the inferentialist account of concept possession that Paul Boghossian takes as a premise in his account of the transmission of justification by deductive reasoning in his paper 'Blind Reasoning'. Unorthodox speakers who reject the inferences in an alleged possession condition can still have the concept by understanding a word for it. In that sense, the inferences are not analytic. Inferentialist accounts of logical constants, theoretical terms (using the Ramsey-Carnap-Lewis method) and pejorative expressions such as 'Boche' are examined (...) and rejected. It is suggested that epistemological questions cannot be reduced to questions in the theory of thought and meaning. (shrink)

The full-text of this article is not currently available in ORA, but you may be able to access the article via the publisher copy link on this record page. N.B. Prof Williamson is now based at the Faculty of Philosophy, University of Oxford.

Isn't probability 1 certainty? If the probability is objective, so is the certainty: whatever has chance 1 of occurring is certain to occur. Equivalently, whatever has chance 0 of occurring is certain not to occur. If the probability is subjective, so is the certainty: if you give credence 1 to an event, you are certain that it will occur. Equivalently, if you give credence 0 to an event, you are certain that it will not occur. And so on for other (...) kinds of probability, such as evidential probability. The formal analogue of this picture is the regularity constraint: a probability distribution over sets of possibilities is regular just in case it assigns probability 0 only to the null set, and therefore probability 1 only to the set of all possibilities. (shrink)

Donald Davidson's theory of mind is widely regarded as a normative theory. This is a something of a confusion. Once a distinction has been made between the categorisation scheme of a norm and the norm's force-maker, it becomes clear that a Davidsonian theory of mind is not a normative theory after all. Making clear the distinction, applying it to Davidson's theory of mind, and showing its significance are the main purposes of this paper. In the concluding paragraphs, a sketch is (...) given of how a truly normative Davidsonian theory of mind might be formulated. (shrink)

1 Department of Philosophy, University of Dundee, Dundee DD1 4HN. [email protected].

Many advocates of non-classical logic for reasons external to mathematics claim that their proposed revisions are consistent with the use of classical logic within pure mathematics. Doubts are raised about such claims, concerning the applicability of pure mathematics to natural and social science. -/- .

Teleosemantics has a problem: it holds that to have a mind one must have a history, often a long evolutionary history. The solution to the problem is for teleosemanticists to give up on natural selection as the source of natural norms (functions) for neural structures, and to find a different source of natural norms which is not essentially history-involving. Such a source in fact exists, in cybernetic governance. This paper argues for the existence of natural norms derived from cybernetic governance, (...) parallel to natural norms derived from natural selection but distinct from them, and shows how such norms could save teleosemantics from its over-emphasis on history. (shrink)

THERE IS WIDESPREAD AGREEMENT among historians that the writings of Robert Boyle (1697-1691) constitute a valuable archive for understanding the concerns of seventeenth-century British natural philosophers. His writings have often been seen as representing, in one fashion or another, all of the leading intellectual currents of his day. ~ There is somewhat less consensus, however, on the proper historiographic method for interpreting these writings, as well as on the specific details of the beliefs expressed in them. Studies seeking to explicate (...) Boyle's thought have been, roughly speaking, of two general sorts. On the one hand there are those studies of a broadly "intellectualist" orientation which situate his natural philosophy within the intellectual context provided by metaphysics, religion, and early modern science. In this connection his corpuscularianism has been shown to be motivated by specific epistemological, theological, as well as empirical concerns. One of the central aims of such studies has been to show that apparently discordant elements in his scientific thought are rendered coherent by referring them to such "non-scientific" commitments. Among studies of this sort might be mentioned the works of John Hedley Brooke, E. A. Burtt, Gary B. Deason, J. E. McGuire, R. Hooykaas, Robert H. Kargon, Eugene M. Klaaren, P. M. Rattansi, and Richard S. Westfall. (shrink)

The propositional attitudes are attitudes such as believing and desiring, taken toward propositions such as the proposition that snow flurries are expected, or that the Prime Minister likes poutine. Collectively, our views about the propositional attitudes make up much of folk psychology, our everyday theory of how the mind works.

This note is to show that a well-known point about David Lewis’s (1986) modal realism applies to Timothy Williamson’s (1998; 2002) theory of necessary existents as well.1 Each theory, together with certain “recombination” principles, generates individuals too numerous to form a set. The simplest version of the argument comes from Daniel Nolan (1996).2 Assume the following recombination principle: for each cardinal number, ν, it’s possible that there exist ν nonsets. Then given Lewis’s modal realism it follows that there (...) can be no set of all (that is, Absolutely All) the nonsets. For suppose for reductio that there were such a set, A; let ν be A’s cardinality; and let µ be any cardinal number larger than ν. By the recombination principle, it’s possible that there exist µ nonsets; by modal realism, there exists a possible world containing, as parts, µ nonsets; each of these nonsets is a member of A. (shrink)

Various reflections on the nature of consciousness, partly inspired by Alastair Hannay's views on the subject, are presented. In particular, its reality as a distinct non-physical existence is defended against such alternatives as have dominated philosophy for many years. The main difficulty in such a defense concerns the contingency it seems to imply as to the relations between consciousness and its expression in behaviour. But it only implies such contingency if some version of the Humean principle that there cannot be (...) necessary connections between distinct existences is assumed. It is more promising to see this relation as the falsification of this Humean principle and thus avoid what seems the main recommendation of behaviourism, functionalism etc. Some final reflections on the nature of the physical suggest that something like consciousness may be the noumenal essence of the physical in general. (shrink)

Dummett's account of the semantic paradoxes in terms of his theory of indefinitely extensible concepts is compared with Bürge's account in terms of indexicality. Dummett's appeal to intuitionistic logic does not block the paradoxes but Bürge's attempt to avoid the Strengthened Liar is unconvincing. It is argued that in order to avoid the Strengthened Liar and other semantic paradoxes involving nonindexical expressions (constants), one must postulate that when we reflect on the paradoxes there are slight shifts in the meaning (not (...) just reference) we ascribe to metalinguistic expressions (in particular 'say', and derivatively 'true' and 'false'). Consideration of metaphor and gradual linguistic change suggests that such semantic shifts are consistent with language-learning and communication. On this account there is no threat to classical logic, bivalence or the fundamental principles governing 'true' and 'false'. (shrink)

This book is intended as a philosophical introduction to geo-ontologies, in response to their increasing diffusion within the contemporary debate, where philosophy plays a fundamental, though still unexplored, role. Accordingly, the first part offers a short overview of the ontological background of geo-ontologies, which comprehends computer science, philosophy and geography. The second part is devoted to describe the ontology of geography, to define notions such as geographical entities and boundaries, and to trace some philosophical tools useful for spatial representation. The (...) third part investigates the emerging of geo-ontologies from the spatial turn and is concerned with a taxonomy for geo-ontologies grounded on some fundamental geographical distinctions. Finally, the last part presents the emergence of Digital Humanities and the consequent proliferation of geographical projects focused on the ancient world, in particular Greek and Roman. (shrink)