Frege's theorem in plural logic

A version of Frege's theorem can be proved in a plural logic with pair abstraction. We talk through this and discuss the philosophical implications of the result

Frege's theorem in plural logic

A version of Frege's theorem can be proved in a plural logic with pair abstraction. We talk through this and discuss the philosophical implications of the result

Plural Logic and Sensitivity to Order

International audienceSentences that exhibit sensitivity to order (e.g. "John and Mary arrived at school in that order" and "Mary and John arrived at school in that order") present a challenge for the standard formulation of plural logic. In response, some authors have advocated new versions of plural logic based on more fine-grained notions of plural reference, such as serial reference [Hewitt 2012] and articulated reference [Ben-Yami 2013]. The aim of this article is to show that sensitivity to order should be accounted for without altering the standard formulation of plural logic. In particular, sensitivity to order does not call for a more fine-grained notion of plural reference. We point out that the phenomenon in question is quite broad and that current proposals are not equipped to deal with the full range of cases in which order plays a role. Then we develop an alternative, unified account, which locates the phenomenon not in the way in which plural terms can refer, but in the meaning of special expressions such as in that order and respectively

Frege's Theorem in Plural Logic

We note that a plural version of logicism about arithmetic is suggested by the standard reading of Hume's Principle in terms of `the number of Fs/Gs'. We lay out the resources needed to prove a version of Frege's principle in plural, rather than second-order, logic. We sketch a proof of the theorem and comment philosophically on the result, which sits well with a metaphysics of natural numbers as plural properties

Executing Gödel’s programme in set theory

The study of set theory (a mathematical theory of infinite collections) has garnered a great deal of philosophical interest since its development. There are several reasons for this, not least because it has a deep foundational role in mathematics; any mathematical statement (with the possible exception of a few controversial examples) can be rendered in set-theoretic terms. However, the fruitfulness of set theory has been tempered by two difficult yet intriguing philosophical problems: (1.) the susceptibility of naive formulations of sets to contradiction, and (2.) the inability of widely accepted set-theoretic axiomatisations to settle many natural questions. Both difficulties have lead scholars to question whether there is a single, maximal Universe of sets in which all set-theoretic statements are determinately true or false (often denoted by ‘V ’). This thesis illuminates this discussion by showing just what is possible on the ‘one Universe’ view. In particular, we show that there are deep relationships between responses to (1.) and the possible tools that can be used in resolving (2.). We argue that an interpretation of extensions of V is desirable for addressing (2.) in a fruitful manner. We then provide critical appraisal of extant philosophical views concerning (1.) and (2.), before motivating a strong mathematical system (known as‘Morse-Kelley’ class theory or ‘MK’). Finally we use MK to provide a coding of discourse involving extensions of V , and argue that it is philosophically virtuous. In more detail, our strategy is as follows: Chapter I (‘Introduction’) outlines some reasons to be interested in set theory from both a philosophical and mathematical perspective. In particular, we describe the current widely accepted conception of set (the ‘Iterative Conception’) on which sets are formed successively in stages, and remark that set-theoretic questions can be resolved on the basis of two dimensions: (i) how ‘high’ V is (i.e. how far we go in forming stages), and (ii) how ‘wide’ V is (i.e. what sets are formed at successor stages). We also provide a very coarse-grained characterisation of the set-theoretic paradoxes and remark that extensions of universes in both height and width are relevant for our understanding of (1.) and (2.). We then present the different motivations for holding either a ‘one Universe’ or ‘many universes’ view of the subject matter of set theory, and argue that there is a stalemate in the dialectic. Instead we advocate filling out each view in its own terms, and adopt the ‘one Universe’ view for the thesis. Chapter II (‘G¨odel’s Programme’) then explains the Universist project for formulating and justifying new axioms concerning V . We argue that extensions of V are relevant to both aspects of G¨odel’s Programme for resolving independence. We also identify a ‘Hilbertian Challenge’ to explain how we should interpret extensions of V , given that we wish to use discourse that makes apparent reference to such nonexistent objects. Chapter III (‘Problematic Principles’) then lends some mathematical precision to the coarse-grained outline of Chapter I, examining mathematical discourse that seems to require talk of extensions of V . Chapter IV (‘Climbing above V ?’), examines some possible interpretations of height extensions of V . We argue that several such accounts are philosophically problematic. However, we point out that these difficulties highlight two constraints on resolution of the Hilbertian Challenge: (i) a Foundational Constraint that we do not appeal to entities not representable using sets from V , and (ii) an Ontological Constraint to interpret extensions of V in such a way that they are clearly different from ordinary sets. 5 Chapter V (‘Broadening V ’s Horizons?’), considers interpretations of width extensions. Again, we argue that many of the extant methods for interpreting this kind of extension face difficulties. Again, however, we point out that a constraint is highlighted; a Methodological Constraint to interpret extensions of V in a manner that makes sense of our naive thinking concerning extensions, and links this thought to truth in V . We also note that there is an apparent tension between the three constraints. Chapter VI (‘A Theory of Classes’) changes tack, and provides a positive characterisation of apparently problematic ‘proper classes’ through the use of plural quantification. It is argued that such a characterisation of proper class discourse performs well with respect to the three constraints, and motivates the use of a relatively strong class theory (namely MK). Chapter VII (‘V -logic and Resolution’) then puts MK to work in interpreting extensions of V . We first expand our logical resources to a system called V -logic, and show how discourse concerning extensions can be thereby represented. We then show how to code the required amount of V -logic usingMK. Finally, we argue that such an interpretation performs well with respect to the three constraints. Chapter VIII (‘Conclusions’) reviews the thesis and makes some points regarding the exact dialectical situation. We argue that there are many different philosophical lessons that one might take from the thesis, and are clear that we do not commit ourselves to any one such conclusion. We finally provide some open questions and indicate directions for future research, remarking that the thesis opens the way for new and exciting philosophical and mathematical discussion

Executing Gödel's Programme in Set Theory

The study of set theory (a mathematical theory of infinite collections) has garnered a great deal of philosophical interest since its development. There are several reasons for this, not least because it has a deep foundational role in mathematics; any mathematical statement (with the possible exception of a few controversial examples) can be rendered in set-theoretic terms. However, the fruitfulness of set theory has been tempered by two difficult yet intriguing philosophical problems: (1.) the susceptibility of naive formulations of sets to contradiction, and (2.) the inability of widely accepted set-theoretic axiomatisations to settle many natural questions. Both difficulties have lead scholars to question whether there is a single, maximal Universe of sets in which all set-theoretic statements are determinately true or false (often denoted by ‘V ’). This thesis illuminates this discussion by showing just what is possible on the ‘one Universe’ view. In particular, we show that there are deep relationships between responses to (1.) and the possible tools that can be used in resolving (2.). We argue that an interpretation of extensions of V is desirable for addressing (2.) in a fruitful manner. We then provide critical appraisal of extant philosophical views concerning (1.) and (2.), before motivating a strong mathematical system (known as‘Morse-Kelley’ class theory or ‘MK’). Finally we use MK to provide a coding of discourse involving extensions of V , and argue that it is philosophically virtuous. In more detail, our strategy is as follows: Chapter I (‘Introduction’) outlines some reasons to be interested in set theory from both a philosophical and mathematical perspective. In particular, we describe the current widely accepted conception of set (the ‘Iterative Conception’) on which sets are formed successively in stages, and remark that set-theoretic questions can be resolved on the basis of two dimensions: (i) how ‘high’ V is (i.e. how far we go in forming stages), and (ii) how ‘wide’ V is (i.e. what sets are formed at successor stages). We also provide a very coarse-grained characterisation of the set-theoretic paradoxes and remark that extensions of universes in both height and width are relevant for our understanding of (1.) and (2.). We then present the different motivations for holding either a ‘one Universe’ or ‘many universes’ view of the subject matter of set theory, and argue that there is a stalemate in the dialectic. Instead we advocate filling out each view in its own terms, and adopt the ‘one Universe’ view for the thesis. Chapter II (‘G¨odel’s Programme’) then explains the Universist project for formulating and justifying new axioms concerning V . We argue that extensions of V are relevant to both aspects of G¨odel’s Programme for resolving independence. We also identify a ‘Hilbertian Challenge’ to explain how we should interpret extensions of V , given that we wish to use discourse that makes apparent reference to such nonexistent objects. Chapter III (‘Problematic Principles’) then lends some mathematical precision to the coarse-grained outline of Chapter I, examining mathematical discourse that seems to require talk of extensions of V . Chapter IV (‘Climbing above V ?’), examines some possible interpretations of height extensions of V . We argue that several such accounts are philosophically problematic. However, we point out that these difficulties highlight two constraints on resolution of the Hilbertian Challenge: (i) a Foundational Constraint that we do not appeal to entities not representable using sets from V , and (ii) an Ontological Constraint to interpret extensions of V in such a way that they are clearly different from ordinary sets. 5 Chapter V (‘Broadening V ’s Horizons?’), considers interpretations of width extensions. Again, we argue that many of the extant methods for interpreting this kind of extension face difficulties. Again, however, we point out that a constraint is highlighted; a Methodological Constraint to interpret extensions of V in a manner that makes sense of our naive thinking concerning extensions, and links this thought to truth in V . We also note that there is an apparent tension between the three constraints. Chapter VI (‘A Theory of Classes’) changes tack, and provides a positive characterisation of apparently problematic ‘proper classes’ through the use of plural quantification. It is argued that such a characterisation of proper class discourse performs well with respect to the three constraints, and motivates the use of a relatively strong class theory (namely MK). Chapter VII (‘V -logic and Resolution’) then puts MK to work in interpreting extensions of V . We first expand our logical resources to a system called V -logic, and show how discourse concerning extensions can be thereby represented. We then show how to code the required amount of V -logic usingMK. Finally, we argue that such an interpretation performs well with respect to the three constraints. Chapter VIII (‘Conclusions’) reviews the thesis and makes some points regarding the exact dialectical situation. We argue that there are many different philosophical lessons that one might take from the thesis, and are clear that we do not commit ourselves to any one such conclusion. We finally provide some open questions and indicate directions for future research, remarking that the thesis opens the way for new and exciting philosophical and mathematical discussion

From plurals to superplurals: in defence of higher-level plural logic

Plural Logic is an extension of First-Order Logic with plural terms and quantifiers. When its plural terms are interpreted as denoting more than one object at once, Plural Logic is usually taken to be ontologically innocent: plural quantifiers do not require a domain of their own, but range plurally over the first-order domain of quantification. Given that Plural Logic is equi-interpretable with Monadic Second-Order Logic, it gives us its expressive power at the low ontological cost of a first-order language. This makes it a valuable tool in various areas of philosophy. Some authors believe that Plural Logic can be extended into an even more expressive logic, Higher-Level Plural Logic, by adding higher-level plural terms and quantifiers to it. The basic idea is that second-level plurals stand to plurals like plurals stand to singulars (analogously for higher levels). Allegedly, Higher-Level Plural Logic enjoys the expressive power of type theory while, again, committing us only to the austere ontology of a first-order language. Were this really the case, Higher-Level Plural Logic would be a very useful tool, extending and strengthening some of the applications of Plural Logic. However, while the notions of plural reference and quantification enjoy widespread acceptance today, their higher-level counterparts have been received with scepticism. The main objection raised against them is that higher-level plural reference is unintelligible. This has been argued, among others, on the grounds that there are no higher-level plurals in natural language and that, if there were any, they could be eliminated. In this thesis, after introducing the debate on plurals in Chapters 1 and 2, I turn to defending the legitimacy of the notion of higher-level plural reference. To this end, in Chapter 3, I present and elucidate the notion. Next, in Chapter 4, I show that some natural languages clearly contain these expressions and that they do so in an ineliminable manner. Finally, in Chapters 5 and 6, I develop a semantics for higher-level plurals that employs only devices previously well-understood by English speakers. To finish, in Chapter 7, I describe an application of Higher-level Plural Logic: a strengthening of the neo-Fregean programme. After describing my proposal, I turn to the issue of the logical status of this formalism and defend an optimistic take on the matter