Structural realism has been proposed as an epistemological position interpolating between realism and sceptical anti-realism about scientific theories. The structural realist who accepts a scientific theory thinks that is empirically correct, and furthermore is a realist about the ‘structural content’ of . But what exactly is ‘structural content’? One proposal is that the ‘structural content’ of a scientific theory may be associated with its Ramsey sentence (). However, Demopoulos and Friedman have argued, using ideas drawn from Newman's earlier criticism of (...) Russell's structuralism, that this move fails to achieve an interesting intermediate position between realism and anti-realism. Rather, () adds little content beyond the instrumentalistically acceptable claim that the theory is empirically adequate. Here, I formulate carefully the crucial claim of Demopoulos and Friedman, and show that the Ramsey sentence () is true just in case possesses a full model which is empirically correct and satisfies a certain cardinality condition on its theoretical domain. This suggests that structural realism is not a position significantly different from the anti-realism it attempts to distinguish itself from. Introduction Technical framework Ramsification Empirical adequacy Ramsification empirical adequacy + cardinality constraint Conclusion. (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)

It is argued that Yablo’s Paradox is not strictly paradoxical, but rather ‘ω-paradoxical’. Under a natural formalization, the list of Yablo sentences may be constructed using a diagonalization argument and can be shown to be ω-inconsistent, but nonetheless consistent. The derivation of an inconsistency requires a uniform fixed-point construction. Moreover, the truth-theoretic disquotational principle required is also uniform, rather than the local disquotational T-scheme. The theory with the local disquotation T-scheme applied to individual sentences from the Yablo list is also (...) consistent. (shrink)

Any (1-)consistent and sufficiently strong system of first-order formal arithmetic fails to decide some independent Gödel sentence. We examine consistent first-order extensions of such systems. Our purpose is to discover what is minimally required by way of such extension in order to be able to prove the Gödel sentence in a nontrivial fashion. The extended methods of formal proof must capture the essentials of the so-called 'semantical argument' for the truth of the Gödel sentence. We are concerned to show that (...) the deflationist has at his disposal such extended methods--methods which make no use or mention of a truth-predicate. (edited). (shrink)

The notion of strict identity is sometimes given an explicit second-order definition: objects with all the same properties are identical. Here, a somewhat different problem is raised: Under what conditions is the identity relation on the domain of a structure first-order definable? A structure may have objects that are distinct, but indiscernible by the strongest means of discerning them given the language (the indiscernibility formula). Here a number of results concerning the indiscernibility formula, and the definability of identity, are collected (...) and a number of applications discussed. (shrink)

This paper aims to study the foundations of applied mathematics, using a formalized base theory for applied mathematics: ZFCAσ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathsf {ZFCA}_{\sigma }$$\end{document} with atoms, where the subscript used refers to a signature specific to the application. Examples are given, illustrating the following five features of applied mathematics: comprehension principles, application conditionals, representation hypotheses, transfer principles and abstract equivalents.

This is a response to a paper “Paradox without satisfaction”, Analysis 63, 152-6 (2003) by Otavio Bueno and Mark Colyvan on Yablo’s paradox. I argue that this paper makes several substantial mathematical errors which vitiate the paper. (For the technical details, see [12] below.).

Tarski’s Indefinability Theorem can be generalized so that it applies to many-valued languages. We introduce a notion of strong semantic self-representation applicable to any (sufficiently rich) interpreted many-valued language L. A sufficiently rich interpreted many-valued language L is SSSR just in case it has a function symbol n(x) such that, for any f Sent(L), the denotation of the term n(“f”) in L is precisely ||f||L, the semantic value of f in L. By a simple diagonal construction (finding a sentence l (...) such that l is equivalent to n(“l”) T), it is shown that no such language strongly represents itself semantically. Hence, no such language can be its own metalanguage. (shrink)

The following inference is valid: There are exactly 101 dalmatians, There are exactly 100 food bowls, Each dalmatian uses exactly one food bowl Hence, at least two dalmatians use the same food bowl. Here, “there are at least 101 dalmatians” is nominalized as, "x1"x2…."x100$y(Dy & y ¹ x1 & y ¹ x2 & … & y ¹ x100) and “there are exactly 101 dalmatians” is nominalized as, "x1"x2…."x100$y(Dy & y ¹ x1 & y ¹ x2 & … & y ¹ (...) x100) & Ø"x1"x2…."x101$y(Dy & y ¹ x1 & y ¹ x2 & … & y ¹ x101). This is abbreviated $101xDx. The validity of the above inference corresponds to the valid formula, PHP(100): [$101xDx & $100xFx & "x(Dx ® Ff(x))] ® $x1$x2(Dx1 & Dx2 & x1 ¹ x2 & f(x1) = f(x2)). More generally, for variable n, the formula PHP(n) is PHP(n): [$n+1xDx & $nxFx & "x(Dx ® Ff(x))] ® $x1$x2(Dx1 & Dx2 & x1 ¹ x2 & f(x1) = f(x2)). A mathematical proof that PHP(n) is valid, for all n > 0, is quite short (less than a page), but refers to numbers, functions and sets. It uses the Pigeonhole Principle. This explains why PHP(n) is valid, for all n>0. However, I estimate that a predicate calculus derivation of PHP(100), using natural deduction, say, would require around 107 symbols. Unfeasibility Problem: nominalism is the radical anti-realist view that there are no numbers, functions or sets. So, how could a nominalist know that PHP(100) is valid, without directly performing the rather long derivation? Can the nominalist “ride piggyback” on the standard mathematical proof? If so, how is this justified? (shrink)

Sometimes structures or theories are formulated with different sets of primitives and yet are definitionally equivalent. In a sense, the transformations between such equivalent formulations are rather like basis transformations in linear algebra or co-ordinate transformations in geometry. Here an analogous idea is investigated. Let a relational signature \ be given. For a set \ of \-formulas, we introduce a corresponding set \ of new relation symbols and a set of explicit definitions of the \ in terms of the \. (...) This is called a definition system, denoted \. A definition system \ determines a translation function\. Any \-structure A can be uniquely definitionally expanded to a model \, called \. The reduct \ to the Q-symbols is called the definitional image\ of A. Likewise, a theory T in \ may be extended a definitional extension \; the restriction of this extension \ to \ is called the definitional image\ of T. If \ and \ are in disjoint signatures and \, we say that \ and \ are definitionally equivalent and \). Some results relating these notions are given, culminating in two characterization theorems for the definitional equivalence of structures and theories. (shrink)

This dialogue concerns metasemantics and language cognition. It defends a Lewisian conception of languages as abstract entities (Lewis 1975), arguing that semantic facts are necessities (Soames 1984), and therefore not naturalistically reducible. It identifies spoken languages as idiolects, in line roughly with Chomskyan I-languages. It relocates traditional metasemantic indeterminacy arguments as indeterminacies of what language an agent speaks or cognizes. Finally, it aims to provide a theoretical analysis of the cognizing relation in terms of the agent's assigning certain meanings to (...) strings. (shrink)

Instrumentalist nominalism responds to the indispensability arguments by rejecting the demand that successful mathematicized scientific theories be nominalized, and instead claiming merely that such theories are nominalistically adequate: the concreta behave ‘as if’ the theory is true. This article examines some definitions of the concept of nominalistic adequacy and concludes with some considerations against instrumentalist nominalism.

Lucas and Redhead ([2007]) announce that they will defend the views of Redhead ([2004]) against the argument by Panu Raatikainen ([2005]). They certainly re-state the main claims of Redhead ([2004]), but they do not give any real arguments in their favour, and do not provide anything that would save Redhead’s argument from the serious problems pointed out in (Raatikainen [2005]). Instead, Lucas and Redhead make a number of seemingly irrelevant points, perhaps indicating a failure to understand the logico-mathematical points at (...) issue. (shrink)

In “Truth – A Traditional Debate Reviewed”, Crispin Wright proposed an inductive definition of “coherence truth” for arithmetic relative to an arithmetic base theory B. Wright’s definition is in fact a notational variant of the usual Tarskian inductive definition, except for the basis clause for atomic sentences. This paper provides a model-theoretic characterization of the resulting sets of sentences "cohering" with a given base theory B. These sets are denoted WB. Roughly, if B satisfies a certain minimal condition, then WB (...) is the Th, where M is the canonical model of the set At of atomic sentences provable in B. The paper also shows that the disquotational T-scheme is provable from Wright’s inductive definition just in case the base theory B is sound and complete for arithmetic atomic sentences. (shrink)

In the Tarskian theory of truth, the strengthened liar sentence is a theorem. More generally, any formalized truth theory which proves the full, self-applicative scheme True(“f”) f will prove the strengthened liar sentence. (This scheme is sometimes called (T-Out).).

In the Tarskian theory of truth, the strengthened liar sentence is a theorem. More generally, any formalized truth theory which proves the full, self-applicative scheme True f will prove the strengthened liar sentence..).

Certain translational T-schemes of the form True « f, where f can be almost any translation you like of f, will be a conservative extension of Peano arithmetic. I have an inkling that this means something philosophically, but I don’t understand my own inkling.

Theorem 1 of Ketland 1999 is not quite correct as stated. The theorem would imply that the disquotational T-scheme – suitably restricted to avoid the liar paradox – is conservative over pure logic. But it has been pointed out (e.g. Halbach 2001, “How Innocent is Deflationism?”, Synthese 126, pp. 179-181) that this is not the case, for one can prove ∃x∃y(x ≠ y) from the T-scheme (lemma 2 below).

Is the restricted, consistent, version of the T-scheme sufficient for an ‘implicit definition’ of truth? In a sense, the answer is yes (Haack 1978 , Quine 1953 ). Section 4 of Ketland 1999 mentions this but gives a result saying that the T-scheme does not implicitly define truth in the stronger sense relevant for Beth’s Definability Theorem. This insinuates that the T-scheme fares worse than the compositional truth theory as an implicit definition. However, the insinuation is mistaken. For, as (...) Bays rightly points out, the result given extends to the compositional truth theory also. So, as regards implicit definability, both kinds of truth theory are equivalent. Some further discussion of this topic is mentioned (Gupta 2008 , Ketland 2003 , McGee 1991 ), all in agreement with Bays’s analysis. (shrink)

We can modify Hume’s Principle in the same manner that George Boolos suggested for modifying Frege’s Basic Law V. This leads to the principle Small Hume. Then, we can show that Small Hume is interderivable with Hume’s Principle.

A standard formalization of a scientific theory is a system of axioms for that theory in a first-order language (possibly many-sorted; possibly with the membership primitive $$\in$$ ). Suppes (in: Carvallo M (ed) Nature, cognition and system II. Kluwer, Dordrecht, 1992) expressed skepticism about whether there is a “simple or elegant method” for presenting mathematicized scientific theories in such a standard formalization, because they “assume a great deal of mathematics as part of their substructure”. The major difficulties amount to these. (...) First, as the theories of interest are mathematicized, one must specify the underlying applied mathematics base theory, which the physical axioms live on top of. Second, such theories are typically geometric, concerning quantities or trajectories in space/time: so, one must specify the underlying physical geometry. Third, the differential equations involved generally refer to coordinate representations of these physical quantities with respect to some implicit coordinate chart, not to the original quantities. These issues may be resolved. Once this is done, constructing standard formalizations is not so difficult—at least for the theories where the mathematics has been worked out rigorously. Here we give what may be claimed to be a simple and elegant means of doing that. This is for mathematicized scientific theories comprising differential equations for $$\mathbb{R}$$ -valued quantities Q (that is, scalar fields), defined on n (“spatial” or “temporal”) dimensions, taken to be isomorphic to the usual Euclidean space $$\mathbb{R}^n$$. For illustration, I give standard (in a sense, “text-book”) formalizations: for the simple harmonic oscillator equation in one-dimension and for the Laplace equation in two dimensions. (shrink)

In Part I of this paper, I assumed we begin with a signature $$P = \{P_i\}$$ P = { P i } and the corresponding language $$L_P$$ L P, and introduced the following notions: a definition system$$d_{\Phi }$$ d Φ for a set of new predicate symbols $$Q_i$$ Q i, given by a set $$\Phi = \{\phi _i\}$$ Φ = { ϕ i } of defining $$L_P$$ L P -formulas \leftrightarrow \phi _i)$$ ∀ x ¯ ↔ ϕ i ) ); (...) a corresponding translation function$$\tau _{\Phi }: L_Q \rightarrow L_P$$ τ Φ : L Q → L P ; the corresponding definitional image operator$$D_{\Phi }$$ D Φ, applicable to $$L_P$$ L P -structures and $$L_P$$ L P -theories; and the notion of definitional equivalence itself: for structures $$A + d_{\Phi } \equiv B + d_{\Theta }$$ A + d Φ ≡ B + d Θ ; for theories, $$T_1 + d_{\Phi } \equiv T_2 + d_{\Theta }$$ T 1 + d Φ ≡ T 2 + d Θ. Some results relating these notions were given, ending with two characterizations for definitional equivalence. In this second part, we explain the notion of a representation basis. Suppose a set $$\Phi = \{\phi _i\}$$ Φ = { ϕ i } of $$L_P$$ L P -formulas is given, and $$\Theta = \{\theta _i\}$$ Θ = { θ i } is a set of $$L_Q$$ L Q -formulas. Then the original set $$\Phi $$ Φ is called a representation basis for an $$L_P$$ L P -structure A with inverse $$\Theta $$ Θ iff an inverse explicit definition $$\forall \overline{x} \leftrightarrow \theta _i)$$ ∀ x ¯ ↔ θ i ) is true in $$A + d_{\Phi }$$ A + d Φ, for each $$P_i$$ P i. Similarly, the set $$\Phi $$ Φ is called a representation basis for a $$L_P$$ L P -theory T with inverse $$\Theta $$ Θ iff each explicit definition $$\forall \overline{x} \leftrightarrow \theta _i)$$ ∀ x ¯ ↔ θ i ) is provable in $$T + d_{\Phi }$$ T + d Φ. Some results about representation bases, the mappings they induce and their relationship with the notion of definitional equivalence are given. In particular, we show that $$T_1$$ T 1 is definitionally equivalent to $$T_2$$ T 2, with respect to $$\Phi $$ Φ and $$\Theta $$ Θ, if and only if $$\Phi $$ Φ is a representation basis for $$T_1$$ T 1 with inverse $$\Theta $$ Θ and $$T_2 \equiv D_{\Phi }T_1$$ T 2 ≡ D Φ T 1. (shrink)

This article provides a computational example of a mathematical explanation within science, concerning computational equivalence of programs. In addition, it outlines the logical structure of the reasoning involved in explanations in applied mathematics. It concludes with a challenge that the nominalist provide a nominalistic explanation for the computational equivalence of certain programs.

Certain translational T-schemes of the form True(“f”) « f(f), where f(f) can be almost any translation you like of f, will be a conservative extension of Peano arithmetic. I have an inkling that this means something philosophically, but I don’t understand my own inkling.

In mathematical logic, Craig’s Theorem states that any recursively enumerable theory is recursively axiomatizable. Its epistemological interest concerns its possible use as a method of eliminating “theoretical content” from scientific theories.

This discusses a mistake (concerning what a definition is) in “Grelling’s revenge”, Analysis 64, 251-6 (2004), by Dale Jacquette, who claims that the simple theory of types is inconsistent.

Second-order logic is the extension of first-order logic obtaining by introducing quantification of predicate and function variables.

Ordinarily, in mathematical and scientific practice, the notion of a “theory” is understood as follows: (SCT) Standard Conception of Theories : A theory T is a collection of statements, propositions, conjectures, etc. A theory claims that things are thus and so. The theory may be true, and may be false. A theory T is true if things are as T says they are, and T is false if things are not as T says they are. One can make this Aristotelian (...) explanation more precise, as Tarski showed, in the cases where we understand how to give precise logical analyses of theories, by identifying an interpreted language L, Á) in which T may be formulated. Here L is some formalized language and Á is an L-interpretation. One can define the satisfaction relation ⊨, holding between L-sentences j and L-interpretations, and then define the notion L-sentence j is true in L, Á)” as Á ⊨ j”. What is essential about this is that theories are truth bearers . They are bearers of semantic properties. (shrink)

That believing truly as a matter of luck does not generally constitute knowing has become epistemic commonplace. Accounts of knowledge incorporating this anti-luck idea frequently rely on one or another of a safety or sensitivity condition. Sensitivity-based accounts of knowledge have a well-known problem with necessary truths, to wit, that any believed necessary truth trivially counts as knowledge on such accounts. In this paper, we argue that safety-based accounts similarly trivialize knowledge of necessary truths and that two ways of responding (...) to this problem for safety, issuing from work by Williamson and Pritchard, are of dubious success. (shrink)

Followers of Wittgenstein allegedly once held that a meaningful claim to know that p could only be made if there was some doubt about the truth of p. The correct response to this thesis involved appealing to the distinction between the semantic content of a sentence and features attaching to its use. It is inappropriate to assert a knowledge-claim unless someone in the audience has doubt about what the speaker claims to know. But this fact has nothing to do with (...) the semantic content of knowledgeascriptions; it is entirely explicable by appeal to pragmatic facts about felicitous assertion. (shrink)

Could space consist entirely of extended regions, without any regions shaped like points, lines, or surfaces? Peter Forrest and Frank Arntzenius have independently raised a paradox of size for space like this, drawing on a construction of Cantor’s. I present a new version of this argument and explore possible lines of response.

The Face of the Other and the Trace of God contain essays on the philosophy of Emmanuel Levinas, and how his philosophy intersects with that of other philosophers, particularly Husserl, Kierkegaard, Sartre, and Derrida. This collection is broadly divided into two parts: relations with the other, and the questions of God.

This book offers a concise survey of basic probability theory from a thoroughly subjective point of view whereby probability is a mode of judgment. Written by one of the greatest figures in the field of probability theory, the book is both a summation and synthesis of a lifetime of wrestling with these problems and issues. After an introduction to basic probability theory, there are chapters on scientific hypothesis-testing, on changing your mind in response to generally uncertain observations, on expectations of (...) the values of random variables, on de Finetti's dissolution of the so-called problem of induction, and on decision theory. (shrink)

Few notions are more central to Aquinas’s thought than those of matter and form. Although he invokes these notions in a number of different contexts, and puts them to a number of different uses, he always assumes that in their primary or basic sense they are correlative both with each other and with the notion of a “hylomorphic compound”—that is, a compound of matter (hyle) and form (morphe). Thus, matter is an entity that can have form, form is an entity (...) that can be had by matter, and a hylomorphic compound is an entity that exists when the potentiality of some matter to have form is actualized.1 What is more, Aquinas assumes that the matter of a hylomorphic compound explains certain of its general characteristics, whereas its form explains certain of its more specific characteristics. Thus, the matter of a bronze statue explains the fact that it is bronze, whereas its form explains the fact that it is a statue. Again, the matter of a human being explains the fact that it is a material object, whereas its form explains the specific type of material object it is (namely, human). My aim in this chapter is to provide a systematic introduction to Aquinas’s primary or basic notions of matter and form. To accomplish this aim, I focus on the two main theoretical contexts in which he deploys them—namely, his theory of change and his theory of individuation. In both contexts, as we shall see, Aquinas appeals to matter and form to account for relations of sameness and difference holding between distinct individuals. (shrink)

Richard Jeffrey is beyond dispute one of the most distinguished and influential philosophers working in the field of decision theory and the theory of knowledge. His work is distinctive in showing the interplay of epistemological concerns with probability and utility theory. Not only has he made use of standard probabilistic and decision theoretic tools to clarify concepts of evidential support and informed choice, he has also proposed significant modifications of the standard Bayesian position in order that it provide a (...) better fit with actual human experience. Probability logic is viewed not as a source of judgment but as a framework for explaining the implications of probabilistic judgments and their mutual compatability. This collection of essays spans a period of some 35 years and includes what have become some of the classic works in the literature. There is also one completely new piece, while in many instances Jeffrey includes afterthoughts on the older essays. (shrink)

Jeffrey Barrett presents the most comprehensive study yet of a problem that has puzzled physicists and philosophers since the 1930s.

Eliminativists sometimes invoke evolutionary debunking arguments against ordinary object beliefs, either to help them establish object skepticism or to soften the appeal of commonsense ontology. I argue that object debunkers face a self-defeat problem: their conclusion undermines the scientific support for one of their premises, because evolutionary biology depends on our object beliefs. Using work on reductionism and multiple realizability from the philosophy of science, I argue that it will not suffice for an eliminativist debunker to simply appeal to some (...) object-free surrogate theory of evolution that results from converting any scientific proposition about some object K into a proposition about simples arranged K-wise. In the process, I examine some hazards peculiar to eliminative reductions of scientific theories, and propose a trilemma for eliminativists who attempt to recoup generality for ontologically sparse reducing theories by appealing to pluralities of simples arranged K-wise. The paper is intended to define and develop the object debunker’s self-defeat problem for further study, and to clarify some of the ways sparse and abundant ontologies interact with scientific theory. (shrink)

There are two main ways in which things with minds, like us, differ from things without minds, like tables and chairs. First, we are conscious--there is something that it is like to be us. We instantiate phenomenal properties. Second, we represent, in various ways, our world as being certain ways. We instantiate representational properties. Jeff Speaks attempts to make progress on three questions: What are phenomenal properties? What are representational properties? How are the phenomenal and the representational related?

What degrees of belief does Sleeping Beauty's evidence support? That depends.