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)

Belief in propositions has had a long and distinguished history in analytic philosophy. Three of the founding fathers of analytic philosophy, Gottlob Frege, Bertrand Russell, and G. E. Moore, believed in propositions. Many philosophers since then have shared this belief; and the belief is widely, though certainly not universally, accepted among philosophers today. Among contemporary philosophers who believe in propositions, many, and perhaps even most, take them to be structured entities with individuals, properties, and relations as constituents. For example, the (...) proposition that Glenn loves Tracy has Glenn, the loving relation, and Tracy as constituents. What is it, then, that binds these constituents together and imposes structure on them? And if the proposition that Glenn loves Tracy is distinct from the proposition that Tracy loves Glenn yet both have the same constituents, what is about the way these constituents are structured or bound together that makes them two different propositions? In The Nature and Structure of Content, Jeffrey C. King formulates a detailed account of the metaphysical nature of propositions, and provides fresh answers to the above questions. In addition to explaining what it is that binds together the constituents of structured propositions and imposes structure on them, King deals with some of the standard objections to accounts of propositions: he shows that there is no mystery about what propositions are; that given certain minimal assumptions, it follows that they exist; and that on his approach, we can see how and why propositions manage to have truth conditions and represent the world as being a certain way. The Nature and Structure of Content also contains a detailed account of the nature of tense and modality, and provides a solution to the paradox of analysis. Scholars and students working in the philosophy of mind and language will find this book rewarding reading. (shrink)

This brief paperback is designed for symbolic/formal logic courses. It features the tree method proof system developed by Jeffrey. The new edition contains many more examples and exercises and is reorganized for greater accessibility.

Epicureanism after the generation of its founders has been characterised as dogmatic, uncreative and static. But this volume brings together work from leading classicists and philosophers that demonstrates the persistent interplay in the school between historical and contemporary influences from outside the school and a commitment to the founders' authority. The interplay begins with Epicurus himself, who made arresting claims of intellectual independence, yet also admitted to taking over important ideas from predecessors, and displayed more receptivity than is usually thought (...) to those of his contemporaries. The same principles of autonomy and openness figure importantly in the three major areas of focus in these essays: theology, politics and the emotions. (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)

Thats logic. LEWIS CARROLL, Through the Looking Glass 1-1. The fundamental problem of this work is the question of the nature of scientific inference.

Data science, and the related field of big data, is an emerging discipline involving the analysis of data to solve problems and develop insights. This rapidly growing domain promises many benefits to both consumers and businesses. However, the use of big data analytics can also introduce many ethical concerns, stemming from, for example, the possible loss of privacy or the harming of a sub-category of the population via a classification algorithm. To help address these potential ethical challenges, this paper maps (...) and describes the main ethical themes that were identified via systematic literature review. It then identifies a possible structure to integrate these themes within a data science project, thus helping to provide some structure in the on-going debate with respect to the possible ethical situations that can arise when using data science analytics. (shrink)

Certain realists about properties and relations identify them with universals. Furthermore, some hold that for a wide range of meaningful predicates, the semantic contribution to the propositions expressed by the sentences in which those predicates figure is the universal expressed by the predicate. I here address ontological issues raised by predicates first introduced to us via works of fiction and whether the universal realist should accept that any such predicates express universals. After assessing arguments by Braun, D. and Sawyer, S. (...) for fictional universal anti-realism, I propose a novel, Kripke-inspired argument for the same conclusion. I ultimately defend the claim that while this argument presents the strongest case for fictional universal anti-realism, it is nonetheless unsound. I conclude that nothing stands in the way of accepting that some fictional predicates express fictional universals. (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.

Direct neurological and especially imaging-driven investigations into the structures essential to naturally occurring cognitive systems in their development and operation have motivated broadening interest in the potential for artificial consciousness modeled on these systems. This first paper in a series of three begins with a brief review of Boltuc’s (2009) “brain-based” thesis on the prospect of artificial consciousness, focusing on his formulation of h-consciousness. We then explore some of the implications of brain research on the structure of consciousness, finding limitations (...) in biological approaches to the study of consciousness. Looking past these limitations, we introduce research in artificial consciousness designed to test for the emergence of consciousness, a phenomenon beyond the purview of the study of existing biological systems. (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)

This third paper locates the synthetic neurorobotics research reviewed in the second paper in terms of themes introduced in the first paper. It begins with biological non-reductionism as understood by Searle. It emphasizes the role of synthetic neurorobotics studies in accessing the dynamic structure essential to consciousness with a focus on system criticality and self, develops a distinction between simulated and formal consciousness based on this emphasis, reviews Tani and colleagues' work in light of this distinction, and ends by forecasting (...) the increasing importance of synthetic neurorobotics studies for cognitive science and philosophy of mind going forward, finally in regards to most- and myth-consciousness. (shrink)