Certain puzzling cases have been discussed in the literature recently which appear to support the thought that knowledge can be obtained by way of deduction from a falsehood; moreover, these cases put pressure, prima facie, on the thesis of counter closure for knowledge. We argue that the cases do not involve knowledge from falsehood; despite appearances, the false beliefs in the cases in question are causally, and therefore epistemologically, incidental, and knowledge is achieved despite falsehood. We also show that (...) the principle of counter closure, and the concomitant denial of knowledge from falsehood, is well motivated by considerations in epistemological theory--in particular, by the view that knowledge is first in the epistemological order of things. (shrink)

This paper proposes the feasibility of a second-order approach in cosmology. It is intended to encourage cosmologists to rethink standard ideas in their field, leading to a broader concept of self-organization and of science itself. It is argued, from a cognitive epistemology perspective, that a first-order approach is inadequate for cosmology; study of the universe as a whole must include study of the scientific observer and the process of theorizing. Otherwise, concepts of self-organization at the cosmological scale (...) remain constrained by unacknowledged assumptions and biases. Examples of limiting notions are discussed in the context of alternatives. To include the role of the theorist does not mean reducing science to subjective or sociological terms. On the contrary, second-order science would provide a more complete portrait of nature. The work of cosmologist Lee Smolin is discussed as a candidate example of second-order cosmology. (shrink)

Higher-order evidence is, roughly, evidence of evidence. The idea is that evidence comes in levels. At the first, or lowest, evidential level is evidence of the familiar type—evidence concerning some proposition that is not itself about evidence. At a higher evidential level the evidence concerns some proposition about the evidence at a lower level. Only in relatively recent years has this less familiar type of evidence been explicitly identified as a subject of epistemological focus, and the work on (...) it remains relegated to a small circle of authors and a short stack of published articles—far disproportionate to the attention it deserves. It deserves to occupy center stage for several reasons. First, higher-order evidence frequently arises in a strikingly diverse range of epistemic contexts, including testimony, disagreement, empirical observation, introspection, and memory, among others. Second, in many of the contexts in which it arises, such evidence plays a crucial epistemic role. Third, the precise role it plays is complex, gives rise to a number of interesting epistemological puzzles, and for these reasons remains controversial and is not yet fully understood. As such, higher-order evidence merits systematic investigation. This thesis undertakes such an investigation. It aims to produce a thorough account of higher-order evidence—what it is, how it works, and its epistemic consequences. Chapter 1 serves as a general introduction to the topic and an overview of the existing literature, but primarily aims to further elucidate the concept of higher-order evidence and build a theoretical framework for later chapters. Chapter 2 develops an account of what I call “higher-order support”: the bearing higher-order evidence has, not on corresponding “lower-order evidence” (roughly, the evidence the higher-order evidence is about), but on corresponding “object-level propositions” (roughly, the propositions the higher-order evidence alleges the lower-order evidence to be about). Chapter 3 develops an account of “levels interaction”: the effect on overall support when the different evidential levels combine. Chapter 4 identifies important consequences of the theoretical results of the previous two chapters and applies the theory to four select cases of current epistemological controversy—testimony, memory, the closure of inquiry, and disagreement. (shrink)

In this paper, I consider the so-called Access Problem for Duncan Pritchard’s Epistemological Disjunctivism (2012). After reconstructing Pritchard’s own response to the Access Problem, I argue that in order to assess whether Pritchard’s response is a satisfying one, we first need an account of the notion of ‘Reflective Access’ that underpins Pritchard’s Epistemological Disjunctivism. I provide three interpretations of the notion of Reflective Access: a metaphysical interpretation, a folk interpretation, and an epistemic interpretation. I argue that none of (...) these three interpretations comes without problems. I conclude that, until we have a clear and unproblematic account of Reflective Access, the Access Problem remains a challenge for Pritchard’s Epistemological Disjunctivism. (shrink)

The logics of formal inconsistency (LFIs, for short) are paraconsistent logics (that is, logics containing contradictory but non-trivial theories) having a consistency connective which allows to recover the ex falso quodlibet principle in a controlled way. The aim of this paper is considering a novel semantical approach to first-order LFIs based on Tarskian structures defined over swap structures, a special class of multialgebras. The proposed semantical framework generalizes previous aproaches to quantified LFIs presented in the literature. The case (...) of QmbC, the simpler quantified LFI expanding classical logic, will be analyzed in detail. An axiomatic extension of QmbC called QLFI1o is also studied, which is equivalent to the quantified version of da Costa and D'Ottaviano 3-valued logic J3. The semantical structures for this logic turn out to be Tarkian structures based on twist structures. The expansion of QmbC and QLFI1o with a standard equality predicate is also considered. (shrink)

Basic Formal Ontology (BFO) is a top-level ontology used in hundreds of active projects in scientific and other domains. BFO has been selected to serve as top-level ontology in the Industrial Ontologies Foundry (IOF), an initiative to create a suite of ontologies to support digital manufacturing on the part of representatives from a number of branches of the advanced manufacturing industries. We here present a first draft set of axioms and definitions of an IOF upper ontology descending from BFO. (...) The axiomatization is designed to capture the meanings of terms commonly used in manufacturing and is designed to serve as starting point for the construction of the IOF ontology suite. (shrink)

Edward Nieznanski developed in 2007 and 2008 two different systems in formal logic which deal with the problem of evil. Particularly, his aim is to refute a version of the logical problem of evil associated with a form of religious determinism. In this paper, we revisit his first system to give a more suitable form to it, reformulating it in first-order modal logic. The new resulting system, called N1, has much of the original basic structure, and many (...) axioms, definitions, and theorems still remain; however, some new results are obtained. If the conclusions attained are correct and true, then N1 solves the problem of evil through the refutation of a version of religious determinism, showing that the attributes of God in Classical Theism, namely, those of omniscience, omnipotence, infallibility, and omnibenevolence, when adequately formalized, are consistent with the existence of evil in the world. We consider that N1 is a good example of how formal systems can be applied in solving interesting philosophical issues, particularly in Philosophy of Religion and Analytic Theology, establishing bridges between such disciplines. (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)

This paper introduces two languages and associated logics designed to afford perspicuous representations of a range of natural language arguments involving adverbs and the like: first-order logic with basic adverbs (FOL-BA) and first-order logic with scoped adverbs (FOL-SA). The guiding logical idea is that an adverb can come between a term and the rest of the statement it is a part of, resulting in a logically stronger statement. I explain various interesting challenges that arise in the (...) attempt to implement the guiding idea, and provide solutions for some but not all of them. I conclude by outlining some directions for further research. (shrink)

In §21 of Grundgesetze der Arithmetik asks us to consider the forms: a a2 = 4 and a a > 0 and notices that they can be obtained from a φ(a) by replacing the function-name placeholder φ(ξ) by names for the functions ξ2 = 4 and ξ > 0 (and the placeholder cannot be replaced by names of objects or of functions of 2 arguments).

Panqualityism, recently defended by Sam Coleman, is a variety of Russellian monism on which the categorical properties of fundamental physical entities are qualities, or, in Coleman’s exposition, unconscious qualia. Coleman defends a quotationalist, higher-order thought version of panqualityism. The aim of this paper is, first, to demonstrate that a first-order representationalist panqualityism is also available, and to argue positively in its favor. For it shall become apparent that quotationalist and first-order representationalist panqualityism are, in (...) spite of their close similarities, radically different theories: quotationalist panqualityism locates qualities in the subject of an experience, while first-order panqualityism locates qualities in the world. I argue that this makes quotationalist panqualityism implausible and first-order, representationalist panqualityism a highly natural, elegant, and intuitive theory. (shrink)

A uniform construction for sequent calculi for finite-valued first-order logics with distribution quantifiers is exhibited. Completeness, cut-elimination and midsequent theorems are established. As an application, an analog of Herbrand’s theorem for the four-valued knowledge-representation logic of Belnap and Ginsberg is presented. It is indicated how this theorem can be used for reasoning about knowledge bases with incomplete and inconsistent information.

Starting from certain metalogical results, I argue that first-order logical truths of classical logic are a priori and necessary. Afterwards, I formulate two arguments for the idea that first-order logical truths are also analytic, namely, I first argue that there is a conceptual connection between aprioricity, necessity, and analyticity, such that aprioricity together with necessity entails analyticity; then, I argue that the structure of natural deduction systems for FOL displays the analyticity of its truths. Consequently, (...) each philosophical approach to these truths should account for this evidence, i.e., that first-order logical truths are a priori, necessary, and analytic, and it is my contention that the semantic account is a better candidate. (shrink)

The first-order temporal logics with □ and ○ of time structures isomorphic to ω (discrete linear time) and trees of ω-segments (linear time with branching gaps) and some of its fragments are compared: the first is not recursively axiomatizable. For the second, a cut-free complete sequent calculus is given, and from this, a resolution system is derived by the method of Maslov.

In this paper paraconsistent first-order logic LP^{#} with infinite hierarchy levels of contradiction is proposed. Corresponding paraconsistent set theory KSth^{#} is discussed.Axiomatical system HST^{#}as paraconsistent generalization of Hrbacek set theory HST is considered.

A first-order logic of belief with identity is proposed, primarily to give an account of possible de re contradictory beliefs, which sometimes occur as consequences of de dicto non-contradictory beliefs. A model has two separate, though interconnected domains: the domain of objects and the domain of appearances. The satisfaction of atomic formulas is defined by a particular S-accessibility relation between worlds. Identity is non-classical, and is conceived as an equivalence relation having the classical identity relation as a subset. (...) A tableau system with labels, signs, and suffixes is defined, extending the basic language $\mathscr{L}_{\mathbf{QB}}$ by quasiformulas (to express the denotations of predicates). The proposed logical system is paraconsistent since $\phi \wedge \neg\phi$ does not ``explode'' with arbitrary syntactic consequences. (shrink)

‘Quantified pure existentials’ are sentences (e.g., ‘Some things do not exist’) which meet these conditions: (i) the verb EXIST is contained in, and is, apart from quantificational BE, the only full (as against auxiliary) verb in the sentence; (ii) no (other) logical predicate features in the sentence; (iii) no name or other sub-sentential referring expression features in the sentence; (iv) the sentence contains a quantifier that is not an occurrence of EXIST. Colin McGinn and Rod Girle have alleged that standard (...) first-order logic cannot adequately deal with some such existentials. The article defends the view that it can. (shrink)

It is shown that the infinite-valued first-order Gödel logic G° based on the set of truth values {1/k: k ε w {0}} U {0} is not r.e. The logic G° is the same as that obtained from the Kripke semantics for first-order intuitionistic logic with constant domains and where the order structure of the model is linear. From this, the unaxiomatizability of Kröger's temporal logic of programs (even of the fragment without the nexttime operator O) (...) and of the authors' temporal logic of linear discrete time with gaps follows. (shrink)

One well known problem regarding quantifiers, in particular the 1storder quantifiers, is connected with their syntactic categories and denotations. The unsatisfactory efforts to establish the syntactic and ontological categories of quantifiers in formalized first-order languages can be solved by means of the so called principle of categorial compatibility formulated by Roman Suszko, referring to some innovative ideas of Gottlob Frege and visible in syntactic and semantic compatibility of language expressions. In the paper the principle is introduced for categorial (...) languages generated by the Ajdukiewicz’s classical categorial grammar. The 1st-order quantifiers are typically ambiguous. Every 1st-order quantifier of the type k > 0 is treated as a two-argument functorfunction defined on the variable standing at this quantifier and its scope (the sentential function with exactly k free variables, including the variable bound by this quantifier); a binary function defined on denotations of its two arguments is its denotation. Denotations of sentential functions, and hence also quantifiers, are defined separately in Fregean and in situational semantics. They belong to the ontological categories that correspond to the syntactic categories of these sentential functions and the considered quantifiers. The main result of the paper is a solution of the problem of categories of the 1st-order quantifiers based on the principle of categorial compatibility. (shrink)

We discuss cases where subjects seem to enjoy conscious experience when the relevant first-order perceptual representations are either missing or too weak to account for the experience. Though these cases are originally considered to be theoretical possibilities that may be problematical for the higher-order view of consciousness, careful considerations of actual empirical examples suggest that this strategy may backfire; these cases may cause more trouble for first-order theories instead. Specifically, these cases suggest that (I) recurrent (...) feedback loops to V1 are most likely not the neural correlate of first-order representations for conscious experience, (II) first-order views seem to have a problem accounting for the phenomenology in these cases, and either (III) a version of the ambitious higher-order approach is superior in that it is the simplest theory that can account for all results at face value, or (IV) a view where phenomenology is jointly determined by both first-order and higher-order states. In our view (III) and (IV) are both live options and the decision between them may ultimately be an empirical question that cannot yet be decided. (shrink)

One well known problem regarding quantifiers, in particular the 1st order quantifiers, is connected with their syntactic categories and denotations.The unsatisfactory efforts to establish the syntactic and ontological categories of quantifiers in formalized first-order languages can be solved by means of the so called principle of categorial compatibility formulated by Roman Suszko, referring to some innovative ideas of Gottlob Frege and visible in syntactic and semantic compatibility of language expressions. In the paper the principle is introduced for (...) categorial languages generated by the Ajdukiewicz’s classical categorial grammar. The 1st-order quantifiers are typically ambiguous. Every 1st-order quantifier of the type k > 0 is treated as a two-argument functor-function defined on the variable standing at this quantifier and its scope (the sentential function with exactly k free variables, including the variable bound by this quantifier); a binary function defined on denotations of its two arguments is its denotation. Denotations of sentential functions, and hence also quantifiers, are defined separately in Fregean and in situational semantics. They belong to the ontological categories that correspond to the syntactic categories of these sentential functions and the considered quantifiers. The main result of the paper is a solution of the problem of categories of the 1st-order quantifiers based on the principle of categorial compatibility. (shrink)

In this article, logical concepts are defined using the internal syntactic and semantic structure of language. For a first-order language, it has been shown that its logical constants are connectives and a certain type of quantifiers for which the universal and existential quantifiers form a functionally complete set of quantifiers. Neither equality nor cardinal quantifiers belong to the logical constants of a first-order language.

We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language. -/- We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'. -/- We then adopt what may (...) be labelled a finitary, evidence-based, `agnostic' perspective and argue that Brouwerian atheism is merely a restricted perspective within the finitary agnostic perspective, whilst Hilbertian theism contradicts the finitary agnostic perspective. -/- We then consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary, evidence-based, definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways. -/- We show that the two definitions correspond to two distinctly different---not necessarily evidence-based but complementary---assignments of satisfaction and truth to the compound formulas of PA over N. -/- We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both the complementary interpretations; and conclude some unsuspected constructive consequences of such complementarity for the foundations of mathematics, logic, philosophy, and the physical sciences. -/- . (shrink)

To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on pro- cedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves (...) found in Leibnizian infinitesimal calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than modern Weierstrassian ones. (shrink)

In 1942 Haskell B. Curry presented what is now called Curry's paradox which can be found in a logic independently of its stand on negation. In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this article the non-classical resolution of Curry’s Paradox and Shaw-Kwei' sparadox without rejection any contraction postulate is proposed. In additional relevant paraconsistent logic C ̌_n^#,1≤n<ω, in fact,provide an effective way of circumventing triviality of da Costa’s paraconsistent Set Theories〖NF〗n^C.

All first-order Gödel logics G_V with globalization operator based on truth value sets V C [0,1] where 0 and 1 lie in the perfect kernel of V are axiomatized by Ciabattoni’s hypersequent calculus HGIF.

Many authors have noted that there are types of English modal sentences cannot be formalized in the language of basic first-order modal logic. Some widely discussed examples include “There could have been things other than there actually are” and “Everyone who is actually rich could have been poor.” In response to this lack of expressive power, many authors have discussed extensions of first-order modal logic with two-dimensional operators. But claims about the relative expressive power of these (...) extensions are often justified only by example rather than by rigorous proof. In this paper, we provide proofs of many of these claims and present a more complete picture of the expressive landscape for such languages. (shrink)

In his book Modal Logic as Metaphysics, Timothy Williamson defends first-order necessitism using simplicity as a powerful argument. However, simplicity is decomposed into two different, even antagonistic, sides: elegance and parsimony. On the one hand, elegance is the property of theories possessing few and simple principles that allow them to deploy all their theoretical power; on the other hand, parsimony is the property of theories having the fair and necessary number of ontological entities that allow such theories give (...) an account of themselves. Since necessitism endorses Barcan Formulae for the sake of elegance, it is committed to a vast number of contingently non-concrete objects, so one may think that it is not qualitatively parsimonious. I argue that necessitism could be viewed as an additive case in the sense that Alan Baker characterizes the adjective so quantitative parsimony should be considered when it comes to necessitism instead of qualitative parsimony. (shrink)

In the present paper, we prove the normalization theorem and the consistency of the first-order classical logic with disjunctive syllogism. First, we propose the natural deduction system SCD for classical propositional logic having rules for conjunction, implication, negation, and disjunction. The rules for disjunctive syllogism are regarded as the rules for disjunction. After we prove the normalization theorem and the consistency of SCD, we extend SCD to the system SPCD for the first-order classical logic with (...) disjunctive syllogism. It can be shown that SPCD is conservative extension to SCD. Then, the normalization theorem and the consistency of SPCD are given. (shrink)

This paper is a study of higher-order contingentism – the view, roughly, that it is contingent what properties and propositions there are. We explore the motivations for this view and various ways in which it might be developed, synthesizing and expanding on work by Kit Fine, Robert Stalnaker, and Timothy Williamson. Special attention is paid to the question of whether the view makes sense by its own lights, or whether articulating the view requires drawing distinctions among possibilities that, according (...) to the view itself, do not exist to be drawn. The paper begins with a non-technical exposition of the main ideas and technical results, which can be read on its own. This exposition is followed by a formal investigation of higher-order contingentism, in which the tools of variable-domain intensional model theory are used to articulate various versions of the view, understood as theories formulated in a higher-order modal language. Our overall assessment is mixed: higher-order contingentism can be fleshed out into an elegant systematic theory, but perhaps only at the cost of abandoning some of its original motivations. (shrink)

It is widely taken that the first-order part of Frege's Begriffsschrift is complete. However, there does not seem to have been a formal verification of this received claim. The general concern is that Frege's system is one axiom short in the first-order predicate calculus comparing to, by now, the standard first-order theory. Yet Frege has one extra inference rule in his system. Then the question is whether Frege's first-order calculus is still deductively (...) sufficient as far as the first-order completeness is concerned. In this short note we confirm that the missing axiom is derivable from his stated axioms and inference rules, and hence the logic system in the Begriffsschrift is indeed first-order complete. (shrink)

Interesting as they are by themselves in philosophy and mathematics, paradoxes can be made even more fascinating when turned into proofs and theorems. For example, Russell’s paradox, which overthrew Frege’s logical edifice, is now a classical theorem in set theory, to the effect that no set contains all sets. Paradoxes can be used in proofs of some other theorems—thus Liar’s paradox has been used in the classical proof of Tarski’s theorem on the undefinability of truth in sufficiently rich languages. This (...) paradox (as well as Richard’s paradox) appears implicitly in Gödel’s proof of his celebrated first incompleteness theorem. In this paper, we study Yablo’s paradox from the viewpoint of first- and second-order logics. We prove that a formalization of Yablo’s paradox (which is second order in nature) is non-first-orderizable in the sense of George Boolos (1984). (shrink)

This document presents a Gentzen-style deductive calculus and proves that it is complete with respect to a 3-valued semantics for a language with quantifiers. The semantics resembles the strong Kleene semantics with respect to conjunction, disjunction and negation. The completeness proof for the sentential fragment fills in the details of a proof sketched in Arnon Avron (2003). The extension to quantifiers is original but uses standard techniques.

This note clarifies an error in the proof of the main theorem of “The Ricean Objection: An Analogue of Rice’s Theorem for First-Order Theories”, Logic Journal of the IGPL, 16(6): 585–590(2008).

This paper argues that, given the representational theory of mind, one cannot know a priori that one knows that p as opposed to being incapable of having any knowledge states; but one can know a priori that one knows that p as opposed to some other proposition q.

Provided here is an account, both syntactic and semantic, of first-order and monadic second-order quantification theory for domains that may be non-atomic. Although the rules of inference largely parallel those of classical logic, there are important differences in connection with the identification of argument places and the significance of the identity relation.

The principle of epistemic closure is the claim that what is known to follow from knowledge is known to be true. This intuitively plausible idea is endorsed by a vast majority of knowledge theorists. There are significant problems, however, that have to be addressed if epistemic closure – closed knowledge – is endorsed. The present essay locates the problem for closed knowledge in the separation it imposes between knowledge and evidence. Although it might appear that all that stands (...) between knowing the truth of the premises of a valid inference and knowledge of its conclusion is inferring it from the premises, the evidence for each of the premises may jointly count against the conclusion. The intuitive view regarding inferred knowledge says one thing, the evidence says another. One epistemological framework that seems to have the resources to resolve this tension endorses the view that knowledge always requires conclusive evidence. A second framework resolves the tension by limiting the scope of the closure principle. Only inferences drawn directly from propositions contained in the scope of a single knowledge operator are considered closed. The aim of the present essay is to revive the unpopular third option, the idea that knowledge is open. The essay proceeds by arguing that in different ways the two former frameworks only succeed in relocating the problem, not in resolving it. The first framework, the infallibilist view, relocates the problem to a sharp separation between knowledge of the occurrences of events from knowledge of their chance of occurring, a separation leading to several significant additional problems. The fallibilist view, the second framework, in endorsing closure neglects to take into full account the ways in which evidence fails to be transitive. For instance, evidence can count in favor of a conjunction while counting against each of its conjuncts. This fact, which is argued for in the essay on probabilistic as well as non-probabilistic grounds, is used as the foundation of an argument against closed knowledge that can be used as a way to understand several of the most fundamental challenges of epistemology. Not only can an open knowledge view that is based on open evidence resolve all these problems in a simple and natural way, it can also respond to formidable challenges that significantly hinder other open knowledge views. There are good reasons, then, to view both knowledge and evidence as open. (shrink)

Several of Thomas Aquinas's proofs for the existence of God rely on the claim that causal series cannot proceed in infinitum. I argue that Aquinas has good reason to hold this claim given his conception of causation. Because he holds that effects are ontologically dependent on their causes, he holds that the relevant causal series are wholly derivative: the later members of such series serve as causes only insofar as they have been caused by and are effects of the earlier (...) members. Because the intermediate causes in such series possess causal powers only by deriving them from all the preceding causes, they need a first and non-derivative cause to serve as the source of their causal powers. (shrink)

This chapter provides an introduction to higher-order metaphysics as well as to the contributions to this volume. We discuss five topics, corresponding to the five parts of this volume, and summarize the contributions to each part. First, we motivate the usefulness of higher-order quantification in metaphysics using a number of examples, and discuss the question of how such quantifiers should be interpreted. We provide a brief introduction to the most common forms of higher-order logics used in (...) metaphysics, and indicate a number of questions which can be raised in such systems using logical vocabulary alone. Using a further example, we return to applications of higher-order logics in metaphysics. We also mention key developments in the history of higher-order logic as it pertains to metaphysics. Finally, we mention certain arguments which have been raised against the use of higher-order logic, and some ways of responding to them. (shrink)

“Second-order Logic” in Anderson, C.A. and Zeleny, M., Eds. Logic, Meaning, and Computation: Essays in Memory of Alonzo Church. Dordrecht: Kluwer, 2001. Pp. 61–76. -/- Abstract. This expository article focuses on the fundamental differences between second- order logic and first-order logic. It is written entirely in ordinary English without logical symbols. It employs second-order propositions and second-order reasoning in a natural way to illustrate the fact that second-order logic is actually a familiar part (...) of our traditional intuitive logical framework and that it is not an artificial formalism created by specialists for technical purposes. To illustrate some of the main relationships between second-order logic and first-order logic, this paper introduces basic logic, a kind of zero-order logic, which is more rudimentary than first-order and which is transcended by first-order in the same way that first-order is transcended by second-order. The heuristic effectiveness and the historical importance of second-order logic are reviewed in the context of the contemporary debate over the legitimacy of second-order logic. Rejection of second-order logic is viewed as radical: an incipient paradigm shift involving radical repudiation of a part of our scientific tradition, a tradition that is defended by classical logicians. But it is also viewed as reactionary: as being analogous to the reactionary repudiation of symbolic logic by supporters of “Aristotelian” traditional logic. But even if “genuine” logic comes to be regarded as excluding second-order reasoning, which seems less likely today than fifty years ago, its effectiveness as a heuristic instrument will remain and its importance for understanding the history of logic and mathematics will not be diminished. Second-order logic may someday be gone, but it will never be forgotten. Technical formalisms have been avoided entirely in an effort to reach a wide audience, but every effort has been made to limit the inevitable sacrifice of rigor. People who do not know second-order logic cannot understand the modern debate over its legitimacy and they are cut-off from the heuristic advantages of second-order logic. And, what may be worse, they are cut-off from an understanding of the history of logic and thus are constrained to have distorted views of the nature of the subject. As Aristotle first said, we do not understand a discipline until we have seen its development. It is a truism that a person's conceptions of what a discipline is and of what it can become are predicated on their conception of what it has been. (shrink)

Higher-order realists about properties express their view that there are properties with the help of higher-order rather than first-order quantifiers. They claim two types of advantages for this way of formulating property realism. First, certain gridlocked debates about the nature of properties, such as the immanentism versus transcendentalism dispute, are taken to be dissolved. Second, a further such debate, the tropes versus universals dispute, is taken to be resolved. In this paper I first argue (...) that higher-order realism does not in fact resolve the tropes versus universals dispute. In a constructive spirit, I then develop higher-order realism in a way that leads to a dissolution, rather than a resolution, of this dispute too. (shrink)

Duncan Pritchard’s Epistemic Angst promises a novel solution to the closure-based sceptical problem that, unlike more traditional solutions, does not entail revising our fundamental epistemological commitments. In order to do this, it appeals to a Wittgensteinian account of rational evaluation, the overarching theme of which is that it neither makes sense to doubt nor to believe in our anti-sceptical hinge commitments. The purpose of this paper is to show that the argument for the claim that there can be (...) no rational basis to believe our anti-sceptical hinge commitments relies upon an implicit assumption about rational support that I label The Pritchensteinian Rational Grounds Principle. I argue that, insofar as this principle is intended to apply to closure-style inferences, it leads to irrational doxastic attitudes. I consider a seemingly plausible modification of the principle that would avoid this result but show that this modified principle faces serious problems of its own. -/- . (shrink)

The idea that knowledge can be extended by inference from what is known seems highly plausible. Yet, as shown by familiar preface paradox and lottery-type cases, the possibility of aggregating uncertainty casts doubt on its tenability. We show that these considerations go much further than previously recognized and significantly restrict the kinds of closure ordinary theories of knowledge can endorse. Meeting the challenge of uncertainty aggregation requires either the restriction of knowledge-extending inferences to single premises, or eliminating epistemic uncertainty (...) in known premises. The first strategy, while effective, retains little of the original idea—conclusions even of modus ponens inferences from known premises are not always known. We then look at the second strategy, inspecting the most elaborate and promising attempt to secure the epistemic role of basic inferences, namely Timothy Williamson’s safety theory of knowledge. We argue that while it indeed has the merit of allowing basic inferences such as modus ponens to extend knowledge, Williamson’s theory faces formidable difficulties. These difficulties, moreover, arise from the very feature responsible for its virtue- the infallibilism of knowledge. (shrink)

Generally, an epistemic fallibilist considers it reasonable to claim, “I know that P, but I may be wrong.” An epistemic infallibilist, on the other hand, would consider this claim absurd. I argue initially that infallibilism presents more advantages in its assertion of the claim’s absurdity than fallibilism does in making the claim. One, infallibilism is not faulted with the propensity for violations of epistemic closure that beleaguers some fallibilist accounts, due in part to the latter’s problematic shunting of fallible (...) epistemic standards across inferential chains. Two, infallibilism is more easily modelled doxastically than fallibilism, as the former’s understanding of certainty is more tractable than the latter’s idea of what counts as a viable standard of fallibility. A rectification of these fallibilist issues may then be called upon to motivate gradualist variants of fallibilism. For epistemic gradualism, the problematic modellability and shunting of standards is curtailed via an awareness that the standard for knowledge is just one out of many along a gradient that includes, at one extreme, the infallibilist standard of certainty. Specifically, first, gradualism, along with infallibilism, can be modelled doxastically through elucidating upon an obtaining relation between doubt and knowledge; next, the problem with violating closure, although not generally applicable to infallibilism, can be answered by gradualism with a reasonable denial of closure altogether that makes precise which epistemic standard is relevant for which part of an inferential chain; lastly, these modelling resources can be appropriated, by both gradualism and infallibilism, to successfully address a doxastically pertinent form of closure violation called rational self-doubt. (shrink)

This paper argues against common views that at least in many cases Robert Nozick is not forced to deny common closure principles. More importantly, Nozick does not – despite first (and second) appearances and despite his own words – deny closure. On the contrary, he is defending a more sophisticated and complex principle of closure. This principle does remarkably well though it is not without problems. It is surprising how rarely Nozick’s principle of closure has (...) been discussed. He should be seen not so much as a denier of closure than as someone who’s proposing an alternative, more complex principle of closure. (shrink)

In response to the claim that certain epistemically defective inferences such as Moore’s argument lead us to the conclusion that we ought to abandon closure, Crispin Wright suggests that we can avoid doing so by distinguishing it from a stronger principle, namely transmission. Where closure says that knowledge of a proposition is a necessary condition on knowledge of anything one knows to entail it, transmission makes a stronger claim, saying that by reasoning deductively from known premises one can (...) thereby acquire knowledge of or justification for the conclusion. Wright’s thought is that in cases such as Moore what is really going on is a failure of transmission, not a failure of closure. The problem with this claim is that it relies on an outdated formulation of closure which few nowadays would find plausible. Once we stipulate that it is the intuitive closure principle that deserves our attention, it becomes far less obvious that the project of diagnosing what is wrong with arguments such as Moore in terms of transmission failure but not closure failure can be vindicated. In order to demonstrate this claim, I consider intuitive closure of propositional justification vs. transmission of propositional justification and concede that perhaps these can come apart in the way that Wright suggests. I then argue that despite this concession, we also need an answer to the question of whether intuitive closure for doxastic justification and transmission for doxastic justification can come apart. We need an answer to this question because our ultimate interest in these issues stems from our interest in whether knowledge obeys closure, not merely whether propositional justification does. I argue that, even on the assumption that there is transmission failure of propositional justification, there is no transmission failure of doxastic justification. And if this is true, then there is no transmission failure of knowledge either. The transmission failure diagnosis of Moore’s argument and its ilk is thus in trouble. (shrink)

Many theories of rational belief give a special place to logic. They say that an ideally rational agent would never be uncertain about logical facts. In short: they say that ideal rationality requires "logical omniscience." Here I argue against the view that ideal rationality requires logical omniscience on the grounds that the requirement of logical omniscience can come into conflict with the requirement to proportion one’s beliefs to the evidence. I proceed in two steps. First, I rehearse an influential (...) line of argument from the "higher-order evidence" debate, which purports to show that it would be dogmatic, even for a cognitively infallible agent, to refuse to revise her beliefs about logical matters in response to evidence indicating that those beliefs are irrational. Second, I defend this "anti-dogmatism" argument against two responses put forth by Declan Smithies and David Christensen. Against Smithies’ response, I argue that it leads to irrational self-ascriptions of epistemic luck, and that it obscures the distinction between propositional and doxastic justification. Against Christensen’s response, I argue that it clashes with one of two attractive deontic principles, and that it is extensionally inadequate. Taken together, these criticisms will suggest that the connection between logic and rationality cannot be what it is standardly taken to be—ideal rationality does not require logical omniscience. (shrink)

Anthony Brueckner argues for a strong connection between the closure and the underdetermination argument for scepticism. Moreover, he claims that both arguments rest on infallibilism: In order to motivate the premises of the arguments, the sceptic has to refer to an infallibility principle. If this were true, fallibilists would be right in not taking the problems posed by these sceptical arguments seriously. As many epistemologists are sympathetic to fallibilism, this would be a very interesting result. However, in this (...) paper I will argue that Brueckner’s claims are wrong: The closure and the underdetermination argument are not as closely related as he assumes and neither rests on infallibilism. Thus even a fallibilist should take these arguments to raise serious problems that must be dealt with somehow. (shrink)