Øystein Linnebo has recently shown that the existence of successors cannot be proven in predicative Frege arithmetic, using Frege’s definitions of arithmetical notions. By contrast, it is shown here that the existence of successor can be proven in ramified predicative Frege arithmetic.

The paper argues, as against Thau and Caplan, that the traditional interpretation that Frege abandoned his earlier views about identity and identity--statements is correct.

Why should one think Frege's definition of the ancestral correct? It can be proven to be extensionally correct, but the argument uses arithmetical induction, and that seems to undermine Frege's claim to have justified induction in purely logical terms. I discuss such circularity objections and then offer a new definition of the ancestral intended to be intensionally correct; its extensional correctness then follows without proof. This new definition can be proven equivalent to Frege's without any use of (...) arithmetical induction. This proves, without any use of arithmetical induction, that Frege's definition is extensionally correct and so answers the circularity objections. (shrink)

An investigation of Frege's various contributions to the study of language, focusing on three of his most famous doctrines: that concepts are unsaturated, that sentences refer to truth-values, and that sense must be distinguished from reference.

As is well-known, the formal system in which Frege works in his Grundgesetze der Arithmetik is formally inconsistent, Russell’s Paradox being derivable in it.This system is, except for minor differ...

An investigation of what Frege means by his doctrine that functions (and so concepts) are 'unsaturated'. We argue that this doctrine is far less peculiar than it is usually taken to be. What makes it hard to understand, oddly enough, is the fact that it is so deeply embedded in our contemporary understanding of logic and language. To see this, we look at how it emerges out of Frege's confrontation with the Booleans and how it expresses a fundamental (...) difference between Frege's approach to logic and theirs. (shrink)

The 'substitution argument' purports to demonstrate the falsity of Russellian accounts of belief-ascription by observing that, e.g., these two sentences: (LC) Lois believes that Clark can fly. (LS) Lois believes that Superman can fly. could have different truth-values. But what is the basis for that claim? It seems widely to be supposed, especially by Russellians, that it is simply an 'intuition', one that could then be 'explained away'. And this supposition plays an especially important role in Jennifer Saul's defense of (...) Russellianism, based upon the existence of an allegedly similar contrast between these two sentences: (PC) Superman is more popular than Clark. (PS) Superman is more popular than Superman. The latter contrast looks pragmatic. But then, Saul asks, why shouldn't we then say the same about the former? The answer to this question is that the two cases simply are not similar. In the case of (PC) and (PS), we have only the facts that these strike us differently, and that people will sometimes say things like (PC), whereas they will never say things like (PS). By contrast, there is an argument to be given that (LS) can be true even if (LC) is false, and this argument does not appeal to anyone's 'intuitions'. The main goal of the paper is to present such a version of the substitution argument, building upon the treatment of the Fregan argument against Russellian accounts of belief itself in "Solving Frege's Puzzle". A subsidiary goal is to contribute to the growing literature arguing that 'intuitions' simply do not play the sort of role in philosophical inquiry that so-called 'experimental philosophers' have supposed they do. (shrink)

A close look at Frege's proof in "Foundations of Arithmetic" that every number has a successor. The examination reveals a surprising gap in the proof, one that Frege would later fill in "Basic Laws of Arithmetic".

In his paper “Flaws of Formal Relationism”, Mahrad Almotahari argues against the sort of response to Frege's Puzzle I have defended elsewhere, which he dubs ‘Formal Relationism’. Almotahari argues that, because of its specifically formal character, this view is vulnerable to objections that cannot be raised against the otherwise similar Semantic Relationism due to Kit Fine. I argue in response that Formal Relationism has neither of the flaws Almotahari claims to identify.

This book provides a concise overview, with excellent historical and systematic coverage, of the problems of the philosophy of language in the analytic tradition. Howard Callaway explains and explores the relation of language to the philosophy of mind and culture, to the theory of knowledge, and to ontology. He places the question of linguistic meaning at the center of his investigations. The teachings of authors who have become classics in the field, including Frege, Russell, Carnap, Quine, Davidson, and Putnam (...) are critically analyzed. I share completely his conviction that contemporary Anglo-American philosophy follows the spirit of the enlightenment in insisting on intellectual sincerity, clarity, and the willingness to meet scientific doubts or objections openly. --Professor Henri Lauener, Editor of Dialectica. (shrink)

Georgalis claimed that when Frege wrote ‘Über Sinn und Bedeutung’ Frege thought that the indirect [ungerade] sense of an expression was identical to its normal [gewöhnlich] sense (Georgalis 2022: e.g. 4, 5, 13). In this paper, I present five arguments for the falsity of Georgalis’ claim which are based on three pieces of apparent counterevidence: a passage from Frege’s letter to Russell dated 28.12.1902; a passage from Frege’s letter to Russell dated 20.10.1902; and a passage from (...) ‘Über Sinn und Bedeutung’. I outline possible counter-arguments and responses. (shrink)

Öz: Frege özel adların (ve diğer dilsel simgelerin) anlamları ve gönderimleri arasında ünlü ayrımını yaptığı “Anlam ve Gönderim Üzerine” (1948) adlı makalesinde, bu ayrımın önemi, gerekliliği ve sonuçları üzerine uzun değerlendirmeler yapar ancak özel adın anlamından tam olarak ne anlaşılması gerektiğinden yalnızca bir dipnotta kısaca söz eder. Örneğin “Aristoteles” özel adının anlamının Platon’un öğrencisi ve Büyük İskender’in öğretmeni ya da Stagira’da doğan Büyük İskender’in öğretmeni olarak alınabileceğini söyler. Burada dikkat çeken nokta örnekteki özel adın olası anlamları olarak gösterilen belirli (...) betimlemelerin de özel ad içeriyor olması. Anlamın Frege için bileşimsel olduğu, bir başka deyişle bir dilsel simgenin anlamının (varsa) parçalarının anlamlarınca belirlendiği düşünülürse, örnekteki belirli betimlemelerin anlamlarının saptanabilmesi için içerdikleri özel adların da anlamları saptanıp betimleme içinde geçtikleri yere konmalıdır. Bu işlem hiçbir özel ad kalmayana kadar sürdürülmelidir. Ancak biraz düşünülünce bir özel adın nesnesini tekil olarak betimleyecek ve içinde özel ad geçmeyecek bir betimleme bulmanın kolay olmadığı görülür. Örneğin yukarıdaki betimlemelerde geçen “Platon”, “Büyük İskender” ve “Stagira” gibi özel adların anlamları olabilecek ancak özel ad içermeyen betimlemeler bulmak pek kolay görünmüyor. Ortaya bir sonsuz gerileme sorunu çıkmış gibi duruyor. Frege’nin anlam-gönderim ayrımını için ciddi sonuçları olabilecek bu sorunu Frege çözebilir mi? Bu yazıda bu sorunun yanıtını arayacağım. Sorunu betimledikten sonra Frege’nin önündeki seçenekleri (örneğin bağlama duyarlı terimlerden (gösterme adılları veya belirteçler) yararlanma veya bağlamı sınırlama gibi) değerlendireceğim. -/- Abstract: In his “Sense and Reference” (1948), Frege makes his famous distinction between the sense and the reference of a proper name (and other signs) and discusses at length the significance, necessity and consequences of the distinction, but he explains how the sense of a proper name should be understood very briefly in a footnote. According to him, for example, the sense of the proper name “Aristotle” can be taken as the pupil of Plato and teacher of Alexander the Great or the teacher of Alexander the Great who was born in Stagira. What is interesting here is that the definite descriptions given as the possible senses of this proper name do themselves contain proper names too. Since the sense is something compositional for Frege, which means the sense of a linguistic sign is determined by its constituents (if there are any), in order to determine the sense of definite descriptions in question, we should first determine the senses of proper names they contain and substitute these senses with the proper names themselves. This process should continue until no proper name remains. However, it does not seem easy to find a definite description which describes the object of a proper name uniquely but contains no proper name itself. For instance, could we find appropriate senses that contain no proper name for “Plato”, “Alexander the Great” and “Stagira”? It does not seem likely. A problem of infinite regress appears to arise. Can Frege solve this problem, which poses a serious threat for his sensereference distinction? I will explore this problem in this paper. After explaining the problem, I will discuss the options (e.g. turning to indexicals or context restriction) Frege can take to deal with it. (shrink)

Abstractionism in the philosophy of mathematics aims at deriving large fragments of mathematics by combining abstraction principles (i.e. the abstract objects $\S e_1, \S e_2$, are identical if, and only if, an equivalence relation $Eq_\S$ holds between the entities $e_1, e_2$) with logic. Still, as highlighted in work on the semantics for relevant logics, there are different ways theories might be combined. In exactly what ways must logic and abstraction be combined in order to get interesting mathematics? In this paper, (...) we investigate the matter by deriving the axioms of second-order Peano Arithmetic from Frege's Basic Law V (the extension of $F$ is identical with the extension of $G$ if, and only if, $F$ and $G$ are extensionally equivalent) in the presence of a relevant higher-order logic. The results are interesting. Not only must we take on logic as true, and not only must we apply our logic to abstraction principles, but also we have to apply our theory of abstraction back to the logic in order to arrive at arithmetic. Thus, what Abstractionism gives us is not simply what we get from abstraction via logic, but also what we get from logic via abstraction. (shrink)

I resolve the major challenge to an Expressivist theory of the meaning of normative discourse: the Frege–Geach Problem. Drawing on considerations from the semantics of directive language (e.g., imperatives), I argue that, although certain forms of Expressivism (like Gibbard’s) do run into at least one version of the Problem, it is reasonably clear that there is a version of Expressivism that does not.

There is no doubt that social interaction plays an important role in language-learning, as well as in concept acquisition. In surprising contrast, social interaction makes only passing appearance in our most promising naturalistic theories of content. This is particularly true in the case of mental content (e.g., Cummins, 1996; Dretske, 1981, 1988; Fodor, 1987, 1990a; Millikan, 1984); and insofar as linguistic content derives from mental content (Grice, 1957), social interaction seems missing from our best naturalistic theories of both.1 In this (...) paper, I explore the ways in which even the most individualistic of theories of mental content can, and should, accommodate social effects. I focus especially on the way in which inferential relations, including those that are socially taught, influence language-learning and concept acquisition. I argue that these factors affect the way subjects conceive of mental and linguistic content. Such effects have a dark side: the social and inferential processes in question give rise to misleading intuitions about content itself. They create the illusion that content and inferential relations are more deeply intertwined than they actually are. This illusion confounds an otherwise attractive solution to what is known as ‘Frege’s puzzle’ (Salmon, 1986). I.. (shrink)

In contemporary philosophy of religion, the two most standard approaches to predicates of God are analogy and univocation. While analogy lacks precision and is best used in liturgical and sacred texts, univocal predicates are problematic because they seem to lead to ontological monism of sameness between God and creatures, which cannot be allowed within metaphysics of Absolute Being. In this article, I examine and contrast G. Frege’s approach to univocal predications and L. Wittgenstein’s notion of language-games, which allows us (...) to move away from univocation as a case of strong identity toward univocation as the extra-linguistic context of human activities. Indeed, that human context can be shared with ‘Goodness,’ ‘Wisdom,’ ‘Simplicity,’ ‘Love’ and many other predicates we attribute to God. I argue that our language of univocal predicates must reflect not the abstract concepts but that same context of divine and human activities in which names of God achieve univocal context. Key words: analogy, univocal predicates, G. Frege, L. Wittgenstein, contextual principle, language-games . (shrink)

"The Third Being" presents texts by A.V. Bronnikov, written from 2011–2019 and is devoted to issues regarding the philosophy of art, creativity and language. If the first being is the being of the eternal and divine and the second being is that of the temporary and human, then the new, third, being appears as the intersection and continuation of the first two. The third being is seen and anticipated in art—in the timeless and indestructible reality created by man, in the (...) reality of a poetic word that raises the world and its things to the level of form, image and idea. The book compares approaches to the philosophy of art and the theory of language by H.-G. Gadamer, M. Heidegger, G. Frege, R. Ingarden, R. Bart, L. Wittgenstein and J. Derrida. An important place is given to Russian religious philosophy, presented by P. A. Florensky, S. N. Bulgakov, N. A. Berdyaev and others. The publication contains essays on the life and work of poets R. M. Rilke, E. Pound and T. S. Eliot, as well as the first Russian translation of J. Derrida's “Advances,” known as the postmodern commentary on Plato's “Timaeus.” -/- Contents: Russia in Rilke / Plato's ideality of text / Word and world / Art as reality / Sacrality of art / Cosmists / Florensky space / Serial music of "Four Quartets" / Philosophy of "The Cantos" / Ezra Pound and his "Cantos" / Poetry: a breakthrough to reality / At the origin of the world / J. Derrida's "Advances". (shrink)

The traditional picture of the development of analytical philosophy, represented especially by such thinkers as G. Frege, G. E. Moore, B. Russell or R. Carnap, whose attitude was generally anti-metaphysical, can, on closer study, be shown to be incomplete. This article treats of the Cracow circle – a group of Polish philosophers among whom are, above all, to be counted J. Salamucha, J. M. Bocheński, J. F. Drewnowski, and B. Sobociński, who were, at the beginning of the twentieth century, (...) fascinated by the development of modern formal logic and its application to philosophical thinking. They also attempted to apply it to Catholic philosophy. The result of their endeavours were many remarkable works introducing not only a defence of the use of modern philosophical approaches in Christian thought, but also the reconstruction, by means of formal logic, of significant proofs given by Scholastic authors. (shrink)

Frege argued for the force-content distinction not only by appealing to the logical and fictional contexts which are most closely associated with the “Frege point", but also based on the fact that an affirmative answer to a yes-no question constitutes an assertion. Supposedly this is only intelligible if the question contains a forceless thought or proposition which an affirmative answer then asserts. Against this I argue that this fact more readily supports the view that questions operate on assertions (...) and other forceful acts themselves. Force is neither added to propositions as on the traditional view, nor is it cancelled as has recently been proposed. Rather higher-level acts such as questioning, but also e.g. conditionalizing, embed and present assertoric or directive acts that are forceful and committal, while suspending commitment to them. The Frege point confounds different varieties of force and the question whether something is merely presented for consideration with the question what is so presented. Force is representational: through assertoric and directive force indicators subjects non-conceptually present positions of theoretical or practical knowledge, while interrogative acts indicate positions of wondering which strive for such knowledge. (shrink)

In Frege’s Puzzle (1986), Salmon analyzed ‘a withholds believing p’ in terms of a ternary relation BEL of x believing a proposition p under a guise g. The proposed analysis is the following: There is a proposition guise g such that a grasps p by means of g but a does not stand in BEL to p and g. Sean Crawford has made a proposal for Millians to evade propositional guises through second-order belief. Specifically, in effect, Crawford’s proposes to (...) analyze the crucial notion of withheld belief instead as believing that one does not believe: a believes that a does not believe p. Crawford’s clever proposal thus avoids explicit quantification over guises. However, it is shown that the proposal is inadequate to avoid guises. The resulting notion is too weak to capture the relevant notion of withheld belief. (shrink)

Bertrand Russell was one of the best-known proponents of logicism: the theory that mathematics reduces to, or is an extension of, logic. Russell argued for this thesis in his 1903 The Principles of Mathematics and attempted to demonstrate it formally in Principia Mathematica (PM 1910–1913; with A. N. Whitehead). Russell later described his work as a further “regressive” step in understanding the foundations of mathematics made possible by the late 19th century “arithmetization” of mathematics and Frege’s logical definitions of (...) arithmetical concepts. The logical system of PM sought to improve on earlier attempts by solving the contradictions found in, e.g., Frege’s system, by employing a theory of types. In this article, I also consider and critically evaluate the most common objections to Russell’s logicism, including the claim that it is undermined by Gödel’s incompleteness results, and Putnam's charge of “if-thenism”. I suggest that if we are willing to accept a slightly revisionist account of what counts as a mathematical truth, these criticisms do not obviously refute Russell’s claim to have established that mathematical truths generally are a species of logical truth. (shrink)

The Frege point to the effect that e.g. the clauses of conditionals are not asserted and therefore cannot be assertions is often taken to establish a dichotomy between the content of a speech act, which is propositional and belongs to logic and semantics, and its force, which belongs to pragmatics. Recently this dichotomy has been questioned by philosophers such as Peter Hanks and Francois Recanati, who propose act-theoretic accounts of propositions, argue that we can’t account for propositional unity independently (...) of the forceful acts of speakers, and respond to the Frege point by appealing to a notion of force cancellation. I argue that the notion of force cancellation is faced with a dilemma and offer an alternative response to the Frege point, which extends the act-theoretic account to logical acts such as conditionalizing or disjoining. Such higher-level acts allow us to present forceful acts while suspending commitment to them. In connecting them, a subject rather commits to an affirmation function of such acts. In contrast, the Frege point confuses a lack of commitment to what is put forward with a lack of commitment or force in what is put forward. (shrink)

This paper shows how the universals of category theory in mathematics provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought. The paper also shows how the always-self-predicative universals of category theory provide the "opposite bookend" to the never-self-predicative universals of iterative set theory and thus that the paradoxes arose from having (...) one theory (e.g., Frege's Paradise) where universals could be either self-predicative or non-self-predicative (instead of being always one or always the other). (shrink)

In addition to the Frege point, Frege also argued for the force-content distinction from the fact that an affirmative answer to a yes-no question constitutes an assertion. I argue that this fact more readily supports the view that questions operate on and present assertions and other forceful acts themselves. Force is neither added to propositions as on the traditional view, nor is it cancelled as has recently been proposed. Rather higher level acts such as questioning, but also e.g. (...) conditionalizing, embed assertive or directive acts that are forceful and committal, while suspending commitment to them. The Frege point confounds different varieties of force and the question whether something is merely presented for consideration with the question what is so presented. Force is representational: through assertoric and directive force indicators subjects non-conceptually present positions of theoretical or practical knowledge, while interrogative acts indicate positions of wondering which strive for such knowledge. (shrink)

Certain advocates of the so-called “neo-logicist” movement in the philosophy of mathematics identify themselves as “neo-Fregeans” (e.g., Hale and Wright), presenting an updated and revised version of Frege’s form of logicism. Russell’s form of logicism is scarcely discussed in this literature and, when it is, often dismissed as not really logicism at all (in light of its assumption of axioms of infinity, reducibility and so on). In this paper I have three aims: firstly, to identify more clearly the primary (...) meta-ontological and methodological differences between Russell’s logicism and the more recent forms; secondly, to argue that Russell’s form of logicism offers more elegant and satisfactory solutions to a variety of problems that continue to plague the neo-logicist movement (the bad company objection, the embarrassment of richness objection, worries about a bloated ontology, etc.); thirdly, to argue that neo-Russellian forms of logicism remain viable positions for current philosophers of mathematics. (shrink)

This article analyzes the historical development of the philosophical logic syntax from the standpoint of the unity of historical and logical methods. According to this perspective, there are three types of logical syntax: the elementary subject-predicate, the modified definitivespecificative, and the standard propositional-functional. These types are generalized in the grammatical and mathematical styles of logical syntax. The main attention is paid to two scientific revolutions in elementary subject-predicate syntax, which led to the emergence of modified definitive-specific and standard propositional-functional syntaxes (...) and created the syntactic conditions for the development of contemporary philosophical logic. The specifics of contemporary philosophical logic and the methodological possibilities of its application to philosophical discourse are studied. The article aims to reevaluate the undeservedly forgotten systems of philosophical logic of the continental tradition, created by such prominent representatives as Aristotle, G.W.F. Hegel, and E. Husserl, and to actualize these logics in the context of contemporary philosophical culture. The potential of the above-mentioned logics is not fully involved in the philosophical discourse of modernity, primarily because they primarily used an imperfect elementary subject-predicate syntax and modified definitive-specificative syntax as its slightly improved version. Both syntaxes have one thing in common: the grammatical style of sentence structure. Nevertheless, they also have one common flaw – a high dependence on grammar formalism. As a result, the interaction between these syntaxes and Frege’s standard propositional-functional syntax is impossible, because the latter is based on mathematical formalism, which operates on the philosophical logic of the analytic tradition. The article substantiates the way to solve this problem by constructing a modified subject-predicate syntax of contemporary philosophical logic. This syntax provides information interaction between Aristotle’s elementary subject-predicate syntax, and Frege’s standard propositional-functional syntax based on Hegel’s modified definitive-specificative syntax. The proposed solution to this problem can create new opportunities for complementarity and mutual enrichment between the philosophical logic of continental and analytical traditions. The theoretical basis for the construction and study of contemporary philosophical logic is a functional analysis of contemporary symbolic logic, which improves the grammatical analysis of traditional formal logic. Functional-grammatical analysis is a way to rehabilitate the philosophical logic of the continental tradition. The novelty of this paper lies in the substantiation of the modified subjectpredicate syntax of contemporary philosophical logic. It makes it possible to establish a dialogue between continental and analytical traditions, which is designed to promote the further development of philosophy. (shrink)

Attention to the conversational role of alethic terms seems to dominate, and even sometimes exhaust, many contemporary analyses of the nature of truth. Yet, because truth plays a role in judgment and assertion regardless of whether alethic terms are expressly used, such analyses cannot be comprehensive or fully adequate. A more general analysis of the nature of truth is therefore required – one which continues to explain the significance of truth independently of the role alethic terms play in discourse. We (...) undertake such an analysis in this paper; in particular, we start with certain elements from Kant and Frege, and develop a construct of truth as a normative modality of cognitive acts (e.g., thought, judgment, assertion). Using the various biconditional T-schemas to sanction the general passage from assertions to (equivalent) assertions of truth, we then suggest that an illocutionary analysis of truth can contribute to its locutionary analysis as well, including the analysis of diverse constructions involving alethic terms that have been largely overlooked in the philosophical literature. Finally, we briefly indicate the importance of distinguishing between alethic and epistemic modalities. (shrink)

This paper discusses certain problems arising within the treatment of the senses of functions in Alonzo Church's Logic of Sense and Denotation. Church understands such senses themselves to be "sense-functions," functions from sense to sense. However, the conditions he lays out under which a sense-function is to be regarded as a sense presenting another function as denotation allow for certain undesirable results given certain unusual or "deviant" sense-functions. Certain absurdities result, e.g., an argument can be found for equating any two (...) senses of the same type. An alternative treatment of the senses of functions is discussed, and is thought to do better justice to Frege's original theory. (shrink)

The conditional interpretation of general categorical statements like ‘All men are animals’ as universally quantified material conditionals ‘For all x, if x is F, then x is G’ suggests that the logical structure of law statements is conditional rather than categorical. Disregarding the problem that the universally quantified material conditional is trivially true whenever there are no xs that are F, there are some reasons to be sceptical of Frege’s equivalence between categorical and conditional expressions. -/- Now many philosophers (...) will claim that the material conditional interpretation of laws statements, dispositions ascriptions, or any causal claim is generally accepted as wrong and outdated. Still, there seem to be some basic logical assumptions that are shared by most of the participants in the debate on causal matters which at least stems from the traditional truth functional interpretation of conditionals. This is indicated by the vocabulary in the philosophical debate on causation, where one often speaks of ‘counterfactuals’, ‘possible worlds’ and ‘necessity’ without being explicit on whether or to what extent one accepts the logical-technical definition of these notions. To guarantee a non-Humean and non-extensional approach to causal relations, it is therefore important to be aware of the logical and metaphysical implications of the technical vocabulary. -/- In this paper we want to show why extensional logic cannot deal with causal relations. Via a logical analysis of law-like statements ‘All Fs are Gs’ we hope to throw some new light on interrelated notions like causation, laws, induction, hypotheticality and modality. If successful, our analysis should be of relevance for a deeper understanding of any type of causal relations, whether we understand them to be laws, dispositions, singulars or categoricals. (shrink)

JOHN CORCORAN AND WILIAM FRANK. Surprises in logic. Bulletin of Symbolic Logic. 19 253. Some people, not just beginning students, are at first surprised to learn that the proposition “If zero is odd, then zero is not odd” is not self-contradictory. Some people are surprised to find out that there are logically equivalent false universal propositions that have no counterexamples in common, i. e., that no counterexample for one is a counterexample for the other. Some people would be surprised to (...) find out that in normal first-order logic existential import is quite common: some universals “Everything that is S is P” —actually quite a few—imply their corresponding existentials “Something that is S is P”. Anyway, perhaps contrary to its title, this paper is not a cataloging of surprises in logic but rather about the mistakes that did or might have or might still lead people to think that there are no surprises in logic. The paper cataloging of surprises in logic is on our “to-do” list. -/- ► JOHN CORCORAN AND WILIAM FRANK, Surprises in logic. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA E-mail: [email protected] There are many surprises in logic. Peirce gave us a few. Russell gave Frege one. Löwenheim gave Zermelo one. Gödel gave some to Hilbert. Tarski gave us several. When we get a surprise, we are often delighted, puzzled, or skeptical. Sometimes we feel or say “Nice!”, “Wow, I didn’t know that!”, “Is that so?”, or the like. Every surprise belongs to someone. There are no disembodied surprises. Saying there are surprises in logic means that logicians experience surprises doing logic—not that among logical propositions some are intrinsically or objectively “surprising”. The expression “That isn’t surprising” often denigrates logical results. Logicians often aim for surprises. In fact, [1] argues that logic’s potential for surprises helps motivate its study and, indeed, helps justify logic’s existence as a discipline. Besides big surprises that change logicians’ perspectives, the logician’s daily life brings little surprises, e.g. that Gödel’s induction axiom alone implies Robinson’s axiom. Sometimes wild guesses succeed. Sometimes promising ideas fail. Perhaps one of the least surprising things about logic is that it is full of surprises. Against the above is Wittgenstein’s surprising conclusion : “Hence there can never be surprises in logic”. This paper unearths basic mistakes in [2] that might help to explain how Wittgenstein arrived at his false conclusion and why he never caught it. The mistakes include: unawareness that surprise is personal, confusing logicians having certainty with propositions having logical necessity, confusing definitions with criteria, and thinking that facts demonstrate truths. People demonstrate truths using their deductive know-how and their knowledge of facts: facts per se are epistemically inert. [1] JOHN CORCORAN, Hidden consequence and hidden independence. This Bulletin, vol.16, p. 443. [2] LUDWIG WITTGENSTEIN, Tractatus Logico-Philosophicus, Kegan Paul, London, 1921. -/-. (shrink)

This is a thesis in support of the conceptual yoking of analytic truth to a priori knowledge. My approach is a semantic one; the primary subject matter throughout the thesis is linguistic objects, such as propositions or sentences. I evaluate arguments, and also forward my own, about how such linguistic objects’ truth is determined, how their meaning is fixed and how we, respectively, know the conditions under which their truth and meaning are obtained. The strategy is to make explicit what (...) is distinctive about analytic truths. The objective is to show that truths, known a priori, are trivial in a highly circumscribed way. My arguments are premised on a language-relative account of analytic truth. The language relative account which underwrites much of what I do has two central tenets: 1. Conventionalism about truth and, 2. Non-factualism about meaning. I argue that one decisive way of establishing conventionalism and non-factualism is to prioritise epistemological questions. Once it is established that some truths are not known empirically an account of truth must follow which precludes factual truths being known non-empirically. The function of Part 1 is, chiefly, to render Carnap’s language-relative account of analytic truth. I do not offer arguments in support of Carnap at this stage, but throughout Parts 2 and 3, by looking at more current literature on a priori knowledge and analytic truth, it becomes quickly evident that I take Carnap to be correct, and why. In order to illustrate the extent to which Carnap’s account is conventionalist and non-factualist I pose his arguments against those of his predecessors, Kant and Frege. Part 1 is a lightly retrospective background to the concepts of ‘analytic’ and ‘a priori’. The strategy therein is more mercenary than exegetical: I select the parts from Kant and Frege most relevant to Carnap’s eventual reaction to them. Hereby I give the reasons why Carnap foregoes a factual and objective basis for logical truth. The upshot of this is an account of analytic truth (i.e. logical truth, to him) which ensures its trivial nature. In opposition to accounts of a priori knowledge, which describe it as knowledge gained from rational apprehension, I argue that it is either knowledge from logical deduction or knowledge of stipulations. I therefore reject, in Part 2, three epistemologies for knowing linguistic conventions (e.g. implicit definitions): 1. intuition, 2. inferential a priori knowledge and, 3. a posteriori knowledge. At base, all three epistemologies are rejected because they are incompatible with conventionalism and non-factualism. I argue this point by signalling that such accounts of knowledge yield unsubstantiated second-order claims and/or they render the relevant linguistic conventions epistemically arrogant. For a convention to be arrogant it must be stipulated to be true. The stipulation is then considered arrogant when its meaning cannot be fixed, and its truth cannot be determined without empirical ‘work’. Once a working explication of ‘a priori’ has been given, partially in Part 1 (as inferential) and then in Part 2 (as non-inferential) I look, in Part 3, at an apriorist account of analytic truth, which, I argue, renders analytic truth non-trivial. The particular subject matter here is the implicit definitions of logical terms. The opposition’s argument holds that logical truths are known a priori (this is part of their identification criteria) and that their meaning is factually based. From here it follows that analytic truth, being determined by factually based meaning, is also factual. I oppose these arguments by exposing the internal inconsistencies; that implicit definition is premised on the arbitrary stipulation of truth which is inconsistent with saying that there are facts which determine the same truth. In doing so, I endorse the standard irrealist position about implicit definition and analytic truth (along with the “early friends of implicit definition” such as Wittgenstein and Carnap). What is it that I am trying to get at by doing all of the abovementioned? Here is a very abstracted explanation. The unmitigated realism of the rationalists of old, e.g. Plato, Descartes, Kant, have stoically borne the brunt of the allegation of yielding ‘synthetic a priori’ claims. The anti-rationalist phase of this accusation I am most interested in is that forwarded by the semantically driven empiricism of the early 20th century. It is here that the charge of the ‘synthetic a priori’ really takes hold. Since then new methods and accusatory terms are employed by, chiefly, non-realist positions. I plan to give these proper attention in due course. However, it seems to me that the reframing of the debate in these new terms has also created the illusion that current philosophical realism, whether naturalistic realism, realism in science, realism in logic and mathematics, is somehow not guilty of the same epistemological and semantic charges levelled against Plato, Descartes and Kant. It is of interest to me that in, particularly, current analytic philosophy (given its rationale) realism in many areas seems to escape the accusation of yielding synthetic priori claims. Yet yielding synthetic a priori claims is something which realism so easily falls prey to. Perhaps this is a function of the fact that the phrase, ‘synthetic a priori’, used as an allegation, is now outmoded. This thesis is nothing other than an indictment of metaphysics, or speculative philosophy (this being the crime), brought against a specific selection of realist arguments. I, therefore, ask of my reader to see my explicit, and perhaps outmoded, charge of the ‘synthetic a priori’ levelled against respective theorists as an attempt to draw a direct comparison with the speculative metaphysics so many analytic philosophers now love to hate. I think the phrase ‘synthetic a priori’ still does a lot of work in this regard, precisely because so many current theorists wrongly think they are immune to this charge. Consequently, I shall say much about what is not permitted. Such is, I suppose, the nature of arguing ‘against’ something. I’ll argue that it is not permitted to be a factualist about logical principles and say that they are known a priori. I’ll argue that it is not permitted to say linguistic conventions are a posteriori, when there is a complete failure in locating such a posteriori conventions. Both such philosophical claims are candidates for the synthetic a priori, for unmitigated rationalism. But on the positive side, we now have these two assets: Firstly, I do not ask us to abandon any of the linguistic practises discussed; merely to adopt the correct attitude towards them. For instance, where we use the laws of logic, let us remember that there are no known/knowable facts about logic. These laws are therefore, to the best of our knowledge, conventions not dissimilar to the rules of a game. And, secondly, once we pass sentence on knowing, a priori, anything but trivial truths we shall have at our disposal the sharpest of philosophical tools. A tool which can only proffer a better brand of empiricism. (shrink)

In a review of Frege's Puzzle1, Graeme Forbes makes the claim that Salmon's account of belief might be seen, under certain conditions, as a mere notational variant of a neo-Fregean theory; and thus that such an account might be reduced to a neo-Fregean one simply by rewriting it in terms of Fregean terminology. With a view to supporting his claim, Forbes offers an outline of an account of belief which, according to him, would satisfy the following conditions: (i) it (...) could be directly obtained from Salmon's own analysis by means of a certain set of substitutions, which presumably would not affect the essential features of Salmon's view; (ii) it could naturally be described as Fregean, in the sense that it would preserve, (at least) the spirit of Frege's doctrines, especially his fundamental intuitions about belief. Of course, the upshot of Forbes's argument is that Salmon's theory would not, at bottom, constitute a genuine alternative to a Fregean semantics for belief ascriptions. In this paper I argue to the effect that Forbes's claim is not in general sound. It seems to me that the sort of indirect argument used by Forbes - that of trying to undermine Salmon's theory by showing that it is just a version of a neo-Fregean account - does not provide someone working within a Fregean framework with an adequate strategy to counter Salmon's neo-Russellian views. It would perhaps be better to concentrate a Fregean attack on certain apparently dubious and highly controversial theses and results which are constitutive of Salmon's view, e.g. the counterintuitive character of a substantial set of consequences which follow from his theory of belief, as well as the associated revisionist stand he is forced to take towards our current patterns of speaking about belief. (shrink)

Two concepts of utmost importance for the analytic philosophy of the twentieth century, “sense-data” and “knowledge by acquaintance”, were introduced by Bertrand Russell under the influence of two idealist philosophers: F. H. Bradley and Alexius Meinong. This paper traces the exact history of their introduction. We shall see that between 1896 and 1898, Russell had a fully-elaborated theory of “sense-data”, which he abandoned after his analytic turn of the summer of 1898. Furthermore, following a subsequent turn of August 1900—-after he (...) became acquainted with the works of Peano and later of Frege—-Russell gradually developed another theory of sense-data. With the collaboration of G. E. Moore, Russell reintroduced the term “sense-data” in 1911. Concomitantly with this move, Russell introduced the epistemological term “knowledge by acquaintance”, which came to designate the grasping of sense-data and universals. (shrink)

In the history of philosophy, the distinction between positive and negative predication has been collapsed. The collapse has caused us to search for a way through Parmenides’ gate: we have constructed scaffolding to see over its boundaries. Kant gave us the distinction between conceptual and non-conceptual knowledge; Hegel gave us determinate negation; Frege gave us the negation stroke; Husserl gave us bracketing and disappointment; G. Spencer-Brown gave us a calculus of distinction. Despite this, we find ourselves—alongside Wittgenstein— wondering how (...) it can be the case that we are able to think what is not the case. We are puzzled by the fact that we can think nothing without the thought lacking content. I suggest that this confusion is unnecessary. Throughout this work, I exercise the thought that the distinction between positive and negative predication is mistaken. I rather suggest that to think what is or is not the case necessarily depends upon an exemplar distinction that is experientially recognized and learned in the world. All the same, our capacity to understand natural distinction likewise depends upon our practical and pervasively conceptual capacities to express it in language. I conclude by suggesting that what makes positive and negative predication distinct is a fundamental unity in conceptual form through its capacity to play distinct functional roles in linguistic practice. (shrink)

This paper argues that conceptual engineering comes in many guises, often depending on what type of concept is being engineered. Engineering a classical concept, one that stems from Plato and Frege, is very different from engineering, e.g., a prototype concept or an exemplar concept. The former are abstract and have necessary and sufficient satisfaction conditions. The latter, on the other hand, can and do differ from one person to the next and thus have the earmarks of conceptions. While it (...) is tempting to argue that classical concepts are of natural kinds while prototype and/or exemplar concepts are of socially constructed kinds, the paper argues that such classification is not tenable. Many of our concepts of natural kinds are prototype concepts, and many socially constructed kinds are specified with necessary and sufficient conditions and so are have the earmarks of classical concepts. While the discussion of conceptual engineering has mostly focused on classical concepts and how scholars can, do, and perhaps should ameliorate them, the paper emphasizes the importance of other types of concepts and how they can be engineered. (shrink)

Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as Hilbert developed systems of logic to formalize mathematics. These systems were meant to serve either as themselves foundational, or at least as formal analogs of mathematical reasoning amenable to mathematical study, e.g., in Hilbert’s consistency program. Similar efforts continue, but have (...) been expanded by the development of sophisticated methods to study the properties of such systems using proof and model theory. In parallel with this evolution of logical formalisms as tools for articulating mathematical theories (broadly speaking), much progress has been made in the quest for a mechanization of logical inference and the investigation of its theoretical limits, culminating recently in the development of new foundational frameworks for mathematics with sophisticated computer-assisted proof systems. In addition, logical formalisms developed by logicians in mathematical and philosophical contexts have proved immensely useful in describing theories and systems of interest to computer scientists, and to some degree, vice versa. Three examples of the influence of logic in computer science are automated reasoning, computer verification, and type systems for programming languages. (shrink)

This paper argues that to account for group speech acts, we should adopt a representationalist account of mode / force. Individual and collective subjects do not only represent what they e.g. assert or order. By asserting or ordering they also indicate their theoretical or practical positions towards what they assert or order. The ‘Frege point’ cannot establish the received dichotomy of force and propositional content. On the contrary, only the representationalist account allows a satisfactory response to it. It also (...) allows us to give a more satisfactory analysis of the speech act of inviting a joint commitment and to answer two important questions Bernhard Schmid has raised about group speech acts, namely whether there are 1st person plural forms of Moore’s paradox and of 1st person authority. (shrink)

Advocates of Expressivism about basically any kind of language are best-served by abandoning a traditional content-centric approach to semantic theorizing, in favor of an update-centric or dynamic approach (or so this paper argues). The type of dynamic approach developed here — in contrast to the content-centric approach — is argued to yield canonical, if not strictly classical, "explanations" of the core semantic properties of the connectives. (The cases on which I focus most here are negation and disjunction.) I end the (...) paper by describing a distinctive sense in which mental states might play a fundamental role in the practice of semantic theorizing (as I understand it), and I connect this to a distinctive account of the pragmatic function of, e.g., a normatively laden claim in discourse. (shrink)

// tl;dr A Proposition is a Way of Thinking // -/- This chapter is about type-theoretic approaches to propositional content. Type-theoretic approaches to propositional content originate with Hintikka, Stalnaker, and Lewis, and involve treating attitude environments (e.g. "Nate thinks") as universal quantifiers over domains of "doxastic possibilities" -- ways things could be, given what the subject thinks. -/- This chapter introduces and motivates a line of a type-theoretic theorizing about content that is an outgrowth of the recent literature on epistemic (...) modality, according to which contentful thought is broadly "informational" in its nature and import. The general idea here is that an object of thought is not a way *the world* could be, but rather a way *one's perspective* could be (with respect to a relevant representational question). I will spend the middle part of this chapter motivating and developing a version of this strategy that is, I’ll argue, well-suited to explaining clear phenomena concerning the attribution of perspectival attitudes -- in particular, attitudes towards loosely information-sensitive propositions -- with which extant approaches struggle. My overarching goal here will be to motivate a distinctive version of the "informational" approach -- the "Flexible Types" approach, which is based on the theory proposed in Charlow (2020). According to the Flexible Types approach, propositional attitude verbs are quantifiers over sets of possibilities, but a possibility is a type-flexible notion -- sometimes a possible world, sometimes a perspective, sometimes a set of possible worlds, sometimes a set of perspectives. -/- After introducing the Flexible Types approach, this chapter circles back to more traditional concerns for the analysis of propositions as types of possibilities -- Frege's Puzzle and the problem of Logical Omniscience. Here too the Flexible Types approach bears fruit. Although there are certainly significant differences -- I note some in the concluding section -- the gist of this theory is Hinitkkan or Lewisian in spirit (if not quite in letter). We can make progress on addressing the challenges for the analysis of propositional content in terms of types of possibilities, through empirically driven refinement of our notion of what kind of thing a "doxastic possibility" is. (shrink)

A plurality or plural object is a single object that is also many, and pluralitism is the thesis that there is such an object. This paper argues that pluralitism and closely related theses (e.g., the many-one identity thesis and the composition as identity thesis) violate logic. To do so, it formulates an approach to the logic and semantics of plural constructions that results in plural logic and relates treatments of plural constructions to accounts of natural number. And it gives a (...) critical examination of Frege’s views of numbers and plural constructions, and sketches the view of plural constructions as devices for talking about the many (as such), and an account of natural numbers as properties of a special kind, plural properties. (shrink)

On Kripke’s intended definition, a term designates an object x rigidly if the term designates x with respect to every possible world in which x exists and does not designate anything else with respect to worlds in which x does not exist. Kripke evidently holds in Naming and Necessity, hereafter N&N (pp. 117–144, passim, and especially at 134, 139–140), that certain general terms – including natural-kind terms like ‘‘water’’ and ‘‘tiger’’, phenomenon terms like ‘‘heat’’ and ‘‘hot’’, and color terms like (...) ‘‘blue’’ – are rigid designators solely as a matter of philosophical semantics (independently of empirical, extra-linguistic facts). As a consequence, Kripke argues, identity statements involving these general terms are like identity statements involving proper names (e.g., ‘‘Clark Kent=Superman’’) in that, solely as a matter of philosophical semantics, they express necessary truths if they are true at all. But whereas it is reasonably clear what it is for a (first-order) singular term to designate, Kripke does not explicitly say what it is for a general term to designate. General terms are standardly treated in modern logic as predicates, usually monadic predicates. There are very forceful reasons – due independently to Church and Godel, and ultimately to Frege – for taking predicates to designate their semantic extensions. But insofar as the extension of the general term ‘‘tiger’’ is the class of actual tigers (or its characteristic function), it is clear that the term does not rigidly designate its extension, since the class of tigers in one possible world may differ from the class of tigers in another. What, then, is it for ‘‘tiger’’ to be rigid? (shrink)

John P. Burgess is the John N. Woodhull Professor of Philosophy at Princeton University. He obtained his Ph.D. from the Logic and Methodology program at the University of California at Berkeley under the supervision of Jack H. Silver with a thesis on descriptive set theory. He is a very distinguished and influential philosopher of mathematics. He has written several books: A Subject with No Object (with G. Rosen, Oxford University Press, 1997), Computability and Logic (with G. Boolos and R. Jeffrey, (...) 5th ed., Cambridge University Press, 2007), Fixing Frege (Princeton University Press, 2005), Mathematics, Models, and Modality (Cambridge University Press, 2007), Philosophical Logic (Princeton University Press, 2009), Truth (with A. G. Burgess, Princeton University Press, 2011), Saul Kripke: Puzzles & Mysteries (Polity Press, 2012), Rigor & Structure (Oxford University Press, 2015), and Set Theory (Cambridge Elements, Forthcoming). In this interview, Professor Burgess talks about how his interests in mathematics and philosophy developed and relate to each other. He then answers questions about specific themes of his philosophical work, with a focus on issues pertaining to philosophy of mathematics. (shrink)

A logic is called higher order if it allows for quantification over higher order objects, such as functions of individuals, relations between individuals, functions of functions, relations between functions, etc. Higher order logic began with Frege, was formalized in Russell [46] and Whitehead and Russell [52] early in the previous century, and received its canonical formulation in Church [14].1 While classical type theory has since long been overshadowed by set theory as a foundation of mathematics, recent decades have shown (...) remarkable comebacks in the fields of mechanized reasoning (see, e.g., Benzm¨. (shrink)

I have three main objectives in this essay. First, in chapter 2, I shall put forward and justify what I call worldlessness, by which I mean the following: All truths (as well as falsehoods) are wholly independent of any circumstances, not only time and place but also possible worlds. It follows from this view that whatever is actually true must be taken as true with respect to every possible world, which means that all truths are (in a sense) necessary. However, (...) the account I shall propound is different from what is known in the trade as necessitarianism, i.e. the view that there is only one possible world, viz. the actual one, for the doctrine of the worldlessness of truth values, despite its commitment to the necessity of truths and falsehoods, is quite compatible with the idea of there being other possible worlds. Another important issue in chapter 2, explored in particular in section 2.12, is the claim that there is no real change in the world. Secondly, in chapter 3 I consider the eminent traditional argument for determinism, deriving from Aristotle, namely, logical determinism, i.e. determinism justified by an appeal to the logical principle of bivalence (that all proper statements, including those concerning the future, are either true or false). In this connection I try to show that, (i), the formulation of the conclusion of this argument as "Whatever will happen will happen of necessity" is implausible, at least from the modern point of view, (ii), the formulation as "Whatever will happen will happen inevitably" is more to the point, and (iii), on the basis of the worldless and timeless aspect advocated in chapter 2, this latter formulation is quite harmless, essentially amounting to the trivial statement, "Whatever will happen will happen". Thirdly, in chapter 4 I study theological determinism, or determinism that arises from God's supposed providential control over everything that happens. In this connection, I shall survey some historical accounts of the relation between human free will and determinism (not only theological but also causal determinism); the philosophers the views of whom I shall attend to include Chrysippus, St. Augustine, Boethius and Aquinas. I shall in particular consider G.W. Leibniz' theodicean aspirations, viz. his solution to the problem of evil and, especially, his compatibilist attempts to reconcile human free will with the strictly deterministic flow of actual events. I think it is important to try to explicate Leibniz' ingenious account of these matters, since it seems that it has not been fully appreciated in the literature, not even by contemporary Leibniz scholars (such as B. Mates, R.C. Sleigh, C. Wilson, R.M. Adams and D. Rutherford). In providing the Leibnizian compatibilist solution of the problem of determinism and freedom in chapter 4, I shall utilize the approach of chapter 2. (shrink)

The theory that Kit Fine calls 'semantic relationism' replaces standard semantic compositionality with an alternative according to which statements of the form '... A … A ...’ and ‘... A … B ...’ (e.g., ‘Cicero admires Cicero’ and ‘Cicero admires Tully’) differ in semantic content—even where the two terms involved are exactly synonymous—simply in virtue of the recurrence that is present in the former statement and absent from the latter. A semantic-relationist alternative to standard compositionality was first explicitly proffered by (...) Hilary Putnam in 1954 and later by others in the latter part of the 20th Century. However, semantic relationism had already been implicitly rejected in Frege’s 1892 landmark masterpiece "Über Sinn und Bedeutung" (“On Sense and Reference”). The present compilation of three previously published essays cuts through the hype and thoroughly rebuts semantic relationism, particularly Fine’s more developed version (2007). A fatal fallacy is exposed in Fine's high-flown purported formal disproof of Millianism with standard compositionality. Standard Millianism does not implode; Fine's overzealous attempt to demolish it does. The failure of his purported disproof renders Fine's central notion of coordination either redundant or empty. So-called coordination is non-semantic if it is anything real. (shrink)

Direct reference theory faces serious prima facie counterexamples which must be explained away (e.g., that it is possible to know a priori that Hesperus = Phosphorus). This is done by means of various forms of pragmatic explanation. But when those explanations that provisionally succeed are generalized to deal with analogous prima facie counterexamples concerning the identity of propositions, a fatal dilemma results. Either identity must be treated as a four-place relation (contradicting what just about everyone, including direct reference theorists, takes (...) to be essential to identity). Or direct reference theorists must incorporate a view that was rejected in pretty much our first lesson about identity—namely, that Hesperus at twilight is not identical to Hesperus at dawn. One way of the other, the direct reference theory is thus inconsistent with basic principles concerning the logic of identity, which nearly everyone, including direct reference theorists, take as starting points. (shrink)

I here propose a hitherto unnoticed possibility of solving embedding problems for noncognitivist expressivists in metaethics by appeal to Conceptual Role Semantics. I show that claims from the latter as to what constitutes various concepts can be used to define functions from states expressed by atomic sentences to states expressed by complex sentences, thereby allowing an expressivist semantics that satisfies a rather strict compositionality constraint. The proposal can be coupled with several different types of concept individuation claim, and is shown (...) to pave the way to novel accounts for, e.g., negation. (shrink)