The paper presents Alfred Tarski’s debate with the semantic antinomies: the basic Liar Paradox, and its more sophisticated versions, which are currently discussed in philosophy: Strengthen Liar Paradox, Cyclical Liar Paradox, Contingent Liar Paradox, Correct Liar Paradox, Card Paradox, Yablo’s Paradox and a few others. Since Tarski, himself did not addressed these paradoxes—neither in his famous work published in 1933, nor in later papers in which he developed the Semantic Theory of Truth—therefore, We try to defend his concept (...) of truth against these antinomies. It is demonstrated that Tarskian theory of truth is resistant to the paradoxes and it is still the best solution to avoid the antinomies and remain within a classical logic, that is, accepting the laws of noncontradiction, excluded middle, and the principle of bivalence. Thus, the goal of the paper is double—firstly, to show that none of the versions of the Liar Paradox’s is a serious threat to Tarski’s concept of truth, and secondly, that Semantic Theory of Truth allows to remain within classical logic, and at the same time, avoid antinomies—which makes it the most attractive among classical theories of truth. (shrink)

It is shown that Russell's Paradox can be solved without advocating the Theory of Types, and also that the Liar's Paradox can be solved in much the same way. Neither solution requires that any of our commonsense-based beliefs be revised, let alone jettisoned. It is also shown that the Theory of Types is false.

By considering the semantic footings of the so-called antinomies of pure reason, this article contributes to the debate about whether Kant was committed to semantic realism or anti-realism. That is, whether verification-transcendent judgements are truth-apt (realism) or not (anti-realism). Against the (empiricist) semantic principle that Strawson, and others, have ascribed to Kant as the “principle of significance,” the bedrock of my article is what I call Kant’s Real Principle of Significance: an extension-based and normative principle stating that (...) a judgement can have no “significance” or “objective validity” (truth-value) without a universally recognizable norm for verifying it. This principle entails semantic anti-realism. I argue that we can extract the principle from the antinomy chapter of KrV, since in there Kant concludes that judgements of the form “the world as such is x” are without “significance” (lack a truth-value) in virtue of being unverifiable as a matter of principle. I propose that Kant’s reference to some of the antinomical judgements as “false” is not incompatible with this anti-realist reading because he operates with two distinct world-concepts: an illegitimate transcendental realist one and a legitimate transcendental idealist one. In contrast to most anti-realist Kant-interpretations, it is furthermore argued that any satisfactory anti-realist construal of Kant’s view must be compatible with his assertion that the thesis in the third antinomy about freedom “can be true.” That requires a thicker conception of “significance” or “objective validity” than what is often ascribed to Kant, which is encapsulated by the Real Principle of Significance. (shrink)

A new semantical antinomy, the antinomy of designation, is introduced into a metalanguage M with respect to a modal object language L. Carnap's device of restricting the principle of interchangeability for L does not suffice to prevent occurrence of this new antinomy. To achieve this result it seems most natural to replace the rules of designation for L by more complicated rules. This replacement suffices to prevent occurrence of the antinomy with respect to L. Moreover, it (...) seems likely that analogous replacement will suffice to prevent occurrence of the antinomy with respect to languages which in means of expression are richer than L. (shrink)

Kit Fine has reawakened a puzzle about variables with a long history in analytic philosophy, labeling it “the antinomy of the variable”. Fine suggests that the antinomy demands a reconceptualization of the role of variables in mathematics, natural language semantics, and first-order logic. The difficulty arises because: (i) the variables ‘x’ and ‘y’ cannot be synonymous, since they make different contributions when they jointly occur within a sentence, but (ii) there is a strong temptation to say that distinct (...) variables ‘x’ and ‘y’ are synonymous, since sentences differing by the total, proper substitution of ‘x’ for ‘y’ always agree in meaning. We offer a precise interpretation of the challenge posed by (i) and (ii). We then develop some neglected passages of Tarski to show that his semantics for variables has the resources to resolve the antinomy without abandoning standard compositional semantics. (shrink)

The essay considers three classic definitions of philosophy, namely those offered by Socrates, Boethius, as semantically enriched by Montaigne, and Kant, in order to reflected on individual and collective death. Kant’s philosophical tool of the antinomies of reason is deployed to think through the antinomies of our pandemic in order to make clear that in a pandemic there is only collective, and not individual or even national, inoculation. The false dichotomies of physical versus social, embodiment versus virtuality, nationalism versus planetarization, (...) and propaganda versus science, are discussed. Lady philosophy appears wearing blue plastic gloves. (shrink)

There is a solution to the antinomy of the variable that does not call for semantic relationism. I argue that if we carefully distinguish between variable types and variable tokens or occurrences, and if we take the number of variable types involved properly into account, then coordination among variable tokens or occurrences is reducible to an intrinsic semantic feature of those tokens or occurrences. The fact that two tokens or occurrences of the same variable type contained in (...) the same sentence or context assume the same value at a time derives from an intrinsic semantic feature of a token or occurrence, namely from being a token or occurrence of a type whose tokens or occurrences must assume the same value at a time if contained in the same sentence or context. (shrink)

Résumé Le débat holisme-réductionnisme se structure autour de trois domaines sémantiques : l 'ontologie, la méthodologie et l'épistémologie. Généralement, une méthodologie analytique s'accompagne d'une ontologie atomiste et de la réduction des lois et théorie des niveaux d'organisation supérieurs aux lois et théorie des niveaux inférieurs. Par contre, une ontologie holiste, relationnelle peut s'accorder au concept d'émergence. En conséquence dans l'élaboration des lois et théories d'un phénomène appartenant à un niveau donné la prise en compte du niveau d'organisation supérieurs se révélera (...) déterminante. Les propositions philosophiques anti-mécanicistes de précurseur tels que Bradley, S. Alexander, A.N. Whitehead, C.L. Morgan, D.C. Broad, même en restant au niveau d'une forme de proto-émergentisme représentent le fondement épistémologique à partir duquel s'est développée une méthodologie véritablement émergentiste. Abstract The holism-reductionism debate is structured around three areas of semantics: ontology, methodology and epistemology. As a general rule, an analytical methodology goes with an atomist ontology, and the reduction of laws and theories from the higher levels of organisation to the laws and theories of lower levels. On the other hand, a holistic and relational ontology can be in unison with the concept of emergence. As a consequence, in elaborating laws and theories of a phenornenon of a given level, taking into account that the higher level of organisation will be determinant. The philosophical and anti-mechanistic propositions of precursors such as F. Bradley, S. Alexander, A.N. Whitehead, C.L. Morgan, D.C. Broad, even remaining at the level of a kind of proto-emergentism represent the epistemological basis on which a true emergentist methodology has been developed. (shrink)

We discuss misinformation about “the liar antinomy” with special reference to Tarski’s 1933 truth-definition paper [1]. Lies are speech-acts, not merely sentences or propositions. Roughly, lies are statements of propositions not believed by their speakers. Speakers who state their false beliefs are often not lying. And speakers who state true propositions that they don’t believe are often lying—regardless of whether the non-belief is disbelief. Persons who state propositions on which they have no opinion are lying as much as those (...) who state propositions they believe to be false. Not all lies are statements of false propositions—some lies are true; some have no truth-value. People who only occasionally lie are not liars: roughly, liars repeatedly and habitually lie. Some half-truths are statements intended to mislead even though the speakers “interpret” the sentences used as expressing true propositions. Others are statements of propositions believed by the speakers to be questionable but without revealing their supposed problematic nature. The two “formulations” of “the antinomy of the liar” in [1], pp.157–8 and 161–2, have nothing to do with lying or liars. The first focuses on an “expression” Tarski calls ‘c’, namely the following. -/- c is not a true sentence -/- The second focuses on another “expression”, also called ‘c’, namely the following. -/- for all p, if c is identical with the sentence ‘p’, then not p -/- Without argumentation or even discussion, Tarski implies that these strange “expressions” are English sentences. [1] Alfred Tarski, The concept of truth in formalized languages, pp. 152–278, Logic, Semantics, Metamathematics, papers from 1923 to 1938, ed. John Corcoran, Hackett, Indianapolis 1983. -/- https://www.academia.edu/12525833/Sentence_Proposition_Judgment_Statement_and_Fact_Speaking_about_th e_Written_English_Used_in_Logic. (shrink)

Resolution of Frege's Puzzle by denying that synonym substitution in logical truths preserves sentence sense and explaining how logical form has semantic import.

Logical form has semantic import. Logical sentences (GG: Greeks are Greeks) and their synonym interceptions (GH: Greeks are Hellenes) state the same fact but different truths with different explanations. Terms retain objectual reference but its role in explaining truth is preempted by syntax or synonymy. Church’s Test exposes puzzles. QMi sentences (GmG: ‘Greeks’ means Greeks), and QTi sentences (p≡it is true that p≡“p” is true) are metalogical necessities, true by syntax. Their interceptions alter syntax and modality, yielding contingent truths (...) (GmH: ‘Greeks’ means Hellenes, HmG: ‘Hellenes’ means Greeks). Meta-logical translation preserves syntax (GmG: ‘Greichen’ bedeutet Greichen), not necessarily objectual reference. Metalogical syntax secures truth by self-referential quotational indexing that identifies quotational referent with an intrasentential replica. (shrink)

Resolution of Frege's Puzzle by denying that synonym substitution in logical truths preserves sentence sense and explaining how logical form has semantic import. Intensional context substitutions needn't preserve truth, because intercepting doesn't preserve sentence meaning. Intercepting is nonuniformly substituting a pivotal term in syntactically secured truth. Logical sentences and their synonym interceptions share factual content. Semantic content is factual content in synthetic predications, but not logical sentences and interceptions. Putnam's Postulate entails interception nonsynonymy. Syntax and vocabulary explain only (...) the factual content of synthetic predications; extrasentential reality explains their truth. Construction of logical factual content explains logical necessity. Terms retain objectual reference, but logical syntax preempts their function in explaining truth. Grasping the facts GG/gg assert entails understanding this. Understanding what GH states requires some recognition that GH must be true just because GmH, and GmH state an empirical fact. GH is standardly used to express that fact. Church's Test exposes puzzles. QMi sentences, and QTi sentences are metalogical necessities, true by syntax. Intercepting QMi creates empirical QM contingencies. Synonymy turns semantic contingencies into metalogical and lexical necessities. That transformation is syntactic, via the syntactic duality of definite descriptions. GmH is a contingent copredication, and a lexically necessary referential identity with rigidly codesignating indexicals. Metalogical sentences may be about expressional matter or what it expresses. GG has GG's semantic content, but the referent expression switches. Metalogical syntax secures truth by self-referential quotational indexing. Metalogically, referents are identified with intrasentential replica. Extrasentential identifications are metalogically irrelevant. (shrink)

According to the received view formal semantics applies to natural language to some extent only. It is so because natural language is inherently indefinite, in particular, its expressions are ambiguous, vague and admits departures from syntactic rule. Moreover, intensional contexts occur in ordinary language—it results in limitations of the principle of compositionality. The ordinary conversation appeals to various principles, for instance, Grice’s maxims which exceed logical formalism. Thus, ordinary language cannot be fully formalized. On the other hand, if L is (...) a formal language, its metalanguage ML, must be partially informal—for instance, it contains, terms of ordinary mathematics, especially set theory. Even, if, for instance, due to the technique of aritmetization, ML can be represented in L, such a representation is only local. In fact, this view can be derived from some Tarski’s remarks on the role played by natural language. It is usually assumed that the universality of natural language, is the source of troubles associated with antinomies. It is so this circumstance requires a solution, for example by distinguishing levels of language. However, even if antinomies are excluded, what is informal is prior with respect to what is formal. It shows that formal semantics has limitations even with respect to formalized languages. (shrink)

The paper is a comment on the formalization of the antinomy of futura contigentia in the form of a (inconsistent) theory formulated by Marcin Tkaczyk in the language of classical predicate calculus. I argue that some features of the formalization in question are controversial from the viewpoint of formal semantics and ontology, and suggest two ways of removing some of those controversies.

In speaking of the reasons for the appearance of semantics, certain writers note the following three problems the solution of which demanded semantic analysis: 1) the problem of antinomies; 2) the problem of refining and explaining the basic concepts of formalized logic and mathematics; and 3) the problem of the complete formalization of logic.

The idea of antithetical human nature and the concept of man as a bivalent creature are heuristic to reveal the problem of human being.The traditional antinomic splitting of anthropological features into one’s own – another’s, immanent – transcendent, freedom – necessity, good – evil, happiness –misfortune, etc. acquires a specific coloring in the Ukrainian realities. Purpose is to reveal the ambivalence of the image of a person, which acquires special features in the Ukrainian realities. Methodological basis are the principles of (...) the phenomenological description of subjective experience, the means of historical and philosophical analysis, hermeneutical methods of understanding and interpretation.Discussion. These or those anthropological modes constitute the variability of the human (O.Bollnov, C.Jung, K.Jaspers, J.-P.Sartre, E.Fromm). The reasons for the mosaic nature of the human image are rooted in the existential inexhaustibility of the anthropological palette. The existence of man in the overtones of antinomies is evident in the studies of both domestic classics (V.Lypynsky, I.Franko) and modern Ukrainian philosophers (V.Malakhov, D.Sepety), and Western thinkers (V.Frankl, V.Tatarkevych, G.Hofstede). Conclusion. Controversial nature of the classical opposition of freedom – necessity turns in the Ukrainian realities into the opposition of “will” as an expressive meaningful constant and depressingly deathly “bondage”, and acquires tension of the binarism “freedom – death”. “Will” as the only state acceptable to a person, the authentically own, native way of being turns to a spectrum of modulations of the anthropological attribute of “oneself” as a native oecumene, a truly familiar and close space, opposed to the “alien” – land, people, language, custom, etc.The idea of parity of both – male and female – cores is quite adequate, since in Ukraine the Cossacks was an embodiment of perfect masculinity, “chivalrous nature”, similar to Western chivalry – with similar to chivalrous cult foundations (for example, the cult of the Mother of God and the cult of the Virgin Mary). After all, the ideology of the Cossacks demanded the convergence of the concepts of “nobleman” and “Cossack” as one who was “granted the liberties of a knightly people”.Controversial nature of the modes of life meaning, happiness disappears, losing expressive signs of confrontation and turning into a tense uniqueness of the constants of meaning of life, fate, finiteness of a person as a whole.The concept of happiness, extrapolating into the opposition of happiness /fate – disaster /sorrow (deprivation, misfortune), powerful for Ukrainian culture, is connotative with “will” (after all, “what kind of freedom is such a fate...”), on the one hand, and “underweight” – an evil fate, on the other hand. However, on the existential plane, the life-meaning ethical component of happiness loses its dramatic controversial vector: happiness with its internal semantic and moral modulations is a selfsufficient value. (shrink)

Human language has the characteristic of being open and in some cases polysemic. The word “infinite” is used often in common speech and more frequently in literary language, but rarely with its precise meaning. In this way the concepts can be used in a vague way but an argument can still be structured so that the central idea is understood and is shared with to the partners. At the same time no precise definition is given to the concepts used and (...) each partner makes his own reading of the text based on previous experience and cultural background. In a language dictionary the first meaning of “infinite” agrees with the etymology: what has no end. We apply the word infinite most often and incorrectly as a synonym for “very large” or something that we do not perceive its completion. In this context, the infinite mentioned in dictionaries refers to the idea or notion of the “immeasurably large” although this is open to what the individual’s means by “immeasurably great.” Based on this linguistic imprecision, the authors present a non Cantorian theory of the potential and actual infinite. For this we have introduced a new concept: the homogon that is the whole set that does not fall within the definition of sets established by Cantor. (shrink)

In this article, I explore the consequences of two commonsensical premises in semantics and epistemology: (1) natural language is a complex system rooted in the communal life of human beings within a given environment; and (2) linguistic knowledge is essentially dependent on natural language. These premises lead me to emphasize the process-socio-environmental character of linguistic meaning and knowledge, from which I proceed to analyse a number of long-standing philosophical problems, attempting to throw new light upon them on these grounds. In (...) particular, I criticize the use of expressions such as ‘absolute truth’, ‘absolute existence’ and ‘the thing in itself’, arguing that they lead to what I call ‘the ultralinguistic paradox’ (a fatal antinomy). In the same way, I review a number of mainstream topics in philosophical semantics, epistemology and metaphysics, reformulating them in terms much more naturalistic – and less mysterious – than usual. (shrink)

First, language and axioms of Church's paper 'Comparison of Russell's Resolution of the Semantical Antinomies with that of Tarski' are slightly modified and a version of the Liar paradox tentatively reconstructed. An obvious natural solution of the paradox leads to a hierarchy of truth predicates which is of a different kind from the one defined by Church: it depends on the enlargement of the semantical vocabulary and its levels do not differ in the ramified-type-theoretical sense. Second, two attempts are made (...) in order to justify the Russellian, and perhaps Churchian, idea that language should not be fragmented beyond what is required by type distinctions. After all, because of reducibility, which seems to allow a semantics without propositions, this comes out to be possible only at the cost of resorting to two disputable theses. (shrink)

G¨odel’s a theorem concerns an arithmetical statement and the truth of this statement does not depend on self-reference; nevertheless its interpretation is of tremendous interest. G¨odel’s theorem allows one to conclude that formal arithmetic is not axiomatizable. But there is another very interesting logico-philosophical result: the possibility of a statement to exist such that it is improvable in the object-theory and at the same time its truth is provable in the metatheory. It seems that in the real history G¨odel’s theorem (...) was absolutely unexpected, and even in retrospect, now it is difficult to point out logico-philosophical reasons for the above possibility. In my opinion though it was still conceivable at that time, but unfortunately the philosophers and logicians missed to notice it. In the terms of recursive arithmetic Hilbert’s programme can be formulated as follows: Build a decidable, complete, consistent logical system such that the class of derivable statements coincides with the class of the mathematical statements which are intuitively true; prove this equivalence by means of an arithmetization of the theory in the framework of the recursive arithmetic! Now, if the system is so strong as to admit the construction of the classical mathematics, then in particular it will contain the classical arithmetic. But Hilbert’s requirements reduce to a proof with the means of the recursive arithmetic, by arithmetization. So, on one hand the system will contain the arithmetic and, on the other hand it will be expressible in it, in order words the system will include its own syntax, as well as its own semantics. Now it is clear that Hilbert’s programme contains a possibility of a situation similar to some well known semantic antinomies, situations of self-reference. (shrink)

In Fichte’s early views of the basic laws of traditional formal logic, primarily the law of identity, there is a tension that has gone surprisingly unexplored: While Fichte holds the statements of these laws to be self-evidently true and absolutely certain, he nevertheless claims that they remain to be justified by his “Science of Knowledge.” The aim of this article is to make sense of this tension and to explore how it translates into the dialectical structure and methodology of Fichte’s (...) first Jena Wissenschaftslehre. This is done by, first, conjecturing—in a somewhat ahistorical, yet Fichte-based, fashion—a reason for Fichte’s justificatory demand. It is argued that the validity of the law of identity can be questioned because our belief in its absolute generality appears to be self-refuting in that it involves an antinomy akin to Grelling’s semantic antinomy of the heterological. This antinomy, when, secondly, related to Fichte’s purported justification of the law of identity, serves as a key to understanding why there is an antinomic conflict between Fichte’s supreme principle—namely, the self-positing pure I—and its adversary, the not-I, in the first place. Tracing their contradiction back to that semantic antinomy inherent in our formal-logical certainties opens up a new way of seeing Fichte as radicalizing Kant’s critical philosophy, understood as the project of the self-preservation of reason against reason’s own antinomies. (shrink)

In Fichte’s early views of the basic laws of traditional formal logic, primarily the law of identity, there is a tension that has gone surprisingly unexplored: While Fichte holds the statements of these laws to be self-evidently true and absolutely certain, he nevertheless claims that they remain to be justified by his “Science of Knowledge.” The aim of this article is to make sense of this tension and to explore how it translates into the dialectical structure and methodology of Fichte’s (...) first Jena Wissenschaftslehre. This is done by, first, conjecturing—in a somewhat ahistorical, yet Fichte-based, fashion—a reason for Fichte’s justificatory demand. It is argued that the validity of the law of identity can be questioned because our belief in its absolute generality appears to be self-refuting in that it involves an antinomy akin to Grelling’s semantic antinomy of the heterological. This antinomy, when, secondly, related to Fichte’s purported justification of the law of identity, serves as a key to understanding why there is an antinomic conflict between Fichte’s supreme principle—namely, the self-positing pure I—and its adversary, the not-I, in the first place. Tracing their contradiction (whose synthetic resolution is the main goal of Fichte’s Wissenschaftslehre) back to that semantic antinomy inherent in our formal-logical certainties opens up a new way of seeing Fichte as radicalizing Kant’s critical philosophy, understood as the project of the self-preservation of reason against reason’s own antinomies. (shrink)

The subject of the present article is the analysis of fundamental logical-semantical terminology of late-medieval nominalistic logician Jean Buridan (c. 1295–1360). The analysis focuses on the concepts of truth conditions and logical consequence, whose clarification presupposes explication of modal terminology as well as a solution of semantical antinomies such as “Liar” (or an attempt to solve them). The analysis of Buridan’s argumentation suggests that Buridan’s project of logic actually fails due to several failures of conceptual analysis of semantical and modal (...) terminology. An alternative solution of the question concerning logical consequence is thus proposed in terms of Buridan’s implicit (and unused) semantical conception of modalities that makes it possible to establish conceptually and therefore explicatively closed logical framework. (shrink)

The aim of the article is to present the results of research conducted as part of the project Polonica Philosophica Orientalia: namely, to give an overview of the most important logical and metaphysical treatises written in Vilnius between the sixteenth and eighteenth centuries that are currently accessible in some of the Vilnius libraries. Although the research focused primarily on the Vilnius University Library and its resources, some interesting results were also obtained while researching the Wróblewski Library of the Lithuanian Academy (...) of Sciences. Most sources containing information about the state and the nature of metaphysical and logical investigations conducted in Vilnius during the period discussed are the manuscripts of lecture courses delivered for the students of Vilnius University and other schools. The analysis of those manuscripts has revealed a strong and long-lasting influence of scholastic Aristotelianism in both logic and metaphysics. Namely, up to the mid-eighteenth century, the main questions discussed in logic were Aristotelian syllogism, the square of opposition and the medieval topics of semantical antinomies, modalities and supposition theory, while investigations into metaphysical issues were limited to commentaries on Aristotle’s theoretical treatises without seriously questioning their main theses. In the second part of the eighteenth century, a moderate shift towards subjectivism and psychologism was observed in both of these disciplines. This was influenced by modern epistemologists. Logicians and metaphysicians of Vilnius University developed an interest in the topics of human understanding and cognition. However, this shift did not take a radical form – while the majority of eighteenth-century authors sought a combination of scholastic and modern ideas, a critical and even hostile stance towards modern philosophy and science is still noticeable in many of the analysed sources. An overview of the research supported by statistical data is followed by some preliminary remarks on how and where investigation of this topic might be continued in the future. (shrink)

Tarski's definition of truth has rehabilitated the application of the word "true" to sentences of formalized languages. But a correspondence theory according to which a sentence is true if, And only if, It is related in the peculiar way of correspondence to the facts, Is incompatible with tarski's definition. Actually no theory of truth, Which claims to make proper assertions about sentences when calling them true, Is compatible with tarski's definition. Hence they all have to find their own solution to (...) the problem of the semantical antinomies and have to explain what it means to say that sentences are true. (shrink)

The observation that the standard solution of the paradox of the Liar is not satisfactory as a pragmatic solution of a semantic problem restores its former status as a semantic antinomy. Since the antinomy originates from Tarski's T scheme, a conservative modification of the standard semantics is looked for, which would prevent applying the scheme T to anomalous statements. Two such modifications are considered. The first is simpler and implies Kleene's weak tables for three-valued logic. The (...) second, more complex but also more intuitive, agrees with Kleene's strong tables. This undermines the seemingly natural intuition that a sentence and the sentence attributing truth to it have the same truth conditions, which is compatible only with the weak tables. (shrink)

In a universal logic containing naive semantics the semantic antinomies will be provable. Although being provable they are not assertiblebecause of some pragmatic constraints on assertion I will argue for. Furthermore, since it is not acceptable that the thesis of dialethism is a dialethiaitself, what it would be according to naive semantics and the prefered logical systems of dialethism, a corresponding restriction on proof theory isnecessary.

Let us consider an arbitrary semantical relation. It holds between some linguistic entities and pieces of reality referred to. We may call it meaning. The controversy between psychologism and antipsychologism therefore may be exhibited as an ontological dilemma:Antipsychologism PsychologismThe relation of meaning is independent of human beings The relation of meaning is established by human beingsWhen we describe the meaning of words we do not need to refer to human behavior When we describe the meaning of words we need to (...) refer to human behaviorIt seems to be safe to assume that reality is not contradictory and a good description of reality cannot lead to the contradiction. All presentations of semantical antinomies rest on antipsychologistic description of semantical relations. Thus we can blame the contradictions called semantical antinomies on the antipsychologistic description. (shrink)

In his recent paper in History and Philosophy of Logic, John Kearns argues for a solution of the Liar paradox using an illocutionary logic (Kearns 2007 ). Paraconsistent approaches, especially dialetheism, which accepts the Liar as being both true and false, are rejected by Kearns as making no ?clear sense? (p. 51). In this critical note, I want to highlight some shortcomings of Kearns' approach that concern a general difficulty for supposed solutions to (semantic) antinomies like the Liar. It (...) is not controversial that there are languages which avoid the Liar. For example, the language which consists of the single sentence ?Benedict XVI was born in Germany? lacks the resources to talk about semantics at all and thus avoids the Liar. Similarly, more interesting languages such as the propositional calculus avoid the Liar by lacking the power to express semantic concepts or to quantify over propositions. Kearns also agrees with the dialetheist claim that natural languages are semantically closed (i.e. are able to talk about their sentences and the semantic concepts and distinctions they employ). Without semantic closure, the Liar would be no real problem for us (speakers of natural languages). But given the claim, the expressive power of natural languages may lead to the semantic antinomies. The dialetheist argues for his position by proposing a general hypothesis (cf. Bremer 2005 , pp. 27?28): ?(Dilemma) A linguistic framework that solves some antinomies and is able to express its linguistic resources is confronted with strengthened versions of the antinomies?. Thus, the dialetheist claims that either some semantic concepts used in a supposed solution to a semantic antinomy are inexpressible in the framework used (and so, in view of the claim, violate the aim of being a model of natural language), or else old antinomies are exchanged for new ones. One horn of the dilemma is having inexpressible semantic properties. The other is having strengthened versions of the antinomies, once all semantic properties used are expressible. This dilemma applies, I claim, to Kearns' approach as well. (shrink)

This chapter contains section titled: The Antinomy of the Variable The Tarskian Approach The Rejection of Semantic Role The Instantial Approach The Algebraic Approach The Relational Approach Relational Semantics for First‐order Logic.

We present a semantic proof of Gödel's second incompleteness theorem, employing Grelling's antinomy of heterological expressions. For a theory T containing ZF, we define the sentence HETT which says intuitively that the predicate “heterological” is itself heterological. We show that this sentence doesn't follow from T and is equivalent to the consistency of T. Finally we show how to construct a similar incompleteness proof for Peano Arithmetic.

The semantic rules that govern ordinary property discourse appear to give rise to a version of Russell's antinomy. Do we therefore have an inconsistent conception of properties? This paper firstly develops a consistent conception of properties and secondly argues that we may indeed interpret ordinary property discourse as expressing the consistent conception rather than an inconsistent one.

ABSTRACT This paper aims to show that intuitionistic Kripke models are a powerful tool for interpreting Kant’s ‘Critical Philosophy’. Part I reviews some old work of mine that applies these models to provide a reading of Kant’s second antinomy about the divisibility of matter and to answer several attacks on Kant’s antinomies. But it also points out three shortcomings of that original application. First, the reading fails to account for Kant’s second antinomy claim that matter is divisible ‘ad (...) infinitum’ and his first antinomy claim that empirical space extends only ‘ad indefinitum’. Secondly, in conformity with the ‘assertability’ heuristics for Kripke models, it attributes an assertability semantics to Kant. In fact, it attributes a pair of assertability semantics to Kant, but neither of these accords with modern assertabilism. And finally, one must wonder the propriety of using contemporary assertability and contemporary formal logic to save an eighteenth-century text. Part II goes historical to solve the problems about assertability. It outlines the Descartes–Leibniz dialectic about the infinite/indefinite distinction and demonstrates how each position reflects larger philosophical positions. It then demonstrates that Kant’s two versions of assertability and his version of the infinite/indefinite dichotomy emerge naturally from his attempts to resolve a tension in Leibniz’s philosophy. In light of this demonstration, Part III adapts the Kripke model interpretation so that it does reflect that dichotomy. It then refines the Kripke semantics in order to model some of Kant’s signature critical doctrines, and it shows how Kant’s critical version of the infinite/indefinite distinction differs from all those of his predecessors and from his own pre-critical version. It also addresses the question of using contemporary logical machinery to interpret Kant. Part IV turns the tables and uses what we have learned about Kant and modern semantics in order to address a pair of issues in the contemporary critical foundations of logic. (shrink)

The issue of the emergence of formal axiomatic logical systems due to the emergence of logical antinomies in formal axiomatic systems, specifically the issue of developing formal logical axiomatics in the calculus of considerations was investigated in the considered research. At the same time, in order to determine the characteristics of the implementation of the logical-methodological principles and provisions of the deductive reasoning obviously, conceptual-logical foundations of the calculus of considerations was studied and the main propositions of the calculus of (...) considerations and the analysis of the initial logical operations on them were given in the study. The basic laws of logic and the expression of one logical act with another one were also explained in the research work by giving the concepst of proportional form and tautology in the calculus of considerations. At the same time, main properties and the procedure of setting of the formal deductive theory in the calculus of considerations were studied in the article. In order to provide an adequate characterization of the conceptual and methodological bases of the calculus of considerations in the research work, the basic logical laws of the calculus of considerations developed by the German mathematician and thinker P. Hilbert were also given. In addition, the main principles and methodological provisions of the deductive method were investigated, and the rules of deriving new conclusions from the axioms were interpreted. At the end of the reviewed article, the main principles and methods of the formal axiomatic mathematical systems built on a deductive basis, specifically the calculus of considerations, were studied and a logical-methodological analysis of the calculus of considerations was carried out on this basis. In this framework, the syntactic and semantic analysis of formalized language in formal axiomatic logical systems was carried out separately, the constituent parts of those systems, including the construction scheme, were given. (shrink)

l. There is an antinomy in Hare's thought between Ought-Implies-Can and No-Indicatives-from-Imperatives. It cannot be resolved by drawing a distinction between implication and entailment. 2. Luther resolved this antinomy in the l6th century, but to understand his solution, we need to understand his problem. He thought the necessity of Divine foreknowledge removed contingency from human acts, thus making it impossible for sinners to do otherwise than sin. 3. Erasmus objected (on behalf of Free Will) that this violates Ought-Implies-Can (...) which he supported with Hare-style ordinary language arguments. 4. Luther a) pointed out the antinomy and b) resolved it by undermining the prescriptivist arguments for Ought-Implies-Can. 5. We can reinforce Luther's argument with an example due to David Lewis. 6. Whatever its merits as a moral principle, Ought-Implies-Can is not a logical truth and should not be included in deontic logics. Most deontic logics, and maybe the discipline itself, should therefore be abandoned. 7. Could it be that Ought-Conversationally-Implies-Can? Yes - in some contexts. But a) even if these contexts are central to the evolution of Ought, the implication is not built into the semantics of the word; b) nor is the parallel implication built into the semantics of orders; and c) in some cases Ought conversationally implies Can, only because Ought-Implies-Can is a background moral belief. d) Points a) and b) suggest a criticism of prescriptivism - that Oughts do not entail imperatives but that the relation is one of conversational implicature. 8. If Ought-Implies-Can is treated as a moral principle, Erasmus' argument for Free Will can be revived (given his Christian assumptions). But it does not 'prove' Pelagianism as Luther supposed. A semi-Pelagian alternative is available. (shrink)

The basic conflict: an initial characterization -- The main arguments against ordinary language philosophy -- Must philosophers rely on intuitions? -- Contextualism and the burden of knowledge -- Contextualism, anti-contextualism, and knowing as being in a position to give assurance -- Conclusion: skepticism and the dialectic of (semantically pure) "knowledge" -- Epilogue: ordinary language philosophy, Kant, and the roots of antinomial thinking.

In §93 of The Principles of Mathematics, Bertrand Russell observes that “the variable is a very complicated logical entity, by no means easy to analyze correctly”. This assessment is borne out by the fact that even now we have no fully satisfactory understanding of the role of variables in a compositional semantics for first-order logic. In standard Tarskian semantics, variables are treated as meaning-bearing entities; moreover, they serve as the basic building blocks of all meanings, which are constructed out of (...) variable assignments. But this has disquieting consequences, including Fine’s antinomy of the variable and an undue dependence of meanings on language. Here I develop an alternative, Fregean version of predicate logic that uses the traditional quantifier–variable apparatus for the expression of generality, possesses a fully compositional, non-representational semantics, and is not subject to the antinomy of the variable. The advantages of Fregean over Tarskian predicate logic are due to the former’s treating variables not as meaningful lexical items, but as mere marks of punctuation, similar to parentheses. I submit that this is indeed how the variables of predicate logic should be construed. (shrink)

Foundations of Set Theory discusses the reconstruction undergone by set theory in the hands of Brouwer, Russell, and Zermelo. Only in the axiomatic foundations, however, have there been such extensive, almost revolutionary, developments. This book tries to avoid a detailed discussion of those topics which would have required heavy technical machinery, while describing the major results obtained in their treatment if these results could be stated in relatively non-technical terms. This book comprises five chapters and begins with a discussion of (...) the antinomies that led to the reconstruction of set theory as it was known before. It then moves to the axiomatic foundations of set theory, including a discussion of the basic notions of equality and extensionality and axioms of comprehension and infinity. The next chapters discuss type-theoretical approaches, including the ideal calculus, the theory of types, and Quine's mathematical logic and new foundations; intuitionistic conceptions of mathematics and its constructive character; and metamathematical and semantical approaches, such as the Hilbert program. This book will be of interest to mathematicians, logicians, and statisticians. (shrink)

Are Fregean thoughts compositionally complex and composed of senses? We argue that, in Begriffsschrift, Frege took 'conceptual contents' to be unstructured, but that he quickly moved away from this position, holding just two years later that conceptual contents divide of themselves into 'function' and 'argument'. This second position is shown to be unstable, however, by Frege's famous substitution puzzle. For Frege, the crucial question the puzzle raises is why "The Morning Star is a planet" and "The Evening Star is a (...) planet" have different contents, but his second position predicts that they should have the same content. Frege's response to this antinomy is of course to distinguish sense from reference, but what has not previously been noticed is that this response also requires thoughts to be compositionally complex, composed of senses. That, however, raises the question just how thoughts are composed from senses. We reconstruct a Fregean answer, one that turns on an insistence that this question must be understood as semantic rather than metaphysical. It is not a question about the intrinsic nature of residents of the third realm but a question about how thoughts are expressed by sentences. (shrink)