Liar Paradox

In this paper a principle of substitutivity of logical equivalents salve veritate and a version of Leibniz’s law are formulated and each is shown to cause problems when combined with naive truth theories.

This is the story of how the noble squire Sancho Panza, while governing what he thought to be an insula, ingeniously solved a paradox not unlike those of modern logic.

This article informally presents a solution to the paradoxes of truth and shows how the solution solves classical paradoxes (such as the original Liar) as well as the paradoxes that were invented as counter-arguments for various proposed solutions (``the revenge of the Liar''). Any solution to the paradoxes of truth necessarily establishes a certain logical concept of truth. This solution complements the classical procedure of determining the truth values of sentences by its own failure and, when the procedure fails, through (...) an appropriate semantic shift allows us to express the failure in a classical two-valued language. Formally speaking, the solution is a language with one meaning of symbols and two valuations of the truth values of sentences. The primary valuation is a classical valuation that is partial in the presence of the truth predicate. It enables us to determine the classical truth value of a sentence or leads to the failure of that determination. The language with the primary valuation is precisely the largest intrinsic fixed point of the strong Kleene three-valued semantics (LIFPSK3). The semantic shift that allows us to express the failure of the primary valuation is precisely the classical closure of LIFPSK3: it extends LIFPSK3 to a classical language in parts where LIFPSK3 is undetermined. Thus, this article provides a content-wise argumentation, which has not been present in contemporary debates so far, for the choice of LIFPSK3 and its classical closure as the right model for the logical concept of truth. In the end, an erroneous critique of Kripke-Feferman axiomatic theory of truth, which is present in contemporary literature, is pointed out. (shrink)

The concept of truth has many aims but only one source. The article describes the primary concept of truth, here called the synthetic concept of truth, according to which truth is the objective result of the synthesis of us and nature in the process of rational cognition. It is shown how various aspects of the concept of truth -- logical, scientific, and mathematical aspect -- arise from the synthetic concept of truth. Also, it is shown how the paradoxes of truth (...) arise. (shrink)

While non-classical theories of truth that take truth to be transparent have some obvious advantages over any classical theory that evidently must take it as non-transparent, several authors have recently argued that there's also a big disadvantage of non-classical theories as compared to their “external” classical counterparts: proof-theoretic strength. While conceding the relevance of this, the paper argues that there is a natural way to beef up extant internal theories so as to remove their proof-theoretic disadvantage. It is suggested that (...) the resulting internal theories should seem preferable to their external counterparts. (shrink)

We can classify the (truth-theoretic) paradoxes according to their degrees of paradoxicality. Roughly speaking, two paradoxes have the same degrees of paradoxicality, if they lead to a contradiction under the same conditions, and one paradox has a (non-strictly) lower degree of paradoxicality than another, if whenever the former leads to a contradiction under a condition, the latter does so under the same condition. In this paper, we outline some results and questions around the degrees of paradoxicality and summarize recent progress.

Section 1 reviews Strawson’s logic of presuppositions. Strawson’s justification is critiqued and a new justification proposed. Section 2 extends the logic of presuppositions to cases when the subject class is necessarily empty, such as (x)((Px & ~Px) → Qx) . The strong similarity of the resulting logic with Richard Diaz’s truth-relevant logic is pointed out. Section 3 further extends the logic of presuppositions to sentences with many variables, and a certain valuation is proposed. It is noted that, given this valuation, (...) Gödel’s sentence becomes neither true nor false. The similarity of this outcome with Goldstein and Gaifman’s solution of the Liar paradox, which is discussed in section 4, is emphasized. Section 5 returns to the definition of meaningfulness; the meaninglessness of certain sentences with empty subjects and of the Liar sentence is discussed. The objective of this paper is to show how all of the above-mentioned concepts are interrelated. (shrink)

The paper studies a cluster of systems for fully disquotational truth based on the restriction of initial sequents. Unlike well-known alternative approaches, such systems display both a simple and intuitive model theory and remarkable proof-theoretic properties. We start by showing that, due to a strong form of invertibility of the truth rules, cut is eliminable in the systems via a standard strategy supplemented by a suitable measure of the number of applications of truth rules to formulas in derivations. Next, we (...) notice that cut remains eliminable when suitable arithmetical axioms are added to the system. Finally, we establish a direct link between cut-free derivability in infinitary formulations of the systems considered and fixed-point semantics. Noticeably, unlike what happens with other background logics, such links are established without imposing any restriction to the premisses of the truth rules. (shrink)

This paper decomposes the Liar Paradox into its semantic atoms using Meaning Postulates (1952) provided by Rudolf Carnap. Formalizing truth values of propositions as Boolean properties of these propositions is a key new insight. This new insight divides the translation of a declarative sentence into its equivalent mathematical proposition into three separate steps. When each of these steps are separately examined the logical error of the Liar Paradox is unequivocally shown.

Minimal Type Theory (MTT) shows exactly how all of the constituent parts of an expression relate to each other (in 2D space) when this expression is formalized using a directed acyclic graph (DAG). This provides substantially greater expressiveness than the 1D space of FOPL syntax. -/- The increase in expressiveness over other formal systems of logic shows the Pathological Self-Reference Error of expressions previously considered to be sentences of formal systems. MTT shows that these expressions were never truth bearers, thus (...) never sentences of any formal logic system. (shrink)

For any natural (human) or formal (mathematical) language L we know that an expression X of language L is true if and only if there are expressions Γ of language L that connect X to known facts. -/- By extending the notion of a Well Formed Formula to include syntactically formalized rules for rejecting semantically incorrect expressions we recognize and reject expressions that evaluate to neither True nor False.

By extending the notion of a Well Formed Formula to include syntactically formalized rules for rejecting semantically incorrect expressions we recognize and reject expressions that have the semantic error of Pathological self-reference(Olcott 2004). The foundation of this system requires the notion of a BaseFact that anchors the semantic notions of True and False. When-so-ever a formal proof from BaseFacts of language L to a closed WFF X or ~X of language L does not exist X is decided to be semantically (...) incorrect. (shrink)

Minimal Type Theory (MTT) is based on type theory in that it is agnostic about Predicate Logic level and expressly disallows the evaluation of incompatible types. It is called Minimal because it has the fewest possible number of fundamental types, and has all of its syntax expressed entirely as the connections in a directed acyclic graph.

Some such as Dean (2014) suggest that Montague's paradox requires the necessitation rule, and that the use of the rule in such a context is contentious. But here, I show that the paradox arises independently of the necessitation rule. A derivation of the paradox is given in modal system T without deploying necessitation; a necessitation-free derivation is also formulated in a significantly weaker system.

As part of an approach to the liar paradox and the other paradoxes affecting truth, I have proposed replacing our concept of truth with two concepts: ascending truth and descending truth.1 I am not going to discuss why I think this is the best approach or how it solves the paradoxes; instead, I concentrate on the theory of ascending and descending truth. I formulate an axiomatic theory of ascending truth and descending truth (ADT) and provide a possible-worlds semantics for it (...) (which I dub xeno semantics). Xeno semantics is a generalization of the familiar neighborhood semantics, which itself is a generalization of the standard relational semantics. Once the details of ADT have been presented, it is easy to show that neither relational semantics nor neighborhood semantics will work for it; thus, the move to a more general framework is required. The main result is a fixed point theorem that guarantees the existence of an acceptable first-order constant-domain xeno model. From this result it follows that ADT is sound with respect to the class of such models. The upshot is that ADT is consistent relative to the background set theory. (shrink)

It seems that the most common strategy to solve the liar paradox is to argue that liar sentences are meaningless and, consequently, truth-valueless. The other main option that has grown in recent years is the dialetheist view that treats liar sentences as meaningful, truth-apt and true. In this paper I will offer a new approach that does not belong in either camp. I hope to show that liar sentences can be interpreted as meaningful, truth-apt and false, but without engendering any (...) contradiction. This seemingly impossible task can be accomplished once the semantic structure of the liar sentence is unpacked by a quantified analysis. The paper will be divided in two sections. In the first section, I present the independent reasons that motivate the quantificational strategy and how it works in the liar sentence. In the second section, I explain how this quantificational analysis allows us to explain the truth teller sentence and a counter-example advanced against truthmaker maximalism, and deal with some potential objections. (shrink)

The paper explores applications of Kripke's theory of truth to semantics for anti-luck epistemology, that is, to subjunctive theories of knowledge. Subjunctive theories put forward modal or subjunctive conditions to rule out knowledge by mere luck as to be found in Gettier-style counterexamples to the analysis of knowledge as justified true belief. Because of the subjunctive nature of these conditions the resulting semantics turns out to be non-monotone, even if it is based on non-classical evaluation schemes such as strong Kleene (...) or FDE. This blocks the usual road to fixed-point results for Kripke's theory of truth within these semantics and consequently the paper is predominantly an exploration of fixed point results for Kripke's theory of truth within non-monotone semantics. Using the theory of quasi-inductive definitions we show that in case of the subjunctive theories of knowledge the so-called Kripke jump will have fixed points despite the non-monotonicity of the semantics: Kripke's theory of truth can be successfully applied in the framework of subjunctive theories of knowledge. (shrink)

This is a discussion of different ways of working out the idea that the semantic paradoxes show that natural languages are somehow “inconsistent”. I take the workable form of the idea to be that there are expressions such that a necessary condition of understanding them is that one be inclined to accept inconsistent claims (an conception also suggested by Matti Eklund). I then distinguish “simple” from “complex” forms of such views. On a simple theory, such expressions are meaningless, while on (...) a complex theory they are not. I argue that complex theories are incompatible with truth conditional semantics and that simple theories are only coherent when the inconsistent claims are metalingusitic attributions of meaning. I close with a discussion of the version of the simple metalinguistic theory I have defended in “Understanding the Liar” and other papers. (shrink)

In spite of the various responses and solutions provided for the paradox of liar, different scholars and thinkers are still trying to discover some new points in this regard. We can say that this paradox has attracted the attention of a lot of philosophers due to its simplicity of formulation, long history and complexity of solution. In fact, the existing diversity in the presented solutions to this paradox is not less than the diversity in the solutions provided for the very (...) puzzling paradox. The clarification of this issue, alongside with explaining the existence or the possibility of existence of different solutions to this paradox on the basis of the analysis provided by Fazil Sarab, will be useful not only in solving the paradox of liar but also in removing ambiguities from other philosophical problems. (shrink)

The thesis that nothing is true has long been thought to be a self-refuting position not worthy of serious philosophical consideration. Recently, however, the thesis of alethic nihilism—that nothing is true—has been explicitly defended (notably by David Liggins). Nihilism is also, I argue, a consequence of other views about truth that have recently been advocated, such as fictionalism about truth and the inconsistency account. After offering an account of alethic nihilism, and how it purports to avoid the self-refutation problem, I (...) argue that it avoids the problem at the expense of changing the subject. I then present other arguments against nihilism and responses to the considerations offered in defense of it. The only tenable position is that something is indeed true. (shrink)

Systems of paraconsistent logics violate the law of explosion: from contradictory premises not every formula follows. One of the philosophical options for interpreting the contradictions allowed as premises in these cases was put forward recently by Carnielli and Rodrigues, with their epistemic approach to paraconsistent logics. In a nutshell, the plan consists in interpreting the contradictions in epistemic terms, as indicating the presence of non-conclusive evidence for both a proposition and its negation. Truth, in this approach, is consistent and is (...) dealt with by classical logic. In this paper we discuss the fate of the Liar paradox in this picture. While this is a paradox about truth, it cannot be accommodated by the classical part of the approach, due to trivialization problems. On the other hand, the paraconsistent part does not seem fit as well, due to the fact that its intended reading is in terms of non-conclusive evidence, not truth. We discuss the difficulties involved in each case and argue that none of the options seems to accommodate the paradox in a satisfactory manner. (shrink)

The liar paradox is still an open philosophical problem. Most contemporary answers to the paradox target the logical principles underlying the reasoning from the liar sentence to the paradoxical conclusion that the liar sentence is both true and false. In contrast to these answers, Buddhist epistemology offers resources to devise a distinctively epistemological approach to the liar paradox. In this paper, I mobilise these resources and argue that the liar sentence is what Buddhist epistemologists call a contradiction with one’s own (...) words. I situate my argument in the works of Dignāga and Dharmakīrti and show how Buddhist epistemology answers the paradox. (shrink)

In the 1951 Gibbs lecture, Gödel asserted his famous dichotomy, where the notion of informal proof is at work. G. Priest developed an argument, grounded on the notion of naïve proof, to the effect that Gödel’s first incompleteness theorem suggests the presence of dialetheias. In this paper, we adopt a plausible ideal notion of naïve proof, in agreement with Gödel’s conception, superseding the criticisms against the usual notion of naïve proof used by real working mathematicians. We explore the connection between (...) Gödel’s theorem and naïve proof so understood, both from a classical and a dialetheic perspective. (shrink)

This paper motivates and defends alethic nihilism, the theory that nothing is true. I first argue that alethic paradoxes like the Liar and Curry motivate nihilism; I then defend the view from objections. The critical discussion has two primary outcomes. First, a proof of concept. Alethic nihilism strikes many as silly or obviously false, even incoherent. I argue that it is in fact well-motivated and internally coherent. Second, I argue that deflationists about truth ought to be nihilists. Deflationists maintain that (...) the utility of the truth predicate is exhausted by its expressive role, and I argue that the truth predicate can still play this expressive role even if nothing is true. As such, deflationists do not stand to lose anything by accepting nihilism. Since they also stand to gain an elegant solution to the alethic paradoxes, on balance deflationists ought to be nihilists. (shrink)

We may suppose that the truth predicate that we utilize in our semantic metalanguage is a two-place predicate relating sentences to contexts, the truth-in-context-X predicate. Seeming paradoxes pertaining to the truth-in-context-X predicate can be blocked by placing restrictions on the structure of contexts. While contexts must specify a domain of contexts, and what a context constant denotes relative to a context must be a context in the context domain of that context, no context may belong to its own context domain. (...) A generalization of that restriction appears to block all of the paradoxes of truth-in-context-X. This restriction entails that, in a certain sense, we cannot talk about the context we are in. This result will be defended, up to a point, on broadly ontological grounds. It will also be conjectured that our semantic metalanguage can be regarded as semantically closed. (shrink)

The Liar sentence L, which reads ‘L is not true’, can be used to produce an apparently valid argument proving that L is not true and that L is true. There has been increasing recognition of the appeal of contextualist solutions to the Liar paradox. Contextualist accounts hold that some step in the reasoning induces a context shift that causes the apparently contradictory claims to occur at different contexts. Attempts at identifying the most promising contextualist account often rely on timing (...) arguments, which seek to isolate a step at which the context cannot be claimed to have shifted or must have shifted. The literature contains a number of timing arguments that draw incompatible conclusions about the location of the context shift. I argue that no existing timing arguments succeed. An alternative strategy for assessing contextualist accounts evaluates the plausibility of their explanations of why the context shifts. However, even this strategy yields no clear verdict about which contextualist account is the most promising. I conclude that there are some grounds for optimism and for pessimism about the potential to adequately motivate contextualism. (shrink)

One of Marx‘s and Engels‘ main claims (hereafter ―original Marxism) in their account of the historical ―inevitability of the collapse of capitalism is that one‘s material (economic) conditions, not one‘s ideas, arguments or philosophy, determines one‘s ―consciousness and actions. However, the self-reference in this characterization of philosophical views generates a paradox analogous to the 7th century B.C. Epimenides ―Liar paradox. The Epimenides-paradox arises when Epimenides, a Cretan, states that all Cretans are liars. Epimenides-statement is paradoxical in the sense that if (...) it is true then it is false. Similarly, the Marxist claim that all philosophers who purport to state the cause of all social changes must be wrong, where Marx and Engels are themselves philosophers stating just such a claim, generates a paradox analogous to the ― Epimenides Liar paradox. This leads to what Engels himself calls a―false consciousness, i.e., a failure to understand the true causes of social change. This paradox can only be escaped by abandoning at least one of original Marxism‘s core doctrines. (shrink)

Kevin Scharp’s influential work on the alethic paradoxes combines an extensively developed inconsistency theory with a substantial conceptual engineering project. I argue that Scharp’s inconsistency theory is in tension with his conceptual engineering project: the inconsistency theory includes an account of concepts that implies that the conceptual engineering project will fail. I recommend that Scharp revises his account of concepts, and show how doing so allows him to resolve the tension. The discussion is important for ongoing work on conceptual engineering. (...) Firstly, it is important to get clear on whether Scharp’s conceptual engineering project is—or could be—successful. Secondly, the issues discussed herein may generalise to other conceptual engineering projects, such as explication and ameliorative projects. In particular, the discussion has implications for how conceptual engineers can think about concepts and their relation to how we use words. (shrink)

The liar and kindred paradoxes show that we can derive contradictions when we reason in accordance with classical logic from the schema (T) about truth: S is true iff p, where ‘p’ is to be replaced with a sentence and ‘S’ with a name of that sentence. The paper presents two arguments to the effect that the blame lies not with (T) but with classical logic. The arguments derive contradictions using classical logic, but instead of appealing to (T), they invoke (...) semantic claims that seem even harder to reject. The first argument relies on two standard semantic principles that are not disquotational and on the claim that if there is such a thing as the property of being true, then ‘true’ expresses that property. The second argument relies on a schema about meaning: S means that p, where ‘S’ and ‘p’ are to be replaced as before. (shrink)

In this paper we show how Dummett-Prawitz-style proof-theoretic semantics has to be modified in order to cope with paradoxical phenomena. It will turn out that one of its basic tenets has to be given up, namely the definition of the correctness of an inference as validity preservation. As a result, the notions of an argument being valid and of an argument being constituted by correct inference rules will no more coincide. The gap between the two notions is accounted for by (...) introducing the distinction between sense and denotation in the proof-theoretic-semantic setting. (shrink)

The aim of the paper is to argue that all—or almost all—logical rules have exceptions. In particular, it is argued that this is a moral that we should draw from the semantic paradoxes. The idea that we should respond to the paradoxes by revising logic in some way is familiar. But previous proposals advocate the replacement of classical logic with some alternative logic. That is, some alternative system of rules, where it is taken for granted that these hold without exception. (...) The present proposal is quite different. According to this, there is no such alternative logic. Rather, classical logic retains the status of the ‘one true logic’, but this status must be reconceived so as to be compatible with (almost) all of its rules admitting of exceptions. This would seem to have significant repercussions for a range of widely held views about logic: e.g. that it is a priori, or that it is necessary. Indeed, if the arguments of the paper succeed, then such views must be given up. (shrink)

Approaches to truth and the Liar paradox seem invariably to face a dilemma: either appeal to some sort of hierarchy, or declare apparently perfectly coherent concepts incoherent. But since both options lead to severe expressive restrictions, neither seems satisfactory. The aim of this paper is a new approach, which avoids the dilemma and the resulting expressive restrictions. Previous approaches tend to appeal to some new sort of semantic value for the truth predicate to take. I argue that such approaches inevitably (...) lead to the dilemma in question. In contrast, the present proposal sticks with classical semantic values, but allows the compositional rules associated with these to admit of exceptions. I show how such an approach can be developed systematically. (shrink)

Although there is a prima facie strong case for a close connection between the meaning and inferential role of certain expressions, this connection seems seriously threatened by the semantic and logical paradoxes which rely on these inferential roles. Some philosophers have drawn radical conclusions from the paradoxes for the theory of meaning in general, and for which sentences in our language are true. I criticize these overreactions, and instead propose to distinguish two conceptions of inferential role. This distinction is closely (...) tied to two conceptions of deductive logic, and it is the key, I argue, for understanding first the connection between meaning and inferential role, and second what the paradoxes show more generally. (shrink)

Some think that logic concerns the “laws of truth”; others that logic concerns the “laws of thought.” This paper presents a way to reconcile both views by building a bridge between truth-maker theory, à la Fine, and normative bilateralism, à la Restall and Ripley. The paper suggests a novel way of understanding consequence in truth-maker theory and shows that this allows us to identify a common structure shared by truth-maker theory and normative bilateralism. We can thus transfer ideas from normative (...) bilateralism to truth-maker theory, such as non-transitive solutions to paradox, and vice versa, such as notions of factual equivalence and containment. (shrink)

Some in the recent literature have claimed that a connection exists between the Liar paradox and _semantic relativism_: the view that the truth values of certain occurrences of sentences depend on the contexts at which they are assessed. Sagi (Erkenntnis 82(4):913–928, 2017) argues that contextualist accounts of the Liar paradox are committed to relativism, and Rudnicki and Łukowski (Synthese 1–20, 2019) propose a new account that they classify as relativist. I argue that a full understanding of how relativism is conceived (...) within theories of natural language shows that neither of the purported connections can be maintained. There is no reason why a solution to the Liar paradox needs to accept relativism. (shrink)

An analysis of--yet another--case of academic failure in multi- and interdisciplinarity. An editorial of the Journal of Knowledge Structures & Systems.

In “Some Remarks on Extending and Interpreting Theories with a Partial Truth Predicate”, Reinhardt [21] famously proposed an instrumentalist interpretation of the truth theory Kripke–Feferman in analogy to Hilbert’s program. Reinhardt suggested to view $\mathrm {KF}$ as a tool for generating “the significant part of $\mathrm {KF}$ ”, that is, as a tool for deriving sentences of the form $\mathrm{Tr}\ulcorner {\varphi }\urcorner $. The constitutive question of Reinhardt’s program was whether it was possible “to justify the use of nonsignificant sentences (...) entirely within the framework of significant sentences”. This question was answered negatively by Halbach & Horsten [10] but we argue that under a more careful interpretation the question may receive a positive answer. To this end, we propose to shift attention from $\mathrm {KF}$ -provably true sentences to $\mathrm {KF}$ -provably true inferences, that is, we shall identify the significant part of $\mathrm {KF}$ with the set of pairs $\langle {\Gamma, \Delta }\rangle $, such that $\mathrm {KF}$ proves that if all members of $\Gamma $ are true, at least one member of $\Delta $ is true. In way of addressing Reinhardt’s question we show that the provably true inferences of suitable $\mathrm {KF}$ -like theories coincide with the provable sequents of matching versions of the theory Partial Kripke–Feferman. (shrink)

The concept of hypodox is dual to the concept of paradox. Whereas a paradox is incompatibly overdetermined, a hypodox is underdetermined. Indeed, many particular paradoxes have dual hypodoxes. So, naively the dual of Russell’s Paradox is whether the set of all sets that are members of themselves is self-membered. The dual of the Liar Paradox is the Truth-teller, and a hypodoxical dual of the Heterological paradox is whether ‘autological’ is autological. I provide some analysis of the duality and I search (...) for modal hypodoxes as duals to known paradoxes. I also try to continue this search further by mapping the relations between some paradoxes and hypodoxes in duality squares. This investigation increases the known extension of the concept of hypodox, and the duality relation between paradoxes and hypodoxes assists in finding some of these. (shrink)

By pooling together exhaustive analyses of certain philosophical paradoxes, we can prove a series of fascinating results regarding philosophical progress, agreement on substantive philosophical claims, knockdown arguments in philosophy, the wisdom of philosophical belief, the epistemic status of metaphysics, and the power of philosophy to refute common sense. As examples, this Element examines the Sorites Paradox, the Liar Paradox, and the Problem of the Many – although many other paradoxes can do the trick too.

The paper presents a truth-maker semantics for Strict/Tolerant Logic (ST), which is the currently most popular logic among advocates of the non-transitive approach to paradoxes. Besides being interesting in itself, the truth-maker presentation of ST offers a new perspective on the recently discovered hierarchy of meta-inferences that, according to some, generalizes the idea behind ST. While fascinating from a mathematical perspective, there is no agreement on the philosophical significance of this hierarchy. I aim to show that there is no clear (...) philosophical significance of meta-inferences above the first level. (shrink)

Non-classical solutions to semantic paradox can be associated with conceptions of paradoxicality understood in terms of entailment facts. In a K3-based theory of truth, for example, it is prima facie natural to say that a sentence φ is paradoxical iff φ ∨ ¬φ entails an absurdity. In a recent paper, Julien Murzi and Lorenzo Rossi exploit this idea to introduce revenge paradoxes for a number of non-classical approaches, including K3. In this paper, I show that on no understanding of ‘is (...) paradoxical’ (for K3) should both rules needed for their paradox be expected to hold unrestrictedly. Just which rule fails, however, depends on various factors, including whether the derivability relation of a target system of reasoning is arithmetically definable. (shrink)

The present first part about the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics. One of the main arguments that it is an axiom consists in the direct contradiction of the axiom of induction in arithmetic and the axiom of infinity in set theory. (...) Thus, the pair of arithmetic and set are to be similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate now, i.e. after replacing it and its negation correspondingly by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. Indeed, the axiom of choice, as far as it is equivalent to the well-ordering “theorem”, transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. So, the Gödel incompleteness statement relies on the logical contradiction of the axiom of induction and the axiom of infinity in the final analysis. Nonetheless, both can be considered as two idempotent versions of the same axiom (analogically to the Fifth postulate) and then unified after logicism and its inherent intensionality since the opposition of finiteness and infinity can be only extensional (i.e., relevant to the elements of any set rather than to the set by itself or its characteristic property being a proposition). So, the pathway for interpreting the Gödel incompleteness statement as an axiom and the originating from that assumption for “Hilbert mathematics” accepting its negation is pioneered. A much wider context relevant to realizing the Gödel incompleteness statement as a metamathematical axiom is consistently built step by step. The horizon of Hilbert mathematics is the proper subject in the third part of the paper, and a reinterpretation of Gödel’s papers (1930; 1931) as an apology of logicism as the only consistent foundations of mathematics is the topic of the next second part. (shrink)

In Formale Logik, published in 1956, J. M. Bocheński presented his first proposal for the solution to the liar paradox, which he related to Paul of Venice's argumentation from Logica Magna. A formalized version of this solution was then presented in Formalisierung einer scholastischen Lösung der Paradoxie des ‘Lügners’ in 1959. The historical references of the resulting formalism turn out to be closer to Albert de Saxon's argument and the later solution by John Buridan. Bocheński did not pose the question (...) of the consistency of his theory. The case was taken up by B. Sobociński in his private letter to Bocheński from 12 August 1954. Sobociński used a smart translation of the language of Bocheński's theory into the classical propositional language with the notions of truth and falsehood inverted. The translation preserves the structure of the formalized solution. We explore Sobociński's idea and reconstruct his original proof. (shrink)

The aim of this chapter is to explain, motivate, and provide the central details of a specific version of what has come to be called alethic fictionalism—namely, a fictionalist account of truth (or, more accurately, of truth-talk, that fragment of discourse that involves the truth-predicate and other alethic-locutions). Our particular brand of alethic fictionalism is sometimes described as a “pretense theory of truth,” and a catchphrase for our view is “truth is a pretense.” But a more precise label for the (...) view that we will present is “semantic pretense-involving fictionalism about truth-talk.” Our endorsement of this view (for short, our SPIF account) stems from our belief that deflationism is the right approach to take on the topic of truth. This already shifts the focus away from any property of truth, since deflationism “about truth” (or, as we will call this view, T-deflationism) is best understood as an approach to analyzing truth-talk. We arrive specifically at our SPIF account of truth-talk because we also think that versions of T-deflationism should be understood as a kind of fictionalism (which, again, puts the focus on discourse, rather than metaphysics) and because we maintain that a SPIF account is the best variety of fictionalism to apply specifically to truth-talk. We will explain some of our reasons for holding these beliefs below, laying out the basics of our SPIF account of truth-talk and highlighting the merits of endorsing our particular account of that talk. (shrink)

When one allows truth to be indeterminate, “fixed point” interpretations can be found even when the language includes sentences such as the liar paradox. In this chapter this kind of account is applied to rational credences, to find non-undermining indeterminate epistemic states even in certain situations which have been discussed as challenges for rationality. In the process of doing this, a deeper understanding of how the supervaluational account of truth works is obtained, especially when one focuses on sets of precisifications.