Logical Semantics and Logical Truth

Just started a new book. The aim is to establish a science of knowledge in the same way that we have a science of physics or a science of materials. This might appear as an overly ambitious, possibly arrogant, objective, but bear with me. On the day I am beginning to write it–June 7th, 2020–, I think I am in possession of a few things that will help me to achieve this objective. Again, bear with me. My aim is well (...) reflected in the title I chose (just now) for this book: Knowledge & Logic: Towards a science of knowledge. Its most important feature is that I shall take logic to be to knowledge science as calculus is to physics or to materials science. I do not intend to reclaim knowledge from the bosom of philosophy, in which, known as epistemology its erudite discussion has hardly progressed since Plato first defined it as true belief with logos. With only a few adjustments, it will actually provide me with the right, science-bound start. More recently, knowledge has been reclaimed by the field of BA, a reclaim that has opened the box of Pandora: Among the evils, and perhaps at the head of the list, is an overly lay, essentially naive, notion of knowledge. But the very idea that one can have something like “knowledge (management) software” puts us on the right track. (shrink)

Georg Cantor was the genuine discoverer of the Mathematical Infinity, and whatever he claimed, suggested, or even surmised should be taken seriously -- albeit not necessary at its face value. Because alongside his exquisite in beauty ordinal construction and his fundamental powerset description of the continuum, Cantor has also left to us his obsessive presumption that the universe of sets should be subjected to laws similar to those governing the set of natural numbers, including the universal principles of cardinal comparability (...) and well-ordering -- and implying an ordinal re-creation of the continuum. During the last hundred years, the mainstream set-theoretical research -- all insights and adjustments due to Kurt G\"odel's revolutionary insights and discoveries notwithstanding -- has compliantly centered its efforts on ad hoc axiomatizations of Cantor's intuitive transfinite design. We demonstrate here that the ontological and epistemic sustainability} of this design has been irremediably compromised by the underlying peremptory, Reductionist mindset of the XIXth century's ideology of science. (shrink)

Homo deceptus is a book that brings together new ideas on language, consciousness and physics into a comprehensive theory that unifies science and philosophy in a different kind of Theory of Everything. The subject of how we are to make sense of the world is addressed in a structured and ordered manner, which starts with a recognition that scientific truths are constructed within a linguistic framework. The author argues that an epistemic foundation of natural language must be understood before laying (...) claim to any notion of reality. This foundation begins with Ludwig Wittgenstein’s Tractatus Logico-Philosophicus and the relationship of language to formal logic. Ultimately, we arrive at an answer to the question of why people believe the things they do. This is effectively a modification of Alfred Tarski’s semantic theory of truth. The second major issue addressed is the ‘dreaded’ Hard Problem of Consciousness as first stated by David Chalmers in 1995. The solution is found in the unification of consciousness, information theory and notions of physicalism. The physical world is shown to be an isomorphic representation of the phenomenological conscious experience. New concepts in understanding how language operates help to explain why this relationship has been so difficult to appreciate. The inclusion of concepts from information theory shows how a digital mechanics resolves heretofore conflicting theories in physics, cognitive science and linguistics. Scientific orthodoxy is supported, but viewed in a different light. Mainstream science is not challenged, but findings are interpreted in a manner that unifies consciousness without contradiction. Digital mechanics and formal systems of logic play central roles in combining language, consciousness and the physical world into a unified theory where all can be understood within a single consistent framework. (shrink)

According to Aristotle if a universal proposition (for example: “All men are white”) is true, its contrary proposition (“All men are not white”) must be false; and, according to Aristotle, if a universal proposition (for example: “All men are white”) is true, its contradictory proposition (“Not all men are white”) must be false. I agree with what Aristotle wrote about universal propositions, but there are universal propositions which have no contrary proposition and have no contradictory proposition. The proposition X “All (...) the propositions that contradict this proposition are true” does not have the contrary proposition and does not have the contradictory proposition. In fact: FEX “All the propositions that contradict this proposition are not true” has a different subject: the subject of the proposition X is constituted by all the propositions that contradict the proposition X; by contrast, the subject of the proposition FEX is constituted by all the propositions that contradict the proposition FEX. And FOX “Not all the propositions that contradict this proposition are true” has a different subject: the subject of the proposition X is constituted by the propositions that contradict the proposition X; by contrast, the subject of the proposition FOX is constituted by the propositions that contradict the proposition FOX. According to Aristotle, a singular proposition (in his example: "Socrates is white") which is true must have its negative proposition ("Socrates is not white") which is false. I agree with Aristotle, but there are singular propositions which do not have the corresponding negative proposition. The proposition (or, rather, the pseudo-proposition) L “This same statement is not true” does not have the negative proposition because the proposition FNL “This same statement is true” has a different subject from the subject of the proposition L: the subject of the proposition L is the proposition L; by contrast, the subject of the proposition FNL is the proposition FNL. L and FNL cannot have the same subject. By contrast, the proposition M “This mount is entirely in Swiss territory” and the proposition NM “This mount is not entirely in Swiss territory” can have the same subject (for example, the Mount Eiger): in case the subject of the proposition M and the subject of the proposition NM is the same, M and NM are opposite propositions, NM is the negative proposition of M. By contrast, the proposition (or, rather, the pseudo-proposition) L "This same statement is not true" cannot have the corresponding negative proposition because FNL "This same statement is true" has a different subject: the subject of L is L ; by contrast, the subject of FNL is FNL. Then the paper continues by analyzing some variants of the liar’s paradox: L1 “The statement L1 is not true”; the so-called liar cycle; and the so-called Yablo’s paradox. (shrink)

Inspired by the early Wittgenstein’s concept of nonsense (meaning that which lies beyond the limits of language), we define two different, yet complementary, types of nonsense: formal nonsense and pragmatic nonsense. The simpler notion of formal nonsense is initially defined within Tarski’s semantic theory of truth; the notion of pragmatic nonsense, by its turn, is formulated within the context of the theory of pragmatic truth, also known as quasi-truth, as formalized by da Costa and his collaborators. While an expression will (...) be considered formally nonsensical if the formal criteria required for the assignment of any truth-value (whether true, false, pragmatically true, or pragmatically false) to such sentence are not met, a (well-formed) formula will be considered pragmatically nonsensical if the pragmatic criteria (inscribed within the context of scientific practice) required for the assignment of any truth-value to such sentence are not met. Thus, in the context of the theory of pragmatic truth, any (well-formed) formula of a formal language interpreted on a simple pragmatic structure will be considered pragmatically nonsensical if the set of primary sentences of such structure is not well-built, that is, if it does not include the relevant observational data and/or theoretical results, or if it does include sentences that are inconsistent with such data. (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)

Textbook for students in mathematical logic. Part 1. Total formalization is possible! Formal theories. First order languages. Axioms of constructive and classical logic. Proving formulas in propositional and predicate logic. Glivenko's theorem and constructive embedding. Axiom independence. Interpretations, models and completeness theorems. Normal forms. Tableaux method. Resolution method. Herbrand's theorem.

Throughout this paper, we are trying to show how and why our Mathematical frame-work seems inappropriate to solve problems in Theory of Computation. More exactly, the concept of turning back in time in paradoxes causes inconsistency in modeling of the concept of Time in some semantic situations. As we see in the first chapter, by introducing a version of “Unexpected Hanging Paradox”,first we attempt to open a new explanation for some paradoxes. In the second step, by applying this paradox, it (...) is demonstrated that any formalized system for the Theory of Computation based on Classical Logic and Turing Model of Computation leads us to a contradiction. We conclude that our mathematical frame work is inappropriate for Theory of Computation. Furthermore, the result provides us a reason that many problems in Complexity Theory resist to be solved.(This work is completed in 2017 -5- 2, it is in vixra in 2017-5-14, presented in Unilog 2018, Vichy). (shrink)

CAT4 is proposed as a general method for representing information, enabling a powerful programming method for large-scale information systems. It enables generalised machine learning, software automation and novel AI capabilities. This is Part 3 of a five-part introduction. The focus here is on explaining the semantic model for CAT4. Points in CAT4 graphs represent facts. We introduce all the formal (data) elements used in the classic semantic model: sense or intension (1st and 2nd joins), reference (3rd join), functions (4th join), (...) time and truth (logical fields), and symbolic content (name/value fields). Concepts are introduced through examples alternating with theoretical discussion. Some concepts are assumed from Part 1 and 2, but key ideas are re-introduced. The purpose is to explain the CAT4 interpretation, and why the data structure and CAT4 axioms have been chosen: to make the semantic model consistent and complete. We start with methods to translate information from database tables into graph DBs and into CAT4. We conclude with a method for translating natural language into CAT4. We conclude with a comparison of the system with an advanced semantic logic, the hyper-intensional logic TIL, which also aims to translate NL into a logical calculus. The CAT4 Natural Language Translator is discussed in further detail in Part 4, when we introduce functions more formally. Part 5 discusses software design considerations. (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.

Is logic empirical? Is logic to be found in the world? Or is logic rather a convention, a product of conventions, part of the many rules that regulate the language game? Answers fall in either camp. We like the linguistic answer. In this paper, we want to analyze how a linguistic community would tackle the problem of developing a logic and show how the linguistic conventions adopted by the community determine the properties of the local logic. Then show how to (...) move from a notion of logic that varies from community to community to a notion of logic that is in a sense universal. The framework is conventional up to a point: we have sentences, atomic and composite, the connectives are interpreted, values are computed, and the value of a composite sentence is a function of the values of its subsentences. Less conventional is the use of a plurality of truth values, and the sharp distinction we draw between sentences and statements, in the spirit of the distinction between proposition and judgment that one may find in proof theory. The linguistic community will face many choices. What are the good ones, the ones to avoid? Are there, in some sense, optimal choices? These are the kind of issues we are addressing. Where do we end up? With some kind of universal bivalent logic, ironically enough. We start from an arbitrarily large number of truth values, atomic sentences and connectives, construct a generic many-valued logic, recover more or less the usual results and issues, and in the end it all comes down to a positive bivalent logic with two connectives, `and' and `or', as if logic is nothing more than a mere accounting of possibilities. (shrink)

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)

When are two formal theories of broadly logical concepts, such as truth, equivalent? The paper investigates a case study, involving two well-known variants Kripke-Feferman truth. The first, KF+CONS, features a consistent but partial truth predicate. The second, KF+COMP, an inconsistent but complete truth predicate. It is well-known that the two truth predicates are dual to each other. We show that this duality reveals a much stricter correspondence between the two theories: they are intertraslatable. Intertranslatability under natural assumptions coincides with definitional (...) equivalence, and is arguably the strictest notion of theoretical equivalence different from logical equivalence. The case of KF+CONS and KF+COMP raises a puzzle: the two theories can be proved to be strictly related, yet they appear to embody remarkably different conceptions of truth. We discuss the significance of the result for the broader debate on formal criteria of conceptual reducibility for theories of truth. (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.

We can simply define Gödel 1931 Incompleteness away by redefining the meaning of the standard definition of Incompleteness: A theory T is incomplete if and only if there is some sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ). This definition construes the existence of self-contradictory expressions in a formal system as proof that this formal system is incomplete because self-contradictory expressions are neither provable nor disprovable in this formal system. Since self-contradictory expressions are neither provable nor disprovable (...) only because they are self-contradictory we could define them as unsound instead of defining the formal system as incomplete. (shrink)

The conventional notion of a formal system is adapted to conform to the sound deductive inference model operating on finite strings. Finite strings stipulated to have the semantic property of Boolean true provide the sound deductive premises. Truth preserving finite string transformation rules provide valid the deductive inference. Conclusions of sound arguments are derived from truth preserving finite string transformations applied to true premises.

Could the intersection of [formal proofs of mathematical logic] and [sound deductive inference] specify formal systems having [deductively sound formal proofs of mathematical logic]? All that we have to do to provide [deductively sound formal proofs of mathematical logic] is select the subset of conventional [formal proofs of mathematical logic] having true premises and now we have [deductively sound formal proofs of mathematical logic].

The conventional notion of a formal system is adapted to conform to the sound deductive inference model operating on finite strings. Finite strings stipulated to have the semantic value of Boolean true provide the sound deductive premises. Truth preserving finite string transformation rules provide the valid deductive inference. Sound deductive conclusions are the result of these finite string transformation rules.

This sentence G ↔ ¬(F ⊢ G) and its negation G ↔ ~(F ⊢ ¬G) are shown to meet the conventional definition of incompleteness: Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)). They meet conventional definition of incompleteness because neither the sentence nor its negation is provable in F (or any other formal system). -- .

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)

Why do we use epistemic modals like 'might'? According to Factualism, the function of 'might' is to exchange information about state-of-affairs in the modal universe. As an alternative to Factualism, this paper offers a game-theoretic rationale for epistemic possibility operators in a Bayesian setting. The background picture is one whereby communication facilitates coordination, but coordination could fail if there's too much uncertainty, since the players' ability to share a belief is undermined. However, 'might' and related expressions can be used to (...) reveal one's uncertainty, and exploit this to coordinate despite the lack of a common epistemic ground. The final result is a way to articulate a non-Factualist view of epistemic possibility modals that builds on their standard semantics. (shrink)

By separating the general concept of truth into syntactic truth and semantic truth, this article proposes a new theory of truth to explain several paradoxes like the Liar paradox, Card paradox, Curry’s paradox, etc. By revealing the relationship between syntactic /semantic truth and being-nothing-becoming which are the core concepts of dialectical logic, it is able to formalize dialectical logic. It also provides a logical basis for complexity theory by transferring all reasoning into a directed (cyclic/acyclic) graph which explains both paradoxical (...) and paradox-free reasoning. The different structure between cyclic graphs and acyclic graphs is the key to understanding paradoxes. By explaining the immanent between logic, paradox, and cellular automata, it also illustrates dialectical logic as ontology. (shrink)

I present a solution to the epistemological or characterisation problem of induction. In part I, Bayesian Confirmation Theory (BCT) is discussed as a good contender for such a solution but with a fundamental explanatory gap (along with other well discussed problems); useful assigned probabilities like priors require substantive degrees of belief about the world. I assert that one does not have such substantive information about the world. Consequently, an explanation is needed for how one can be licensed to act as (...) if one has substantive information about the world when one does not. I sketch the outlines of a solution in part I, showing how it differs from others, with full details to follow in subsequent parts. The solution is pragmatic in sentiment (though differs in specifics to arguments from, for example, William James); the conceptions we use to guide our actions are and should be at least partly determined by preferences. This is cashed out in a reformulation of decision theory motivated by a non-reductive formulation of hypotheses and logic. A distinction emerges between initial assumptions--that can be non-dogmatic--and effective assumptions that can simultaneously be substantive. An explanation is provided for the plausibility arguments used to explain assigned probabilities in BCT. -/- In subsequent parts, logic is constructed from principles independent of language and mind. In particular, propositions are defined to not have form. Probabilities are logical and uniquely determined by assumptions. The problems considered fatal to logical probabilities--Goodman's `grue' problem and the uniqueness of priors problem are dissolved due to the particular formulation of logic used. Other problems such as the zero-prior problem are also solved. -/- A universal theory of (non-linguistic) meaning is developed. Problems with counterfactual conditionals are solved by developing concepts of abstractions and corresponding pictures that make up hypotheses. Spaces of hypotheses and the version of Bayes' theorem that utilises them emerge from first principles. -/- Theoretical virtues for hypotheses emerge from the theory. Explanatory force is explicated. The significance of effective assumptions is partly determined by combinatoric factors relating to the structure of hypotheses. I conjecture that this is the origin of simplicity. (shrink)

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. (...) The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines. (shrink)

This paper is a short introduction to Carnap’s writings on semantics with an emphasis on the transition from the syntactic period to the semantic one. I claim that one of Carnap’s main aims was to investigate the possibility of the symmetry between the syntactic and the semantic methods of approaching philosophical problems, both in logic and in the philosophy of science. This ideal of methodological symmetry could be described as an attempt to obtain categorical logical systems, i.e., systems that allow (...) only the intended semantical interpretation. (shrink)

Extended abstract presented at the AISC 2018 Conference, 15th International Conference of the Italian Association of Cognitive Science, Pavia.

Tarski’s Convention T—presenting his notion of adequate definition of truth (sic)—contains two conditions: alpha and beta. Alpha requires that all instances of a certain T Schema be provable. Beta requires in effect the provability of ‘every truth is a sentence’. Beta formally recognizes the fact, repeatedly emphasized by Tarski, that sentences (devoid of free variable occurrences)—as opposed to pre-sentences (having free occurrences of variables)—exhaust the range of significance of is true. In Tarski’s preferred usage, it is part of the meaning (...) of true that attribution of being true to a given thing presupposes the thing is a sentence. Beta’s importance is further highlighted by the fact that alpha can be satisfied using the recursively definable concept of being satisfied by every infinite sequence, which Tarski explicitly rejects. Moreover, in Definition 23, the famous truth-definition, Tarski supplements “being satisfied by every infinite sequence” by adding the condition “being a sentence”. Even where truth is undefinable and treated by Tarski axiomatically, he adds as an explicit axiom a sentence to the effect that every truth is a sentence. Surprisingly, the sentence just before the presentation of Convention T seems to imply that alpha alone might be sufficient. Even more surprising is the sentence just after Convention T saying beta “is not essential”. Why include a condition if it is not essential? Tarski says nothing about this dissonance. Considering the broader context, the Polish original, the German translation from which the English was derived, and other sources, we attempt to determine what Tarski might have intended by the two troubling sentences which, as they stand, are contrary to the spirit, if not the letter, of several other passages in Tarski’s corpus. (shrink)

Some philosophers argue that a theory of logical validity should not interpret its own language, because a Russellian argument shows that self-applicability is inconsistent with the ability to capture all the interpretations of its own language. First, I set up a formal system to examine the Russellian argument. I then defend the need for self-applicability. I argue that self-applicability seems to be implied by generality, and that the Russellian argument rests on a test for meaning that is biased against self-applicability. (...) I propose an alternative test which is unproblematic for self-applicability. I conclude that self-applicability can be vindicated and the Russellian argument can be resisted. (shrink)

This paper raises some questions about the formalization of sentences containing ‘if’ or similar expressions. In particular, it focuses on three kinds of sentences that resemble conditionals in some respects but exhibit distinctive logical features that deserve separate consideration: whether-or-not sentences, biscuit conditionals, and concessive conditionals. As will be suggested, the examples discussed show in different ways that an adequate formalization of a sentence must take into account the content expressed by the sentence. This upshot is arguably what one should (...) expect on the view that logical form is determined by truth conditions. (shrink)

Truthmaker semantics is a recent development in formal and philosophical semantics, with similar motivation and scope to possible worlds semantics. The technical background is rather different, however, and results in a more fine-grained hyperintensional notion of content, allowing us to distinguish between classically equivalent propositions. After briefly introducing the main ideas, this entry will describe the technical apparatus of state spaces and the central notions of content and partial content. It will then outline applications of truthmaker semantics in language, logic, (...) and other areas of philosophy. (shrink)

A general mathematical framework, based on countable partitions of Natural Numbers [1], is presented, that allows to provide a Semantics to propositional languages. It has the particularity of allowing both the valuations and the interpretation Sets for the connectives to discriminate complexity of the formulas. This allows different adequacy criteria to be used to assess formulas associated with the same connective, but that differ in their complexity. The presented method can be adapted potentially infinite number of connectives and truth values, (...) therefore, it can be considered a general framework to provide semantics to several of the known logic systems (eg, LC, L3 LP, FDE). The presented semantics allow to converge to different standard semantics if the separation complexity procedure is annulled. Therefore, it can be understood as a framework that allows greater precision (in complexity terms) with respect to formula satisfaction. Naturally, because of how it is built, it can be incorporated into non-deterministic semantics. The presented procedure also allows generating valuations that grant a different truth value to each formula of propositional language. As a positive side effect, our method allows a constructive proof of the equipotence between N and N^n for all Natural n. (shrink)

In this paper, we provide a semantics for a range of positive substructural logics, including both logics with and logics without modal connectives. The semantics is novel insofar as it is meant to explicitly capture the computational flavor of these logics, and to do so in a way that builds in both nondeterministic and nonconcurrent computational processes.

This paper outlines a new pluralistic theory of normative reasons for belief, welfarist pluralism, which aims to explain how there can be basic alethic/epistemic reasons for belief and basic pragmatic/non-alethic reasons for belief that can combine to determine what one ought to believe. The paper shows how this non-derivative first-order pluralism arises from a purely welfarist account of the foundations of theoretical normativity, thereby combining foundational pragmatism with non-derivative pluralism about normative reasons for belief. In addition, this paper outlines how (...) welfarist pluralism provides a new and potentialy attractive account of the value of truth (or its indicators), namely that the value of truth (or its indicators) is its direct contribution to individuals' personal good or wellbeing. (shrink)

I propose an approach to liar and Curry paradoxes inspired by the work of Roger Swyneshed in his treatise on insolubles (1330-1335). The keystone of the account is the idea that liar sentences and their ilk are false (and only false) and that the so-called ''capture'' direction of the T-schema should be restricted. The proposed account retains what I take to be the attractive features of Swyneshed's approach without leading to some worrying consequences Swyneshed accepts. The approach and the resulting (...) logic (called ''Swynish Logic'') are non-classical, but are consistent and compatible with many elements of the classical picture including modus ponens, modus tollens, and double-negation elimination and introduction. It is also compatible with bivalence and contravalence. My approach to these paradoxes is also immune to an important kind of revenge challenge that plagues some of its rivals. (shrink)

Minimalism about truth is one of the main contenders for our best theory of truth, but minimalists face the charge of being unable to properly state their theory. Donald Davidson incisively pointed out that minimalists must generalize over occurrences of the same expression placed in two different contexts, which is futile. In order to meet the challenge, Paul Horwich argues that one can nevertheless characterize the axioms of the minimalist theory. Sten Lindström and Tim Button have independently argued that Horwich’s (...) attempt to formulate minimalism remains unsuccessful. We show how to properly state Horwich’s axioms by appealing to propositional functions that are given by definite descriptions. Both Lindström and Button discuss proposals similar to ours and conclude that they are unsuccessful. Our new suggestion avoids these objections. (shrink)

In proof-theoretic semantics, meaning is based on inference. It may be seen as the mathematical expression of the inferentialist interpretation of logic. Much recent work has focused on base-extension semantics, in which the validity of formulas is given by an inductive definition generated by provability in a ‘base’ of atomic rules. Base-extension semantics for classical and intuitionistic propositional logic have been explored by several authors. In this paper, we develop base-extension semantics for the classical propositional modal systems K, KT , (...) K4, and S4, with □ as the primary modal operator. We establish appropriate soundness and completeness theorems and establish the duality between □ and a natural presentation of ♢. We also show that our semantics is in its current form not complete with respect to euclidean modal logics. Our formulation makes essential use of relational structures on bases. (shrink)

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. (...) The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines. (shrink)

Due to Gӧdel’s incompleteness results, the categoricity of a sufficiently rich mathematical theory and the semantic completeness of its underlying logic are two mutually exclusive ideals. For first- and second-order logics we obtain one of them with the cost of losing the other. In addition, in both these logics the rules of deduction for their quantifiers are non-categorical. In this paper I examine two recent arguments –Warren (2020), Murzi and Topey (2021)– for the idea that the natural deduction rules for (...) the first-order universal quantifier are categorical, i.e., they uniquely determine its semantic intended meaning. Both of them make use of McGee’s open-endedness requirement and the second one uses in addition Garson’s (2013) local models for defining the validity of these rules. I argue that the success of both these arguments is relative to their semantic or infinitary assumptions, which could be easily discharged if the introduction rule for the universal quantifier is taken to be an infinitary rule, i.e. non-compact. Consequently, I reconsider the use of the ω-rule and I show that the addition of the ω-rule to the standard formalizations of first-order logic is categorical. In addition, I argue that the open-endedness requirement does not make the first-order Peano Arithmetic categorical and I advance an argument for its categoricity based on the inferential conservativity requirement. (shrink)

Non-well-founded set theories allow set-theoretic exotica that standard ZFC will not allow, such as a set that has itself as its sole member. We can distinguish plenitudinous non-well-founded set theories, such as Boffa set theory, that allow infinitely many such sets, from restrictive theories, such as Finsler-Aczel or AFA, that allow exactly one. Plenitudinous non-well-founded set theories face a puzzle: nothing seems to explain the identity or distinctness of various of the sets they countenance. In this paper I aim to (...) sharpen this puzzle, make clear who it does and does not apply to and, ultimately, to argue in favor of a plenitudinous theory like Boffa. (shrink)

Hyperlogic is a hyperintensional system designed to regiment metalogical claims (e.g., “Intuitionistic logic is correct” or “The law of excluded middle holds”) into the object language, including within embedded environments such as attitude reports and counterfactuals. This paper is the first of a two-part series exploring the logic of hyperlogic. This part presents a minimal logic of hyperlogic and proves its completeness. It consists of two interdefined axiomatic systems: one for classical consequence (truth preservation under a classical interpretation of the (...) connectives) and one for “universal” consequence (truth preservation under any interpretation). The sequel to this paper explores stronger logics that are sound and complete over various restricted classes of models as well as languages with hyperintensional operators. (shrink)

Beall’s (e.g., 2009, 2021) transparency theory of truth is recognized as a prominent, deflationist solution to the liar paradox. However, it has been neglected by truth theorists who have attempted to show that a deflationist theory of truth can (or cannot) account for truth dependence, i.e., the claim that the truth of a proposition depends on how things described by the proposition are, but how these things are does not depend on the truth of the proposition. Truth theorists interested in (...) truth dependence have, instead, been focused on Horwich’s Minimalism (e.g., 1998). The goal of this paper is twofold. First, I construct what versions of the transparency theory would say about truth dependence. Second, I argue that even the best version of transparent truth ultimately fails to account for truth dependence. On the assumption that accounting for truth dependence is an adequacy condition on any theory of truth, the paper rejects transparency theory as an adequate theory of truth. (shrink)

Aristotle counts as the founder of formal logic. The logic he develops dominated until Frege and others introduced a new logic. This new logic is taken to be more powerful and better capable of capturing inference patterns. The new logic differs from Aristotelian logic in significant respects. It has been argued by Fred Sommers and Hanoch Ben-Yami that the new logic is not well equipped as a logic of natural language, and that a logic closer to Aristotle's is better suited (...) for this task. Each of them developed their own formalism - Sommers in form of term logic, Ben-Yami in form of his Quantified Argument Calculus (QUARC). I discuss Aristotle's logic - a term logic - and attempt a comparison between Aristotelian logic and (i) the new logic, (ii) Sommers' term logic, and (iii) Ben-Yami's QUARC. I consider differences between the systems, and show how they are related to and diverge from the new logic. (shrink)

En este breve ensayo me interesa analizar el concepto de lenguaje privado desarrollado por Ludwig Wittgenstein. La importancia radica en el desafío que plantea esta noción cuando nos ponemos a pensar sobre las posibilidades reales de un lenguaje puramente privado. En este sentido, el argumento del enfoque wittgensteiniano sostiene que el significado de las palabras no se origina en experiencias subjetivas individuales, sino en la interacción social dentro de una comunidad de hablantes. De modo que, si cada persona tuviera un (...) lenguaje exclusivamente privado, la comunicación sería imposible. Por ello, el objetivo de este escrito se centra en debatir si efectivamente el lenguaje es un fenómeno colectivo, arraigado en prácticas compartidas y acuerdos tácitos, o bien, puede tratarse de una construcción personal subjetiva. En síntesis, para examinar este planteo me centraré en el argumento propuesto por Kripke contra el lenguaje privado, y desde allí, analizaré la paradoja escéptica de Wittgenstein. (shrink)

Our standard model-theoretic definition of logical consequence is originally based on Alfred Tarski’s (1936) semantic definition, which, in turn, is based on Rudolf Carnap’s (1934) similar definition. In recent literature, Tarski’s definition is described as a conceptual analysis of the intuitive ‘everyday’ concept of consequence or as an explication of it, but the use of these terms is loose and largely unaccounted for. I argue that the definition is not an analysis but an explication, in the Carnapian sense: the replacement (...) of the inexact everyday concept with an exact one. Some everyday intuitions were thus brought into a precise form, others were ignored and forgotten. How exactly did the concept of logical consequence change in this process? I suggest that we could find some of the forgotten intuitions in Bernard Bolzano’s (1837) definition of ‘deducibility’, which is traditionally viewed as the main precursor of Tarski’s definition from a time before formalized languages. It turns out that Bolzano’s definition is subject to just the kind of natural features—paradoxicality of everyday language, Platonism about propositions, and dependence on the external world—that Tarski sought to tame by constructing an artificial concept for the special needs of mathematical logic. (shrink)

Without directly addressing the Demarcation Problem for logic—the problem of distinguishing logical vocabulary from others—we focus on distinctive aspects of logical vocabulary in pursuit of a second goal in the philosophy of logic, namely, proposing criteria for the justification of logical rules. Our preferred approach has three components. Two of these are effectively Belnap’s, but with a twist. We agree with Belnap’s response to Prior’s challenge to inferentialist characterisations of the meanings of logical constants. Belnap argued that for a logical (...) constant to exist, its rules must be conservative over a previously given consequence relation and guarantee the uniqueness of the constant. The twist is that we require logical vocabulary to be provably conservative, not over a previously given formal consequence relation, but over a previously given meaningful vocabulary of a language. Uniqueness is also a feature defined in terms of provability: if two syntactically distinguished expressions are governed by the same rules, then formulas with them as main operators are interderivable. Belnap’s criteria are not only those for the existence of a logical constant, but more: they are what distinguishes logical vocabulary from all other expressions. It is the defining mark of a logical constant that it is provably conservative over the fragment of the language which excludes it and that its rules guarantee its uniqueness. The third component is the topic neutrality of logic. The provable conservativeness of logic over a previously given vocabulary of a language is motivated, in part, by appeal to molecularism in the theory of meaning. Molecularity is a feature endorsed by the theory of meaning as a whole, so it does not distinguish logical constants from other expressions. The same is true for conservativeness. We argue that molecularity, and hence conservativeness, are implicit presuppositions of speakers’ use of language: the addition of new vocabulary to a language is presupposed to be conservative. But this presupposition is one that we should be able reflectively to endorse (such as when attempting a rational reconstruction of the use of the vocabulary). Thus, where conservativeness is found to fail, this defect should be remedied and the use of the vocabulary corrected (as in classical negation) or excised altogether (as in pejoratives). We go on to characterise a notion of topic neutrality, which we argue applies to logical vocabulary. We then note that reflective endorsement of the conservativeness of a topic neutral vocabulary requires a proof that the vocabulary is conservative relative to any base vocabulary. Thus we require that logical vocabulary be demonstrably conservative. Allying this requirement, distinctive of logic, to a general consideration about the commitments of assertion yields a mode of justification for logical constants akin to some conceptions of Harmony, i.e., to the idea that the consequences of assertion of a logical complex need to be warranted by the grounds and ought to be the strongest consequences warranted by the grounds. Intuitionistic logic acquires a somewhat familiar justification but emanating from a new motivation. (shrink)

¿Cómo puede la lógica representar expresiones indéxicas como “yo”, “aquí” y “ahora”? ¿Cómo no debe representarlas? Examino estas dos preguntas a partir de la Lógica de los Demostrativos (LD) de Kaplan y su impopular prohibición de operadores monstruosos. A pesar de algunos defectos de formulación, sostengo que dicha prohibición está guiada por una poderosa visión de las relaciones lógicas de validez entre oraciones con indéxicos que desafía la concepción tradicional de consecuencia lógica como preservación de la verdad y resalta el (...) papel fundamental de la información semántica en la construcción de sistemas formales concebidos como modelos de las lenguas naturales. (shrink)

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 counterarguments for various proposed solutions (“the revenge of the Liar”). 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 an 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 truth predicate. In the end, an erroneous critique of Kripke-Feferman axiomatic theory of truth, which is present in contemporary literature, is pointed out. (shrink)

This paper establishes a sound and complete semantics for the impure logic of ground. Fine (Review of Symbolic Logic, 5(1), 1–25, 2012a) sets out a system for the pure logic of ground, one in which the formulas between which ground-theoretic claims hold have no internal logical complexity; and it provides a sound and complete semantics for the system. Fine (2012b) [§§6-8] sets out a system for an impure logic of ground, one that extends the rules of the original pure system (...) with rules for the truth-functional connectives, the first-order quantifiers, and λ-abstraction. However, no semantics has yet been provided for this system. The present paper partly fills this lacuna by providing a sound and complete semantics for a system GG containing the truth-functional operators that is closely related to the truth-functional part of the system of Fine (2012b). (shrink)

By elaborating on the results presented in Lógica cuántica, Nmatrices y adecuación I, here we discuss the notions of adequacy and truth functionality in quantum logic from the point of view of a non-deterministic semantics based on Nmatrices. We present a proof of the impossibility of providing a functional semantics for the quantum lattice. An advantage of our proof is that it is independent of the number of truth values involved, generalizing previous works. Due to the impossibility of defining adequate (...) interpreting sets for the conjunction and disjunction (as was shown in our previous work), it follows that a homomorphism between the lattice of quantum projections and a Boolean algebra of n-elements cannot exist. (shrink)