We study, from a classical point of view, how the truth of a statement about higher type functionals depends on the underlying model. The models considered are the classical set-theoretic finite type hierarchy and the constructively more meaningful models of continuous functionals, hereditarily effective operations, as well as the closed term model of Gödel's system T. The main results are characterisations of prenex classes for which truth in the full set-theoretic model transfers to truth in (...) the other models. As a corollary we obtain that the axiom of choice is not conservative over Gödel's system T with classical logic for closed ∃2-formulas. We hope that this study will contribute to a clearer delineation and perception of constructive mathematics from a classical perspective. (shrink)

The general fact of the impossibility of a bivalent, truth-functional semantics for the propositional structures determined by quantum mechanics should be more subtly demarcated according to whether the structures are taken to be orthomodular latticesP L or partial-Boolean algebrasP A; according to whether the semantic mappings are required to be truth-functional or truth-functional ; and according to whether two-or-higher dimensional Hilbert spaceP structures or three-or-higher dimensional Hilbert spaceP structures are being considered. If the quantumP structures are taken (...) to be orthomodular latticesP L, then bivalent mappings which preserve the operations and relations of aP L must be truth-functional . Then as suggested by von Neumann and Jauch-Piron and as proven in this paper, the mere presence of incompatible elements in aP L is sufficient to rule out any semantical or hidden-variable proposal which imposes this strong condition, for anytwo-or-higher dimensional Hilbert spaceP L structure. Thus from the orthomodular lattice perspective, the peculiarly non-classical feature of quantum mechanics and the peculiarly non-Boolean feature of the quantum propositional structures is the existence of incompatible magnitudes and propositions. However, the weaker truth-functionality condition can instead be imposed upon the semantic or hidden-variable mappings on theP L structures, although such mappings ignore the lattice meets and joins of incompatibles and preserve only the partial-Boolean algebra structural features of theP L structures. Or alternatively, the quantum propositional structures can be taken to be partial-Boolean algebrasP A, where bivalent mappings which preserve the operations and relations of aP A need only be truth-functional (c). In either case, the Gleason, Kochen-Specker proofs show that any semantical or hidden variable proposal which imposes this truth-functionality (c) condition is impossible for anythree-or-higher dimensional Hilbert spaceP A orP L structures. But such semantical or hidden-variable proposals are possible for any two dimensional Hilbert spaceP A orP L structures, in spite of the presence of incompatibles in these structures, in spite of the fact that Heisenberg's Uncertainty Principle applies to the incompatible elements in these structures, and in spite of the fact that these structures are non-Boolean in the Piron sense. (shrink)

We develop a bottom-up approach to truth-value semantics for classical logic of partial terms based on equality and apply it to prove the conservativity of the addition of partial description and selection functions, independently of any strictness assumption.

Methods available for the axiomatization of arbitrary finite-valued logics can be applied to obtain sound and complete intelim rules for all truth-functional connectives of classical logic including the Sheffer stroke and Peirce’s arrow. The restriction to a single conclusion in standard systems of natural deduction requires the introduction of additional rules to make the resulting systems complete; these rules are nevertheless still simple and correspond straightforwardly to the classical absurdity rule. Omitting these rules results in systems for (...) intuitionistic versions of the connectives in question. (shrink)

In this note we present an axiomatization of the classical two-valued propositional calculus, for which proofs of decidability, consistency, completeness, and independence, are almost trivial (given an understanding of truth tables).

§1. Introduction. When truth-theoretic paradoxes are generated, two factors seem to be at play: the behaviour that truth intuitively has; and the facts about which singular terms refer to which sentences, and so on. For example, paradoxicality might be partially attributed to the contingent fact that the singular term, "the italicized sentence on page one", refers to the sentence, The italicized sentence on page one is not true. Factors of this second kind might be represented by a ground (...) model: an interpretation of all the names, function symbols, and predicates in the potentially self-referential language under study, with the exception of the predicate "x is true". Formally, suppose that L is an uninterpreted first order language. M = 〈D, I〉 is a classical model for L if D is a nonempty set; and I is a function assigning to each name of L a member of D, to each n-place function symbol of L a function from Dn to D, and to each nonlogical n-place predicate of L a function from Dn to {t, f}. Suppose furthermore that L+ is obtained by adding a new one-place predicate T to L, and that L has a quote name ‘A’ for each sentence A of L+. We follow Gupta and Belnap [5] in defining S =df {A: A is a sentences of L+}. A classical model M = 〈D, I〉 for L is a ground model for L iff both S ⊆ D, and I(‘A’) = A for each A ∈ S. A classical ground model for L is a representation of the supposedly unproblematic fragment of L+: a representation of which terms refer to which objects, and of which objects have which nonsemantic properties. (shrink)

There are two different ways to introduce the notion of truthin constructive mathematics. The first one is to use a Tarskian definition of truth in aconstructive (meta)language. According to some authors, (Kreisel, van Dalen, Troelstra ... ),this definition is entirely similar to the Tarskian definition of classical truth (thesis A).The second one, due essentially to Heyting and Kolmogorov, and known as theBrouwer–Heyting–Kolmogorov interpretation, is to explain informally what it means fora mathematical proposition to be constructively proved. According (...) to other authors (Martin-Löfand Shapiro), this interpretation and the Tarskian definition of truth amount to thesame (thesis B). My aim in this paper is to show that thesis A is only reasonable, that thesis Bis false and to answer the following question: what is defined by the Tarskian definition ofconstructive truth? (shrink)

In many real-life applications of logic it is useful to interpret a particular sentence as true together with its negation. If we are talking about classical logic, this situation would force all other sentences to be equally interpreted as true. Paraconsistent logics are exactly those logics that escape this explosive effect of the presence of inconsistencies and allow for sensible reasoning still to take effect. To provide reasonably intuitive semantics for paraconsistent logics has traditionally proven to be a challenge. (...) Possible-translations semantics can meet that challenge by allowing for each interpretation to be composed of multiple scenarios. Using that idea, a logic with a complex semantic behavior can be understood as an appropriate combination of ingredient logics with simpler semantic behaviors into which the original logic is given a collection of translations preserving its soundness. Completeness is then achieved through the judicious choice of the admissible translating mappings. The present note provides interpretation by way of possible-translations semantics for a group of fundamental paraconsistent logics extending the positive fragment of classical propositional logic. The logics PI, Cmin, mbC, bC, mCi and Ci, among others, are all initially presented through their non-truth-functional bivaluation semantics and sequent versions and then split by way of possible-translations semantics based on 3-valued ingredients. (shrink)

One of the problems we face in many-valued logic is the difficulty of capturing the intuitive meaning of the connectives introduced through truth tables. At the same time, however, some logics have nice ways to capture the intended meaning of connectives easily, such as four-valued logic studied by Belnap and Dunn. Inspired by Dunn’s discovery, we first describe a mechanical procedure, in expansions of Belnap-Dunn logic, to obtain truth conditions in terms of the behavior of the Truth (...) and the False, which gives us intuitive readings of connectives, out of truth tables. Then, we revisit the notion of functional completeness, which is one of the key notions in many-valued logic, in view of Dunn’s idea. More concretely, we introduce a generalized notion of functional completeness which naturally arises in the spirit of Dunn’s idea, and prove some fundamental results corresponding to the classical results proved by Post and Słupecki. (shrink)

In this paper we study 12 four-valued logics arisen from Belnap's truth and/or knowledge four-valued lattices, with or without constants, by adding one or both or none of two new non-regular operations—classical negation and natural implication. We prove that the secondary connectives of the bilattice four-valued logic with bilattice constants are exactly the regular four-valued operations. Moreover, we prove that its expansion by any non-regular connective (such as, e.g., classical negation or natural implication) is strictly functionally complete. (...) Further, finding axiomatizations of the quasi varieties generated by the 12 logics involved (that prove to be varieties), we find naturell equational axiomatizations of these logics. Finally, applying Pynko's general theory of algebraizable sequential consequence operations, we also find equivalent natural sequentiell axiomatizations of the logics under consideration that expand either of two Pynko's sequential calculi for the constant-free truth-lattice four-valued logic. (shrink)

In this paper, we use a ‘normality operator’ in order to generate logics of formal inconsistency and logics of formal undeterminedness from any subclassical many-valued logic that enjoys a truth-functional semantics. Normality operators express, in any many-valued logic, that a given formula has a classical truth value. In the first part of the paper we provide some setup and focus on many-valued logics that satisfy some of the three properties, namely subclassicality and two properties that we call (...) fixed-point negation property and conservativeness. In the second part of the paper, we introduce normality operators and explore their formal behaviour. In the third and final part of the paper, we establish a number of classical recapture results for systems of formal inconsistency and formal undeterminedness that satisfy some or all the properties above. These are the main formal results of the paper. Also, we illustrate concrete cases of recapture by discussing the logics $\mathsf{K}^{\circledast }_{3}$, $\mathsf{LP}^{\circledast }$, $\mathsf{K}^{w\circledast }_{3}$, $\mathsf{PWK}^{\circledast }$ and $\mathsf{E_{fde}}^{\circledast }$, that are in turn extensions of $\mathsf{{K}_{3}}$, $\mathsf{LP}$, $\mathsf{K}^{w}_{3}$, $\mathsf{PWK}$ and $\mathsf{E_{fde}}$, respectively. (shrink)

This paper begins the study of first-order functions, which are a generalization of truth-functions. The concepts of truth-table and systems of truth-functions, both introduced in propositional logic by Post, are also generalized and studied in the quantificational setting. The general facts about these concepts are given in the first five sections, and constitute a “general theory” of first-order functions. The central theme of this paper is the relation of definition among notions expressed by (...) formulas of first-order logic. We emphasize that logic is not concerned only with the consequence relation among notions expressed by formulas. It also attends to the relation of definition among notions, where a notion is defined from other notions. Sections 5 and 6 deal exclusively with the relation of definition among notions expressed by formulas of first-order logic. In these sections, we study the systems of first-order functions, which are the sets of first-order functions closed under definitions. Sections 7 and 8 are concerned with the relativization of first-order functions to a class of structures. The relativization to a class of structures is a fundamental operation which is used in order to relate the theory of first-order functions with set theory and first-order model theory, a subject which we have barely scratched the surface. The apparatus developed in this paper enables us to define what is a vehicle for the foundation of classical mathematics in set theory, and, in Sect. 8, we prove that first-order logic with one binary predicate variable is not a minimal vehicle for the foundation of classical mathematics in set theory. Sections 9 and 10 introduce further operations and ideals of first-order functions. Besides some results on the influence of the arguments of a first-order function, a result about definability is proved in Sect. 10.1. It is this theorem that provides necessary and sufficient conditions for a first-order function to be in a finitely generated ideal. In Sect. 11, this result is applied to the problem of predicate definability in classes of structures, the problem with which Beth’s theorem dealt in the case of elementary classes. (shrink)

Classical propositional logic can be characterized, indirectly, by means of a complementary formal system whose theorems are exactly those formulas that are not classical tautologies, i.e., contradictions and truth-functional contingencies. Since a formula is contingent if and only if its negation is also contingent, the system in question is paraconsistent. Hence classical propositional logic itself admits of a paraconsistent characterization, albeit “in the negative”. More generally, any decidable logic with a syntactically incomplete proof theory allows for (...) a paraconsistent characterization of its set of theorems. This, we note, has important bearing on the very nature of paraconsistency as standardly characterized. (shrink)

Socratic dialogue is nowadays often invoked as a model in education, and several educational approaches explicitly claim to be inspired by it (Naccari, 2003; Shah, 2008; Delić & Bećirović, 2016). However, contemporary versions depart profoundly from its original form as expressed in the Platonic dialogues, reinterpreting the role of the teacher in the dialogical exchange and referring to different epistemological criteria (cf. Shields, 1953; Reich, 1998; Dinkins, & Cangelosi, 2019; Marshall, 2019). The aim of this paper is to highlight, on (...) the one hand, how such a distortion of the classical model results in the renunciation of the heuristic and transformative functions of Socratic dialogue, and on the other hand, to propose an alternative model of dialogic teaching that restores these functions while maintaining its practicability in contemporary school contexts. Specifically, the role of epistemological relativism and the facilitator in constructivist education are identified as the critical issues in the construction of effective dialogical confrontation in the post-truth era. (shrink)

Dyadic semantics is a sort of non-truth-functional bivalued semantics introduced in Caleiro et al. Logica Universalis, Birkhäuser, Basel, pp 169–189, 2005). Here we introduce an algorithmic procedure for constructing conservative translations of logics characterised by dyadic semantics into classical propositional logic. The procedure uses fresh propositional variables, which we call hidden variables, to represent the indeterminism of dyadic semantics. An alternative algorithmic procedure for constructing conservative translations of any finite-valued logic into classical logic is also introduced. In (...) this alternative procedure hidden variables are also used, but in this case to represent the degree of true or falsehood of propositions. (shrink)

The article is about one of the vital problem for analytic philosophy which is how to define truth value for sentences which include their own truth predicate. The aim of the article is to determine Saul Kripke’s approach to widen epistemological truth to create a systemic model of truth. Despite a lot of work on the subject, the theme of truth is no less relevant to modern philosophy. With the help of S. Kripke’s article “Outline (...) of the Theory of Truth” and R. L. Kirkham’s work “Theories of Truth: A Critical Introduction” the author tried to represent a non-classical approach which is now known as Saul Kripke’s truth value gaps theory. Considering his solving of the Liar Paradox, as well as the Strenghened Liar, it is obvious that there could be the circumstances (or facts) that make sentences have no truth value. In that case the solving of the paradox attracts Kleene’s non-classical three-valued logic which contains the third way and the solution falls into the gap between truth and falsity. Hence there are no relevant and full satisfactory ways to solve the Liar’s Paradox by the language-levels ap- proach. The task consists in an extension of the approach to natural languages and elimination of an ad hoc element as well. Besides that, the language-levels approach refers to the speaker that means that only the last one is allowed to define the hierarchy of the levels and truth value of sentence, but often he does not know what level he is speaking of. As Kripke stated, contingent facts can lead to a paradox which isn’t allowed for a logical point of view. However usage of mathematical apparatus allows the theory in question to be universalized and moreover paves the way to futher epistemological develop- ment of the problem in question. As Kripke’s theory is formalized and presented as a functional series, it gives access to a system model of truth whose elements would be epistemological theories (concepts) of truth. Such a model, for instance, confirms the validity of the deflationary concept of truth. Refs 7. (shrink)

A classical result by Słupecki states that a logic L is functionally complete for the 3-element set of truth-values THREE if, in addition to functionally including Łukasiewicz’s 3-valued logic Ł3, what he names the ‘$T$-function’ is definable in L. By leaning upon this classical result, we prove a general theorem for defining binary expansions of Kleene’s strong logic that are functionally complete for THREE.

This paper generalises an argument for probabilism due to Lindley [9]. I extend the argument to a number of non-classical logical settings whose truth-values, seen here as ideal aims for belief, are in the set $\{0,1\}$, and where logical consequence $\models $ is given the “no-drop” characterization. First I will show that, in each of these settings, an agent’s credence can only avoid accuracy-domination if its canonical transform is a (possibly non-classical) probability function. In other words, if (...) an agent values accuracy as the fundamental epistemic virtue, it is a necessary requirement for rationality that her credence have some probabilistic structure. Then I show that for a certain class of reasonable measures of inaccuracy, having such a probabilistic structure is sufficient to avoid accuracy-domination in these non-classical settings. (shrink)

In a finitary closure space, irreducible sets behave like two-valued models, with membership playing the role of satisfaction. If f is a function on such a space and the membership of in an irreducible set is determined by the presence or absence of the inputs in that set, then f is a kind of truth function. The existence of some of these truth functions is enough to guarantee that every irreducible set is maximally consistent. The closure space (...) is then said to be expressive. This paper identifies the two-valued truth functional conditions that guarantee expressiveness. (shrink)

Different formal tools are useful for different purposes. For example, when it comes to modelling degrees of belief, probability theory is a better tool than classical logic; when it comes to modelling the truth of mathematical claims, classical logic is a better tool than probability theory. In this paper I focus on a widely used formal tool and argue that it does not provide a good model of a phenomenon of which many think it does provide a (...) good model: I shall argue that while supervaluationism may provide a model of probability of truth, or of assertability, it cannot provide a good model of truth -- supertruth cannot be truth. The core of the argument is that an adequate model of truth must render certain connectives truth-functional (at least in certain circumstances) -- and supervaluationism does not do so (in those circumstances). (shrink)

Nonmonotonic causal logic is a knowledge representation language designed for describing domains that involve actions and change. The process of literal completion, similar to program completion familiar from the theory of logic programming, can be used to translate some nonmonotonic causal theories into classical logic. Its applicability is restricted, however, to theories that deal with truth-valued fluents, represented by predicate symbols. In this note we introduce functional completion—a more general process that can be applied to causal theories in (...) which fluents are treated as functions. (shrink)

We offer a defense of one aspect of Paul Horwich’s response to the Liar paradox—more specifically, of his move to preserve classical logic. Horwich’s response requires that the full intersubstitutivity of ‘ ‘A’ is true’ and A be abandoned. It is thus open to the objection, due to Hartry Field, that it undermines the generalization function of truth. We defend Horwich’s move by isolating the grade of intersubstitutivity required by the generalization function and by providing a new reading (...) of the biconditionals of the form “ ‘A’ is true iff A.”. (shrink)

Carnap in the 1930s discovered that there were non-normal interpretations of classical logic - ones for which negation and conjunction are not truth-functional so that a statement and its negation could have the same truth value, and a disjunction of two false sentences could be true. Church ar-gued that this did not call for a revision of classical logic. More recent writers seem to disa-gree. We provide a definition of "non-normal interpretation" and argue that Church was (...) right, and in fact, the existence of non-normal interpretations tells us something important about the condi-tions of extensionality of the classical logical operators. (shrink)

Propositional logic, also known as sentential logic and statement logic, is the branch of logic that studies ways of joining and/or modifying entire propositions, statements or sentences to form more complicated propositions, statements or sentences, as well as the logical relationships and properties that are derived from these methods of combining or altering statements. In propositional logic, the simplest statements are considered as indivisible units, and hence, propositional logic does not study those logical properties and relations that depend upon parts (...) of statements that are not themselves statements on their own, such as the subject and predicate of a statement. The most thoroughly researched branch of propositional logic is classical truth-functional propositional logic, which studies logical operators and connectives that are used to produce complex statements whose truth-value depends entirely on the truth-values of the simpler statements making them up, and in which it is assumed that every statement is either true or false and not both. However, there are other forms of propositional logic in which other truth-values are considered, or in which there is consideration of connectives that are used to produce statements whose truth-values depend not simply on the truth-values of the parts, but additional things such as their necessity, possibility or relatedness to one another. (shrink)

In this paper we argue that Revision Rules, introduced by Anil Gupta and Nuel Belnap as a tool for the analysis of the concept of truth, also provide a useful tool for defining computable functions. This also makes good on Gupta's and Belnap's claim that Revision Rules provide a general theory of definition, a claim for which they supply only the example of truth. In particular we show how Revision Rules arise naturally from relaxing and generalizing a (...) classical construction due to Kleene, and indicate how they can be employed to reconstruct the class of the general recursive functions. We also point at how Revision Rules can be employed to access non-minimal fixed points of partially defined computing procedures. (shrink)

We investigate introduction and elimination rules for truth-functional connectives, focusing on the general questions of the existence, for a given connective, of at least one such rule that it satisfies, and the uniqueness of a connective with respect to the set of all of them. The answers are straightforward in the context of rules using general set/set sequents of formulae, but rather complex and asymmetric in the restricted (but more often used) context of set/formula sequents, as also in the (...) intermediate set/formula-or-empty context. (shrink)

We investigate introduction and elimination rules for truth-functional connectives, focusing on the general questions of the existence, for a given connective, of at least one such rule that it satisfies, and the uniqueness of a connective with respect to the set of all of them. The answers are straightforward in the context of rules using general set/set sequents of formulae, but rather complex and asymmetric in the restricted (but more often used) context of set/formula sequents, as also in the (...) intermediate set/formula-or-empty context. (shrink)

The article examines main competing conceptions of the cor­respondence theory of truth. First, the author investigates pos­sible candidates for the role of truth-bearers. Among those he examines following entities: instances of sentences as concrete sequences of symbols (sounds or letters), which should satisfy wide scope of requirements, such as to be grammatical, mean­ingful, affirmative and so on; abstract propositions, which are ex­pressed by concrete sentences; utterances (either explicit or in lingua mentalis); beliefs of agents as their special mental (...) states. Then he turns to the study of possible candidates for the role of truth-makers, i.e. of those entities to which truth-bearers should correspond to be true. He observes states of affairs, situations, facts and mereological sums of individuals. Then he shows that a notion of correspondence is a functional relationship of interpre­tation of linguistic expressions, where certain fragment of real­ity is taken as a model, i.e. as a finite set of elements on which some functions and relations can be operationally defined. He shows how in some simple cases such interpretation function can be built in operationalist’s manner. After that, he considers some natural objections to this approach, which point that he has no direct cognitive access to the actual world, so it is not possible to take its fragments as a model for any theory. Such objections lead us to the coherentist’s approach, but he shows that the question about origin and status of so-called “specified set” implicitly either leads to infinite regress or requires certain correspondence on some level of investigation and reasoning. Finally, the authore presents some reflections on the meaning of correspondence approach for scientific realism and all other versions of realistic philosophy and metaphysics. (shrink)

We present some proof-theoretic results for the normal modal logic whose characteristic axiom is \. We present a sequent system for this logic and a hypersequent system for its first-order form and show that these are equivalent to Hilbert-style axiomatizations. We show that the question of validity for these logics reduces to that of classical tautologyhood and first-order logical truth, respectively. We close by proving equivalences with a Fitch-style proof system for revision theory.

Is it a "conceptual truth" or only a logically contingent fact that, in any given kind of case, an event x which asymmetrically causes ("produces") an event y likewise temporally precedes y or at least does not temporally succeed y? A bona fide physical example in which the cause retroproduces the effect would show that backward causation is no less conceptually possible than forward causation. And it has been claimed ([9], p. 151; [4], p. 41) that in Dirac's (...) class='Hi'>classical electrodynamics (relativistic and non-relativistic), the preacceleration of charged particles before any forces are applied to them furnishes a genuine case of retrocausation by later forces. An exposition of the pertinent physics furnishes the basis for arguing the following: Whereas the non-zero acceleration of a neutral NEWTONIAN mass particle is, of course, causally connected as such to the simultaneously applied non-zero force, the non-zero acceleration of a DIRACIAN charged particle is not causally connected at all as such to the applied forces. A fortiori, in Dirac's electrodynamics, the applied forces do not qualify asymmetrically as the causes of a non-zero acceleration as such; nor does a non-zero acceleration as such qualify as an effect produced by forces. This is shown by means of two considerations as follows: (1) In Dirac's theory, no functional dependence of the value of a non-zero acceleration on the weighted time-average of the forces is vouchsafed as a matter of physical LAW alone without any value of a constant of integration, just as no Newtonian law(s) alone can guarantee a functional dependence of the non-zero value of the velocity of a Newtonian mass particle on the applied forces. Instead, both functional dependencies alike are vouchsafed only with the crucial aid of some de facto boundary condition pertaining to either the past or to the furute, so that (2) Non-zero preaccelerations of Diracian charged particles can no more be causally attributed as such to the retrocausal action of later forces than non-zero "prevelocities" of Newtonian mass particles can be held to be caused as such by later applied forces. The retrocausal interpretation of Dirac's preaccelerations is just as invalid as the retrocausal interpretation of Newtonian prevelocities. (shrink)

American-plan semantics with 4 values 1, 0, { {1, 0}} {{}}, interpretable as True, False, Both and Neither, are furnished for a range of logics, including relevant affixing systems. The evaluation rules for extensional connectives take a classical form: in particular, those for negation assume the form 1 (A, a) iff 0 (A, a) and 0 (A, a) iff 1 (A, a), so eliminating the star function *, on which much criticism of relevant logic semantics has focussed. The cost (...) of these classical features is a further relation (or operation), required in evaluating falsity assignments of implication formulae.Two styles of 4 valued relational semantics are developed; firstly a semantics using notions of double truth and double validity for basic relevant systemB and some extensions of it; and secondly, since the first semantics makes heavy weather of validating negation principles such as Contraposition, a reduced semantics using more complex implicational rules for relevant systemC and various of its extensions. To deal satisfactorily with elite systemsR,E andT, however, further complication is inevitable; and a relation of mateship (suggested by the Australian plan) is introduced to permit cross-over from 1 to 0 values and vice versa. (shrink)

We present a realizability interpretation for classical analysis–an association of a term to every proof so that the terms assigned to existential formulas represent witnesses to the truth of that formula. For classical proofs of Π2 sentences ∀x∃yA(x,y), this provides a recursive type 1 function which computes the function given by f(x)=y iff y is the least number such that A(x,y).

In a previous paper, we have proposed assigning as the value of a physical quantity in quantum theory, a certain kind of set (a sieve) of quantities that are functions of the given quantity. The motivation was in part physical---such a valuation illuminates the Kochen-Specker theorem; and in part mathematical---the valuation arises naturally in the topos theory of presheaves. This paper discusses the conceptual aspects of this proposal. We also undertake two other tasks. First, we explain how the proposed (...) valuations could arise much more generally than just in quantum physics; in particular, they arise as naturally in classical physics. Second, we give another motivation for such valuations (that applies equally to classical and quantum physics). This arises from applying to propositions about the values of physical quantities some general axioms governing partial truth for any kind of proposition. (shrink)

In this paper, I examine the concept of truth in classical Chinese philosophy, beginning with a critical examination of Chad Hansen’s claim that it has no such concept. By using certain passages that emphasize analogous concepts in the philosophy of language of the Later Mohist Canons, I argue that while there is no word in classical Chinese that functions as truth generally does in Western philosophy for grammatical reasons, the Later Mohists were certainly working with (...) a notion of semantic adequacy in which a language-to-world relationship is made an object of investigation, challenging Hansen’s position that classical Chinese functions within a primarily pragmatic linguistic framework in which a language-to-user relationship determines the meaning of words. (shrink)

In their recent paper Bi-facial truth: a case for generalized truth values Zaitsev and Shramko [7] distinguish between an ontological and an epistemic interpretation of classical truth values. By taking the Cartesian product of the two disjoint sets of values thus obtained, they arrive at four generalized truth values and consider two “semi-classical negations” on them. The resulting semantics is used to define three novel logics which are closely related to Belnap’s well-known four valued (...) logic. A syntactic characterization of these logics is left for further work. In this paper, based on our previous work on a functionally complete extension of Belnap’s logic, we present a sound and complete tableau calculus for these logics. It crucially exploits the Cartesian nature of the four values, which is reflected in the fact that each proof consists of two tableaux. The bi-facial notion of truth of Z&S is thus augmented with a bi-facial notion of proof. We also provide translations between the logics for semi-classical negation and classical logic and show that an argument is valid in a logic for semi-classical negation just in case its translation is valid in classical logic. (shrink)

Geoffrey Hellman has argued that non-truth-functionality entails a change of meaning between classical and quantum logical connectives. This paper criticizes and significantly improves the argument.

In this chapter, I argue that Aristotle’s doctrine of hylomorphism, which conceived the natural world as consisting of substances which are metaphysically composed of matter and form, is ripe for rehabilitation in the light of quantum physics. I begin by discussing Aristotle’s conception of matter and form, as it was understood by Aquinas, and how Aristotle’s doctrine of hylomorphism was ‘physicalised’ and eventually abandoned with the rise of microphysicalism. I argue that the phenomenon of quantum entanglement, and the emergence of (...) irreducibly macroscopic phenomena in finite temperature quantum systems, have given us good reasons to doubt the truth of microphysicalism. In support of my argument, I show how to construct a hylomorphic interpretation of the de Broglie-Bohm theory that posits a single Cosmic Substance. I then show how to construct a hylomorphic interpretation of an alternative ‘contextual’ wave function collapse theory (recently proposed by the physicists Barbara Drossel and George Ellis) which posits a plurality of thermal substances. Both of these neo-Aristotelian ontologies reject the microphysicalist dogma that nature consists solely of some set of microscopic constituents. (shrink)

Four weak positional calculi are constructed and examined. They refer to the use of the connective of negation within the scope of the positional connective “R” of realization. The connective of negation may be fully classical, partially analogical or independent from the classical, truth-functional negation. It has been also proved that the strongest system, containing fully classical connective of negation, is deductively equivalent to the system MR from Jarmużek and Pietruszczak.

In this paper we discuss in which sense truth is considered as a mathematical object in propositional logic. After clarifying how this concept is used in classical logic, through the notions of truth-table, truth-function and bivaluation, we examine some generalizations of it in non-classical logics: many-valued matrix semantics with three and four values, non-truth-functional bivalent semantics, Kripke possible world semantics. • DOI:10.5007/1808-1711.2010v14n1p31.

The majority of formal accounts attribute to Stoic logicians the classical truth-functional understanding of the material conditional and exclusive disjunction.These interpretations were disputed,...

Simple partial logic (=SPL) is, broadly speaking, an extensional logic which allows for the truth-value gap. First I give a system of propositional SPL by partializing classical logic, as well as extending it with several non-classical truth-functional operators. Second I show a way based on SPL to construct a system of tensed ontology, by representing tensed statements as two kinds of necessary statements in a linear model that consists of the present and future worlds. Finally I (...) compare that way with other two ways based on Łukasiewicz’s three-valued logic and branching temporal logic. (shrink)

Up in the last one in the United States comes to understand the general process: it is linguistics and history, and it requires a priori understanding that In order to understand the current situation before, prior to a full understanding effectiveness and bias as understanding the meaning is from whole to part and from part to whole , in the understanding of the history and heritage hermeneutic circle is always in operation, understanding related to the sight of the fusion eventually (...) like Q & A dialogue. In order to interpret this paper, the classic text of the task. Gadamer on this topic already has a large amount of experience, and to make you pleasantly surprised or not surprised the way described in this topic. This presentation will define and into the road to discuss seven key words, these seven key words will help us to understand Gadamer are talking about the interpretation of the classic text. Each of these seven words on a word, the speaker will be asked whether you find in your own experience is true. 1. Gleichzeitigkeit-both of the words appear in the "real management and methods" up to the U.S. for the aesthetic experience in the timing of the discussion. He found that when visual field interaction, the reader found In order to interpret the ancient texts and aging time distance between the disappearance of the activities in reading, and text with our own game to attract you into the presence. In this part of the final, which also is mentioned for the moral, economic, regardless of language and other classics: They attract you to the significance of the cycle, like they were written today. 2. Applicatio-application In order to make up the United States is not used in accordance with such assertions: "understand something is to understand how it applies to pre-existing condition." So when the text is not only common, as in # 1, it means that something in the present before. In order to understand the application will be leading. History of the United States up to review the timing of the application was excluded, as well as understand the text is treated as just understanding and interpretation of events, but up to the United States want to re-restored early priority in the 19th and 20th centuries, the elements of hermeneutics. 3. Erfahrung-up experience in the United States defined by the qualification before a person experiences some of the things. Its typical form, is involved in the Lok empty hope, therefore, sentence description if not in the pit, not wisdom. Gadamer claims previous experience in the concept: "not to consider the experience of the inner history of." But up to the United States said that experience, "teaches us to recognize the reality." Experience, the result is "to know What is the What." The other result is that we "Experiences", it is not arbitrary and open to new experiences. In order to find you first read the classic text commands were prohibitive, and dismissed out of hand In order to your expectations, but in the end so make sure you are not the more arbitrary and more open to new experiences it? 4. Wahrheit-up to the United States is not really reasonable satisfaction of reasonable scientific definition of truth. His concern is that we insisted that the "true" regardless question. Description is described in "What things are the way" - a real management of ontology definitions. He followed Heidegger aletheia true management point of view, is to expose shield . According to this definition, even works of art can be "real". That is, it can expose the shield there things through other means in a way not possible. So the question is, under the classic in the sense that the writing is true? Gadamer description is. 5. Selbst-Verständnis-up to their own understanding to the United States in this裡In order to further understand more. It is not more deeply understand the true management classic, it is found that work experience change a person's sight and view of life, and it changed the understanding of a person's own. When the text of the event experience, we can integrate it into the "existence of a continuous nature." We can find the concept of other games in this裡: Application, experience and true management. Events with the classic encounter with a person unable to read the classics before he was the same person. 6. Verwandlung ins Gebilde-conversion into the structural changes that are described in works of art up to the way the United States. Some things have been transformed into the experience and continued with the description of future generations made ​​the structure裡. This is one of the functions of art: something to save future generations, but is really something. This is Why is it up to the United States in terms of tradition is so important. This structure is special, it no longer be measured by our standards, but from its own standards. Finally, the significance of change is more true for the mediocre, reading the text is found in its own transformation, a person will not stay in place. 7. Solidarität, Fest-one of the festival in the late works with up to learn to find beauty in the human basis of his artistic point of view. Therefore, "the United States related" has three parts, the game, , symbol , and festivals. In later works, he found time not only festivals are important, it's time from the day In order to inspire us, but it also brought together a community and the joint, to celebrate the event in its creation裡. Festival "to improve our sense of life." Arts festival to share with so many things, when we have nature and art associated, the more we are able to understand art. The end of this lecture will ask you to read the classic self-experience, Gadamer's concept of the Concept for you is true. An introductory section explains that the lecture will present seven key terms, representing seven aspects of the universal process of understanding. It is noted that a published lecture in Chinese by this author under the title "Seven Key Terms in the Philosophy of Hans-Georg Gadamer "is not the same as this, and indeed only one of the terms in that lecture is used in this one, although four of them are used in tomorrow's lecture. (shrink)

Qiyās, or “correlational inference” (often glossed as “analogy”), comprises a primary set of methodological tools recognized by a majority of premodern Sunnī jurists. Its elements, valid modes, and proper applications were the focus of continual argument and refinement. A particular area of debate was the methodology of determining or justifying the ʿilla: the legal cause (or occasioning factor, or ratio legis) giving rise to a ruling in God’s Law. This was most often discussed (and disputed) under the rubric of “the (...) modes of causal justification” (masālik al-taʿlīl). Among these modes was the much debated test of dawarān (concomitance of presumed cause and effect). In brief, proponents of dawarān employed it to justify claims that a property (waṣf) occasioned the ruling (ḥukm) in an authoritative source-case (aṣl). In concert with other considerations, the demonstrated co-presence (ṭard) and co-absence (ʿaks) of property and ruling—that is, their concomitance “in existence” (wujūdan) and “in nonexistence” (ʿadaman)— was taken as an indication that the property was the ruling’s ʿilla. Delving further into dawarān and causation (ʿilliyya), the current study interprets “in existence” and “in nonexistence” not as a kind of metaphor for true and false (within the framework of a classical truth-functional formal semantics), but as an accurate terminology vis-à-vis the meaning of causality statements, fully compatible with dominant Islamicate views on causal agency. In brief, a deeper logical and linguistic analysis of the different existential modes of dawarān strongly suggests that we should distinguish property (or phenomenon) and ruling (or effect) as types (concepts or propositions linguistically expressed by a sentence) as opposed to tokens (instantiations of the type; the real, ontological events that verify the proposition). Our reading of dawarān as shaped by a finer-grained structure not only allows us to identify the efficient occasioning process as a function which takes some particular token of the ʿilla (arguably, the property or properties which provide the ruling’s material cause) and renders a token of the general ruling type, but it allows us to elucidate the role of taʿlīl (causal justification) in shaping an epistemological theory of argument to the best explanation: a sophisticated, premodern manifestation of abductive reasoning. (shrink)

A semantics of vagueness should reject the principle that every statement has a truth-value yet retain the classical tautologies. A many-value, non-truth-functional semantics and a semantics of super-valuations each have this result. According to the super-valuation approach, 'if a man with n hairs on his head is bald, then a man with n plus one hairs on his head is also bald' is false because it comes out false no matter how the vague predicate 'is bald' is (...) appropriately made precise. But why should a sentence in which components actually remain imprecise be regarded as actually false just because it would be false if its components were precise? On one of the alternative treatments of quantification allowed by the many-value approach, the sentence in question is assigned an intermediate value closer to 'false' than to 'true'. Despite the elegance of the super-valuation approach, there are reasons to prefer the many-value approach. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:American Journal of Philology 123.3 (2002) 513-516 [Access article in PDF] Deborah Tarn Steiner. Images in Mind: Statues in Archaic and Classical Greek Literature and Thought. Princeton: Princeton University Press, 2001. xviii + 360 pp. 28 black-and-white figures. Cloth, $39.50. The production of sculpture in metal, stone, and other materials was a craft that virtually disappeared from the Greek world for several centuries after the end of the (...) Bronze Age. Small, solid cast bronze figurines reappeared in the ninth century B.C.E., but it was not until the middle seventh century that the first large-scale stone statues of women and men were created. From that point on, freestanding and architectural sculpture played a central role in Greek society and its artistic expression. While the stylistic evolution of Greek sculpture has been traced many times, in this provocative and ambitious book, Deborah Steiner sets out to examine, from a literary standpoint, "how archaic and classical Greek poets, philosophers, dramatists, and historians introduce statues as cognitive and hermeneutic devices in their texts" (xi). The investigation leads her to analyze what the statues "looked like, how they functioned," and "what they were credited with doing." The resulting book "aims to tell two stories, one about objects, the other about texts, and to show how each proves crucial to the other." The examination is somewhat lopsided, since the intellectual use of the visual material is given far greater emphasis than the statues themselves.The book is divided into five lengthy chapters and an epilogue. The first chapter, "Replacement and Replication," focuses on the physical resemblance of statues to people but not their inner natures. Chapter 2, "Inside and Out," examines, instead, the ways that a statue's exterior can relate more or less exactly to its inner truth. Chapter 3, "The Quick and the Dead," suggests that statues, by their very inertia, closely resemble the dead and therefore mediate in our imagination between the two states. Statues as erotic (or anti-erotic) art objects form the subject of chapter 4, "For Love of a Statue." The fifth and final chapter, "The Image in the Text," returns to the literary relation of sculpture to the production of memory: the similarity between victory odes and victors' statues, funerary epigrams and monuments, and the creation and regulation (on both the social and political levels) of honorary statuary and decrees. The retrospective epilogue is followed by twenty-eight black-and-white illustrations: Most of them depict well-known pieces of sculpture, but there are also several photographs of vase paintings and one of a drachma.In all the chapters, Steiner continually draws attention to the multivalent aspects of statues. The Greeks developed and applied a large and nuanced vocabulary to describe different types of image that embody differences only [End Page 513] poorly translated by our term, "statue": agalma, eidolon, eikon, kolossos, xoanon, and sema. Most of these have provoked extended discussion among modern scholars. The functions that statues played are even more diverse: they could serve as replacements for the absent or deceased individual, as votive offerings, as cult images, as commemorative monuments, as apotropaic devices, as protective symbols, as minatory agents, and occasionally as deadly entities (some statues could madden their viewer and even kill those who abused them or their creators by toppling over). They were also focusing devices that could evoke a wide range of emotional responses, from pity to lust. As exempla they served to instruct, and some could speak with their own voices, sometimes comically (e.g., the herms in Attic comedy) or poignantly through the words inscribed on their bases (e.g., the famous lines of Phrasikleia, the dead maiden who will never be a bride). The repeated use of the first person by funerary figures is particularly evocative. The inscriptions on other figures (like the very early Mantiklos Apollo statuette) make it clear that images could serve as bargaining chips with the gods for past services rendered or debts owed for the future.Other statues defy simple categorization—the Athena Parthenos, for example... (shrink)

It is argued that truth value of a sentence containing free variables in a context of use, just as the reference of the free variables concerned, depends on the assumptions and posits given by the context. However, context may under-determine the reference of a free variable and the truth value of sentences in which it occurs. It is argued that in such cases a free variable has indeterminate reference and a sentence in which it occurs may have indeterminate (...) truth value. On letting, say, x be such that \, the sentence ‘Either \ or \’ is true but the sentence ‘\’ has an indeterminate truth value: it is determinate that the variable x refers to either 2 or \, but it is indeterminate which of the two it refers to, as a result ‘\’ has a truth value but its truth value is indeterminate. The semantic indeterminacy is analysed in a ‘radically’ supervaluational semantic framework closely analogous to the treatment of vagueness in McGee and McLaughlin and Smith, which saves bivalence, the T-schema and the truth-functional analysis of the boolean connectives. It is shown that on such an analysis the modality ‘determinately’ is quite clearly not an epistemic modality, avoiding a potential objection raised by Williamson against such ‘radically’ supervaluational treatments of vagueness, and that determinate truth is the semantic value preserved in classically valid arguments. The analysis is contrasted with the epistemicist proposal of Breckenridge and Magidor which implies that ‘\’ has a determinate but unknowable truth value. (shrink)

Wittgenstein's Tractatus contains a wide range of profound insights into the nature of logic and language – insights which will survive the particular theories of the Tractatus and seem to me to mark definitive and unassailable landmarks in our understanding of some of the deepest questions of philosophy. And yet alongside these insights there is a theory of the nature of the relation between language and reality which appears both to be impossible to work out in detail in a way (...) which is completely satisfactory, and to be bizarre and incredible. I am referring to the so-called logical atomism of the Tractatus . The main outlines of this theory at least are clear and familiar: there are elementary propositions which gain their sense from being models of possible states of affairs; such propositions are configurations of names of simple objects, signifying that those simples are analogously configured; every proposition has its sense through being analysable as a truth-functional compound of elementary propositions, thus deriving its sense from the sense of the elementary propositions when this view is taken in conjunction with the idea that the sense of a proposition is completely specified by specifying its truth-conditions. In this way the Tractatus incorporates in its working out a philosophical system analogous to the classical philosophical systems of Leibniz or Spinoza which are regarded by many people, in a sense rightly, as the prehistoric monsters of philosophy which are not to be studied as living organisms, but studied as the curiosities of human thought. And we may here agree that in the end we must simply reject a philosophy which incorporates such features as its postulation of simple eternal objects, or of a possibility of an analysis of a proposition which was presented as a pre-condition for the propositions that we ordinarily utter to make sense, and yet the specific form of which we are unaware of, and so on. (shrink)

Any set of truth-functional connectives has sequent calculus rules that can be generated systematically from the truth tables of the connectives. Such a sequent calculus gives rise to a multi-conclusion natural deduction system and to a version of Parigot’s free deduction. The elimination rules are “general,” but can be systematically simplified. Cut-elimination and normalization hold. Restriction to a single formula in the succedent yields intuitionistic versions of these systems. The rules also yield generalized lambda calculi providing proof terms (...) for natural deduction proofs as in the Curry-Howard isomorphism. Addition of an indirect proof rule yields classical single-conclusion versions of these systems. Gentzen’s standard systems arise as special cases. (shrink)

Halbach has argued that Tarski biconditionals are not ontologically conservative over classical logic, but his argument is undermined by the fact that he cannot include a theory of arithmetic, which functions as a theory of syntax. This article is an improvement on Halbach's argument. By adding the Tarski biconditionals to inclusive negative free logic and the universal closure of minimal arithmetic, which is by itself an ontologically neutral combination, one can prove that at least one thing exists. The (...) result can then be strengthened to the conclusion that infinitely many things exist. Those things are not just all Gödel codes of sentences but rather all natural numbers. Against this background inclusive negative free logic collapses into noninclusive free logic, which collapses into classical logic. The consequences for ontological deflationism with respect to truth are discussed. (shrink)

The first systematic exposition of all the central topics in the philosophy of logic, Susan Haack's book has established an international reputation for its accessibility, clarity, conciseness, orderliness, and range as well as for its thorough scholarship and careful analyses. Haack discusses the scope and purpose of logic, validity, truth-functions, quantification and ontology, names, descriptions, truth, truth-bearers, the set-theoretical and semantic paradoxes, and modality. She also explores the motivations for a whole range of non-classical systems (...) of logic, including many-valued logics, fuzzy logic, moddal and tense logics, and relevance logics. Persupposing only an elementary knowledge of formal logic, this book includes many useful summary tables and diagrams, as well as a helpful glossary of technical terms. Wide-ranging, informative, and eminently readable, this book has proven a valuable resource for generations of students and scholars in a variety of disciplines outside philosophy needing guidance on the philosophy of logic. (shrink)