Tarski and Mautner proposed to characterize the "logical" operations on a given domain as those invariant under arbitrary permutations. These operations are the ones that can be obtained as combinations of the operations on the following list: identity; substitution of variables; negation; finite or infinite disjunction; and existential quantification with respect to a finite or infinite block of variables. Inasmuch as every operation on this list is intuitively "logical", this lends support to the Tarski-Mautner proposal.

I present a notion of invariance under arbitrary surjective mappings for operators on a relational finite type hierarchy generalizing the so-called Tarski-Sher criterion for logicality and I characterize the invariant operators as definable in a fragment of the first-order language. These results are compared with those obtained by Feferman and it is argued that further clarification of the notion of invariance is needed if one wants to use it to characterize logicality.

We revisit a typological puzzle due to Horn (Doctoral Dissertation, UCLA, 1972) regarding the lexicalization of logical operators: in instantiations of the traditional square of opposition across categories and languages, the O corner, corresponding to ‘nand’ (= not and), ‘nevery’ (= not every), etc., is never lexicalized. We discuss Horn’s proposal, which involves the interaction of two economy conditions, one that relies on scalar implicatures and one that relies on markedness. We observe that in order to express markedness (...) and to account for a bigger typological puzzle, namely the absence of lexicalizations of ‘XOR’ (= exclusive or), ‘all-or-none’, and many other imaginable logical operators, one must restrict the basic lexicalizable elements to a small set of primitives. We suggest that an ordering based perspective, following Keenan and Faltz (Boolean semantics for natural language, 1985), makes the stipulated primitives that we arrive at more natural. We also propose a modification to Horn’s proposal, based on recent work on implicatures, in which only the implicature condition is operative and in which markedness is part of the definition of the alternatives for scalar implicatures rather than an independent condition. (shrink)

Tautology is interpreted as a necessary condition for the workability of an operations system. This condition suggests the following possibilities: the stable solvability of balance equations between available resources and requests for them; the calculation of potential and kinetic of the system together with the estimation of the contribution of every operation to the kinetic of the system; the construction of a deadlockless infinite cyclic process for performance of some works.

We show that logic has more to offer to ontologists than standard first order and modal operators. We first describe some operators of linear logic which we believe are particularly suitable for ontological modeling, and suggest how to interpret them within an ontological framework. After showing how they can coexist with those of classical logic, we analyze three notions of artifact from the literature to conclude that these linear operators allow for reducing the ontological commitment needed for (...) their formalization, and even simplify their logical formulation. (shrink)

We generalize, by a progressive procedure, the notions of conjunction and disjunction of two conditional events to the case of n conditional events. In our coherence-based approach, conjunctions and disjunctions are suitable conditional random quantities. We define the notion of negation, by verifying De Morgan’s Laws. We also show that conjunction and disjunction satisfy the associative and commutative properties, and a monotonicity property. Then, we give some results on coherence of prevision assessments for some families of compounded conditionals; in particular (...) we examine the Fréchet-Hoeffding bounds. Moreover, we study the reverse probabilistic inference from the conjunction Cn+1 of n + 1 conditional events to the family {Cn,En+1|Hn+1}. We consider the relation with the notion of quasi-conjunction and we examine in detail the coherence of the prevision assessments related with the conjunction of three conditional events. Based on conjunction, we also give a characterization of p-consistency and of p-entailment, with applications to several inference rules in probabilistic nonmonotonic reasoning. Finally, we examine some non p-valid inference rules; then, we illustrate by an example two methods which allow to suitably modify non p-valid inference rules in order to get inferences which are p-valid. (shrink)

I consider how complex logical operations might self-assemble in a signalling-game context via composition of simpler underlying dispositions. On the one hand, agents may take advantage of pre-evolved dispositions; on the other hand, they may co-evolve dispositions as they simultaneously learn to combine them to display more complex behaviour. In either case, the evolution of complex logical operations can be more efficient than evolving such capacities from scratch. Showing how complex phenomena like these might evolve provides an additional (...) path to the possibility of evolving more or less rich notions of compositionality. This helps provide another facet of the evolutionary story of how sufficiently rich, human-level cognitive or linguistic capacities may arise from simpler precursors. 1Signalling and Self-assembly2Simple Unary Logic Games3Composing Unary Functions for Binary Inputs 3.1Utilizing pre-evolved dispositions3.2Co-evolving logical dispositions3.3Learning appropriate outputs3.4Taking account of the full state-space of unary games3.5Role-free composition4Discussion 4.1Efficacy and efficiency of learning complex dispositions4.2Other binary operations4.3To infinity and beyond5Conclusion. (shrink)

This is a purely conceptual paper. It aims at presenting and putting into perspective the idea of a proof-theoretic semantics of the logical operations. The first section briefly surveys various semantic paradigms, and Section 2 focuses on one particular paradigm, namely the proof-theoretic semantics of the logical operations.

In John Dewey’s logical theory, qualities or qualitative relations account for the capacity to distinguish and associate the objects of reflective thought; they are antecedent to reflective analysis and necessary for coherent processes of inquiry. In Dewey’s writings that are specifically “metaphysical” in orientation, he is much more vague about the function of qualities, but does call them “generic traits of existence.” As such, they appear to be central to his mature ontological theory. In order to more fully understand (...) the metaphysical import of qualitative relations, I first examine the details of Dewey’s logical theory, and then generalize those details in accordance with Dewey’s larger theory of nature. The end result is a novel interpretation of Dewey’s metaphysics. (shrink)

In what follows, the difference between Frege’s and Schröder’s understanding of logical connectives will be investigated. It will be argued that Frege thought of logical connectives as concepts, whereas Schröder thought of them as operations. For Frege, logical connectives can themselves be connected. There is no substantial difference between the connectives and the concepts they connect. Frege’s distinction between concepts and objects is central to this conception, because it allows a method of concept formation which enables us (...) to form concepts from the logical connectives alone. Schröder in contrast unifies the distinction between concepts and objects, but keeps the distinction between logical connectives and what they connect. It will be argued that Frege’s particular way of perceiving logical connectives is crucial for his foundational project. (shrink)

In this paper, we define for the first time three neutrosophic actions and their properties. We then introduce the prevalence order on {T, I, F} with respect to a given neutrosophic operator “o”, which may be subjective - as defined by the neutrosophic experts; and the refinement of neutrosophic entities <A>, <neutA>, and <antiA> . Then we extend the classical logical operators to neutrosophic literal logical operators and to refined literal logical operators, and we (...) define the refinement neutrosophic literal space. (shrink)

The paper proposes a semantic value for the logical constants (connectives and quantifiers) within the framework of proof-theoretic semantics, basic meaning on the introduction rules of a meaning conferring natural deduction proof system. The semantic value is defined based on Fregecontributions” to sentential meanings as determined by the function-argument structure as induced by a type-logical grammar. In doing so, the paper proposes a novel proof-theoretic interpretation of the semantic types, traditionally interpreted in Henkin models. The compositionality of the (...) resulting attribution of semantic values is discussed. Elsewhere, the same method was used for defining proof-theoretic meaning of subsentential phrases in a fragment of natural language. Doing the same for (the simpler and clearer case of) logic sheds more light on the proposal. (shrink)

We formally investigate immediate and mediate grounding operators from an inferential perspective. We discuss the differences in behaviour displayed by several grounding operators and consider a general distinction between grounding and logical operators. Without fixing a particular notion of grounding or grounding relation, we present inferential rules that define, once a base grounding calculus has been fixed, three grounding operators: an operator for immediate grounding, one for mediate grounding – corresponding to the transitive closure of (...) the immediate grounding one – and a grounding tree operator, which enables us to internalise chains of immediate grounding claims without losing any information about them. We then present an in-depth proof-theoretical study of the introduced rules by focussing, in particular, on the question whether grounding operators can be considered as logical operators and whether balanced rules for grounding operators can be defined. (shrink)

Vector logic is a mathematical model of the propositional calculus in which the logical variables are represented by vectors and the logical operations by matrices. In this framework, many tautologies of classical logic are intrinsic identities between operators and, consequently, they are valid beyond the bivalued domain. The operators can be expressed as Kronecker polynomials. These polynomials allow us to show that many important tautologies of classical logic are generated from basic operators via the operations (...) called Type I and Type II products. Finally, it is described a matrix version of the Fredkin gate that extends its properties to the many-valued domain, and it is proved that the filtered Fredkin operators are second degree Kronecker polynomials that cannot be generated by Type I or Type II products. (shrink)

where F is a contradiction (I use his numbering). Tim says about these equivalences: (1) “modulo the implicit recognition of this equivalence, the epistemology of metaphysically modal thinking is a special case of the epistemology of counterfactual thinking. Whoever has what it takes to understand the counterfactual conditional and the elementary logical auxiliaries ~ and F has what it takes to understand possibility and necessity operators.” (158) (2) The idea that we evaluate metaphysically modal claims “by some quite (...) different means [from those we use to evaluate counterfactuals – CP] is highly fanciful, since it indicates a bizarre lack of cognitive economy and has no plausible explanation of where the alternative cognitive resources might come from” (162) (3) “the capacity to handle metaphysical modality is an “accidental” byproduct of the cognitive mechanisms that provide our capacity to handle counterfactual conditionals” (162). The biconditional corresponding to a proposed definition of a concept C may be necessary and a priori. But that leaves open these questions: (1) Does the definition contribute to an explanation of why what are in fact truths containing C are true? (The Explanation Question) (2) Does the definition contribute to an explanation of our knowledge of certain contents containing C? (Epistemic Question) (3) Does the definition contribute to an explanation of our understanding or grasp of contents containing C? (Understanding Question) I argue that applied to (17) and (18), considered as definitions, the answers to these three.. (shrink)

I examine the ideas leading up to Wittgenstein's pronouncement at Tractatus 5.4611 that signs for logical operations are punctuation marks.

Accretive and monotone operator theory are central branches of nonlinear functional analysis and constitute the abstract study of certain set-valued mappings between function spaces. This paper deals with the computational properties of these accretive and (generalized) monotone set-valued operators. In particular, we develop (and extend) for this field the theoretical framework of proof mining, a program in mathematical logic that seeks to extract computational information from prima facie “non-computational” proofs from the mainstream literature. To this end, we establish (...) class='Hi'>logical metatheorems that guarantee and quantify the computational content of theorems pertaining to accretive and (generalized) monotone set-valued operators. On the one hand, our results unify a number of recent case studies, while they also provide characterizations of central analytical notions in terms of proof theoretic ones on the other, which provides a crucial perspective on needed quantitative assumptions in future applications of proof mining to these branches. (shrink)

The logic of proofs was introduced by Artemov in order to analize the formalization of the concept of proof rather than the concept of provability. In this context, some operations on proofs play a very important role. In this paper, we investigate some very natural operations, paying attention not only to positive information, but also to negative information (i.e. information saying that something cannot be a proof). We give a formalization for a fragment of such a logic of proofs, and (...) we prove that our fragment is complete and decidable. (shrink)

The central aim of this paper is to present a Boolean algebraic approach to the classical Aristotelian Relations of Opposition, namely Contradiction and (Sub)contrariety, and to provide a 3D visualisation of those relations based on the geometrical properties of Platonic and Archimedean solids. In the first part we start from the standard Generalized Quantifier analysis of expressions for comparative quantification to build the Comparative Quantifier Algebra CQA. The underlying scalar structure allows us to define the Aristotelian relations in Boolean terms (...) and to propose a 3D visualisation by transforming a cube into an octahedron. In part two, the architecture of the CQA is shown to carry over, both to the classical quantifiers of Predicate Calculus and to the modal operators—which are given a Generalized Quantifier style re-interpretation. In this way we provide an algebraic foundation for Blanché’s Aristotelian hexagon as well as a 3D alternative to his 2D star-like visualisation. In a final part, a richer scalar structure is argued to underly the realm of Modality, thus generalizing the 3D algebra with eight (2 3 ) operators to a 4D algebra with sixteen (2 4 ) operators. The visual representation of the latter structure involves a transformation of the hypercube to a rhombic dodecahedron. The resulting 3D visualisation allows a straightforward embedding, not only of the classical Blanché star of Aristotelian relations or the paracomplete and paraconsistent stars of Béziau (Log Investig 10, 218–232, 2003) but also of three additional isomorphic Aristotelian constellations. (shrink)

In the paper, there is presented the theory of logical consequence operators indexed with taboo functions. It describes the mechanisms of logical inference in the environment of forbidden sentences. This kind of processes take place in ideological discourses within which their participants create various narrative worlds (mental worlds). A peculiar feature of ideological discourses is their association with taboo structures of deduction which penalize speech acts. The development of discourse involves, among others, transforming its deduction structure towards (...) the proliferation of consequence operators and modifying penalty functions. The presented theory enables to define various processes of these transformations in the precise way. It may be used in analyses of conflicts between competing elm experts acting within a discourse. (shrink)

The properties of the ${\forall^{1}}$ quantifier defined by Kontinen and Väänänen in [13] are studied, and its definition is generalized to that of a family of quantifiers ${\forall^{n}}$ . Furthermore, some epistemic operators δ n for Dependence Logic are also introduced, and the relationship between these ${\forall^{n}}$ quantifiers and the δ n operators are investigated.The Game Theoretic Semantics for Dependence Logic and the corresponding Ehrenfeucht- Fraissé game are then adapted to these new connectives.Finally, it is proved that the (...) ${\forall^{1}}$ quantifier is not uniformly definable in Dependence Logic, thus answering a question posed by Kontinen and Väänänen in the above mentioned paper. (shrink)

Classical logic, of first or higher order, is extended with sentential operators and quantifiers, interpreted substitutionally over unrestricted substitution class. Operators mark a single layered, consistent metalanguage. Self-reference, arising from substitutional quantification over sentences, allows to express paradoxes which, unlike contradictions, do not lead to explosion. Semantics of the resulting language, using semi-kernels of digraphs, is non-explosive yet two-valued and has classical semantics as a special case for clasically consistent theories. A complete reasoning is obtained by extending LK (...) with two rules for sentential quantifiers. Adding (cut) yields a complete system for the explosive semantics. (shrink)

The logic with independent truth and falsehood operators TFL is proposed. In TFL(→) standard truth-conditions for the implication are adopted. Nevertheless the laws of classical logic are not valid. In this language more then 107 different binary connectives can be defined. So this logic can be treated as universal logic relatively to the class of sentential logics.

In theTractatus, Wittgenstein advocates two major notational innovations in logic. First, identity is to be expressed by identity of the sign only, not by a sign for identity. Secondly, only one logical operator, called “N” by Wittgenstein, should be employed in the construction of compound formulas. We show that, despite claims to the contrary in the literature, both of these proposals can be realized, severally and jointly, in expressively complete systems of first-order logic. Building on early work of Hintikka’s, (...) we identify three ways in which the first notational convention can be implemented, show that two of these are compatible with the text of theTractatus, and argue on systematic and historical grounds, adducing posthumous work of Ramsey’s, for one of these as Wittgenstein’s envisaged method. With respect to the second Tractarian proposal, we discuss how Wittgenstein distinguished between general and non-general propositions and argue that, claims to the contrary notwithstanding, an expressively adequate N-operator notation is implicit in theTractatuswhen taken in its intellectual environment. We finally introduce a variety of sound and complete tableau calculi for first-order logics formulated in a Wittgensteinian notation. The first of these is based on the contemporary notion of logical truth as truth in all structures. The others take into account the Tractarian notion of logical truth as truth in all structures over one fixed universe of objects. Here the appropriate tableau rules depend on whether this universe is infinite or finite in size, and in the latter case on its exact finite cardinality.As it is obviously easy to express how propositions can be constructed by means of this operation and how propositions are not to be constructed by means of it, this must be capable of exact expression.5.503. (shrink)

In 1990 J-L. Krivine introduced the notion of storage operators. They are $\lambda$ -terms which simulate call-by-value in the call-by-name strategy and they can be used in order to modelize assignment instructions. J-L. Krivine has shown that there is a very simple second order type in AF2 type system for storage operators using Gödel translation of classical to intuitionistic logic. In order to modelize the control operators, J-L. Krivine has extended the system AF2 to the classical logic. (...) In his system the property of the unicity of integers representation is lost, but he has shown that storage operators typable in the system AF2 can be used to find the values of classical integers. In this paper, we present a new classical type system based on a logical system called mixed logic. We prove that in this system we can characterize, by types, the storage operators and the control operators. (shrink)

The objective of this paper is to present the truth tables method of the propositional calculus of Tractatus Logico-Philosophicus as a result of computational procedures involving recursive operations in mathematics, since the secondary literature that is involved with such a problem fails to demonstrate such aspect of the work. The proposal is to demonstrate the base calculation of the truth operations as a consequence of the application of mathematical resources that involve the notion of recursivity, inspired both in the natural (...) numbers as well as in the factorial calculus, as well as in the procedures recommended by the combinatorial analysis and the calculation of probability. It is hoped, therefore, to present the truth operations of the propositional calculus as coming from the application of arithmetic to the logical resources. (shrink)

In this paper the language of first-order modal logic is enriched with an operator @ ('actually') such that, in any model, the evaluation of a formula @A at a possible world depends on the evaluation of A at the actual world. The models have world-variable domains. All the logics that are discussed extend the classical predicate calculus, with or without identity, and conform to the philosophical principle known as serious actualism. The basic logic relies on the system K, whereas others (...) correspond to various properties that the actual world may have. All the logics are axiomatized. (shrink)

The main result established in this paper is the following: If the base normed spaceV of completely additive weights is a norm-determining subspace of the space of finitely additive weights V acting on the order unit space spanning the operational logic, thenV has the ε-Jordan-Hahn property iff V has the approximate Jordan-Hahn property. Several examples illustrating the theory are given.

We shall describe the set of strongly meet irreducible logics in the lattice ϵLin.t of normal tense logics of weak orderings. Based on this description it is shown that all logics in ϵLin.t are independently axiomatizable. Then the description is used in order to investigate tense logics with respect to decidability, finite axiomatizability, axiomatization problems and completeness with respect to Kripke semantics. The main tool for the investigation is a translation of bimodal formulas into a language talking about partitions of (...) general frames into intervals so that relative to both Kripke frames and descriptive frames the expressive power of both languages coincides. (shrink)

We describe here a simple method in order to obtain programs from proofs in second-order classical logic. Then we extend to classical logic the results about storage operators proved by Krivine for intuitionistic logic. This work generalizes previous results of Parigot.

In this paper, I provide a new reading of Wittgenstein’s N operator, and of its significance within his early logical philosophy. I thereby aim to resolve a longstanding scholarly controversy concerning the expressive completeness of N. Within the debate between Fogelin and Geach in particular, an apparent dilemma emerged to the effect that we must either concede Fogelin’s claim that N is expressively incomplete, or reject certain fundamental tenets within Wittgenstein’s logical philosophy. Despite their various points of disagreement, (...) however, Fogelin and Geach nevertheless share several common and problematic assumptions regarding Wittgenstein’s logical philosophy, and it is these mistaken assumptions which are the source of the dilemma. Once we recognize and correct these, and other, associated expository errors, it will become clear how to reconcile the expressive completeness of Wittgenstein’s N operator, with several commonly recognized features of, and fundamental theses within, the Tractarian logical system. (shrink)

Two connectives are of special interest in metalogical investigations — the connective of implication which is important due to its connections to the notion of inference, and the connective of equivalence. The latter connective expresses, in the material sense, the fact that two sentences have the same logical value while in the strict sense it expresses the fact that two sentences are interderivable on the basis of a given logic. The process of identification of equivalent sentences relative to theories (...) of a logic C defines a class of abstract algebras. The members of the class are called Lindenbaum-Tarski algebras of the logic C. One may abstract from the origin of these algebras and examine them by means of purely algebraic methods. (shrink)

We show that in quantum logic of closed subspaces of Hilbert space one cannot substitute quantum operations for classical (standard Hilbert space) ones and treat them as primitive operations. We consider two possible ways of such a substitution and arrive at operation algebras that are not lattices what proves the claim. We devise algorithms and programs which write down any two-variable expression in an orthomodular lattice by means of classical and quantum operations in an identical form. Our results show that (...) lattice structure and classical operations uniquely determine quantum logic underlying Hilbert space. As a consequence of our result, recent proposals for a deduction theorem with quantum operations in an orthomodular lattice as well as a, substitution of quantum operations for the usual standard Hilbert space ones in quantum logic prove to be misleading. Quantum computer quantum logic is also discussed. (shrink)

We formulate a Curry-typed logic with fine-grained intensionality within Turner’s typed predicate logic. This allows for an elegant presentation of a theory that corresponds to Fox and Lappin’s property theory with curry typing, but without the need for a federation of languages. We then consider how the fine-grained intensionality of this theory can be given an operational interpretation. This interpretation suggests itself as expressions in the theory can be viewed as terms in the untyped lambda-calculus, which provides a model of (...) computation. (shrink)

The concept of an operator is used in a variety of practical and theoretical areas. Operators, as both conceptual and physical entities, are found throughout the world as subsystems in nature, the human mind, and the manmade world. Operators, and what they operate, i.e., their substrates, targets, or operands, have a wide variety of forms, functions, and properties. Operators have explicit philosophical significance. On the one hand, they represent important ontological issues of reality. On the other hand, (...) epistemological operators form the basic mechanism of cognition. At the same time, there is no unified theory of the nature and functions of operators. In this work, we elaborate a detailed analysis of operators, which range from the most abstract formal structures and symbols in mathematics and logic to real entities, human and machine, and are responsible for effecting changes at both the individual and collective human levels. Our goal is to find what is common in physical objects called operators and abstract mathematical structures, with the name operator providing foundations for building a unified but flexible theory of operators. The paper concludes with some reflections on functionalism and other philosophical aspects of the ‘operation’ of operators. (shrink)

Research within the operational approach to the logical foundations of physics has recently pointed out a new perspective in which quantum logic can be viewed as an intuitionistic logic with an additional operator to capture its essential, i.e., non-distributive, properties. In this paper we will offer an introduction to this approach. We will focus further on why quantum logic has an inherent dynamic nature which is captured in the meaning of "orthomodularity" and on how it motivates physically the introduction (...) of dynamic implication operators, each for which a deduction theorem holds with respect to a dynamic conjunction. As such we can offer a positive answer to the many who pondered about whether quantum logic should really be called a logic. Doubts to answer the question positively were in first instance due to the former lack of an implication connective which satisfies the deduction theorem within quantum logic. (shrink)

We present a temporal logic of branching time with four primitive operators: |$\exists {\mathcal {C}}$| – it may change whether; |$\forall {\mathcal {C}} $| – it must change whether; |$\exists \Box $| – it may be endlessly unchangeable that; and |$\forall \Box $| – it must be endlessly unchangeable that. Semantically, operator |$\forall {\mathcal {C}}$| expresses a change in the logical value of the given formula in every state that may be an immediate successor of the one considered, (...) while |$\exists {\mathcal {C}}$| expresses a change in the logical value of the given formula in some state that is a possible immediate successor. |$\forall {\mathcal {C}} $| and |$\exists {\mathcal {C}}$| are not normal operators, they are not mutually definable and have unusual properties: |$\forall {\mathcal {C}} A\leftrightarrow \forall {\mathcal {C}} \neg A$| and |$\exists {\mathcal {C}} A \leftrightarrow \exists {\mathcal {C}} \neg A$| are axioms. Operators |$\exists \Box $| and |$\forall \Box $| express endless unchangeability in some and all paths, respectively. Our axiomatization contains two infinitary rules that allow one to obtain |$\exists \Box A$| from infinitely many combinations of formulas with |$A, \neg, \forall {\mathcal {C}}, \land $|⁠, and to get |$\forall \Box A$| from infinitely many combinations of formulas with |$A, \neg, \exists {\mathcal {C}}, \land $|⁠. We show that our formalism is complete by presenting a bridge between our logic and logic |$\textsf {UB}$|⁠, which is a fragment of |$\textsf {CTL}$| in which until operators does not occur. (shrink)