In this book, I will discuss three main topics: the roots and aims of scientific knowledge, scientific knowledge in society, and science and values I understand scientific knowledge as being a planned and continuous production of the general and common knowledge of scientific communities. I begin my discussion with a brief analysis of the main differences between sciences, on the one hand, and everyday experience, philosophies, religions, and ideologies, on the other. I define the concept of science as a set (...) of family resemblances (in Wittgenstein's sense). The sciences can be distinguished along three lines, in terms of the difference between empirical and non-empirical sciences, differences in degrees of theoretical and methodological structure among individual sciences and scientific disciplines. Scientific knowledge is based upon some grounds, paradigmatic knowledge bases. Scientific knowledge is founded on a unified paradigmatic knowledge base. The latter includes fundamental theories, basic facts, generally accepted empirical regularities (laws), scientific research methods, scientific explanations, and established relations to other sciences. Scientific knowledge in general is defined well in classical terms by the definition that knowledge is a true and rationally-justified belief. In this connection, scientific truth is relative, depending on the given paradigmatic knowledge base and on the precision and accessibility of the measurements and observations involved. In the book's first part, I consider the production and testing of scientific hypotheses, scientific explanations, the structure of scientific theories, and the problem of scientific realism. I consider explanation an integral part of a wider nexus of heuristic and theoretical activities in science like giving reasons, understanding, clarification, and founding. Continuous production of scientific explanations is a necessary condition for the systematic reproduction of scientific knowledge. Accordingly, it does not suffice to describe them in terms of the logical structure of the arguments on which they are based, but rather, scientific explanations ought to be given complete with an epistemic and pragmatic context of the explanation. I consider especially deductive-nomological, inductive-statistical, dispositional, teleological-functional, and genetic explanations. I emphasize the ever-increasing importance of explanations by analogy and by model. Orthogonal to these distinctions lies the distinction between causal and epistemic explanation. Epistemic explanations relate scientific discoveries to epistemic progress, and causal explanations are these epistemic explanations that relate scientific discoveries to reality too. Causal explanations inform us that certain changes occur in the world as agents of other changes, or rather, that certain changes are real consequences of other changes. Thus, by having explanations of causes, we can intervene in the world to induce by design certain changes in some course of phenomena. I consider scientific theories as logically and epistemically ordered wholeness of scientific knowledge in a realm of research that enables us to construct useful models of reality. After the discussion of different 'statement' views on theories and different interpretations of the theory-experience relation, I introduce the 'model' theoretical (non-statement) views on theories, especially the structural theory of science. The model views on theories transcend the realism-antirealism dualism by introducing models between theory and reality. Therefore, the question of scientific realism is not merely whether and, if so, which element(s), of scientific theories represent reality, but rather which conditions suffice to infer from a theory that one of its models truly expresses reality. In other words, under which conditions do models of a theory represent models of reality? Generally, models are just as much images of reality in a theory as they are images of a theory in reality. The actual problem of scientific realism doesn't originate in the observable-unobservable distinction (even if this distinction is otherwise theoretically loaded), but rather in ontological quandaries, of the sort especially encountered in modern physical theories such as quantum mechanics, relativity theory, and cosmological theories, although they are known to arise in other sciences as well. Scientific theories may be real, even if they describe nothing about the world. However, such theories can be given this status only if their formal, mathematical postulates exhibit some formal homology between reality and mathematical forms. The very fact that it is possible to describe reality with the help of mathematics even in cases, where every (representational, conceptual) language and thought fall short, would indicate that even mathematics itself is, in some sense, a part of reality, although 'outside' of the empirical world. It follows we have to distinguish the non-empirical reality and the empirical world. Formal objectivity and non-arbitrariness of theories do not emerge from either our language or thoughts, or from the world of facts, causes, and effects, but rather from non-empirical reality, which frames the world as some kind of objective transcendental condition for the possibility of the existence of the world, thoughts, and language. In the second part of the book, I consider the incorporation of scientific work and scientific knowledge into modern society and its relations to values and ethics. Production of scientific knowledge goes on as a parallel distributed process of a special kind of social (shared) cognition. The central part of this cognition is the presentation of parts of the world by scientific (theoretical) models which are warranted by the shared and public practice of justification, and in reflective acting in the world as it would. I consider various formal concepts of collective propositional knowledge (distributed, mutual, common knowledge). Scientific knowledge is an achievement to be ascribed to a scientific community or, indirectly, to all those who apply or expand it, rather than to individuals per se. It presents the continuous and partially planned production of non-trivial common knowledge of the research communities in modern societies. However, besides common knowledge, shared dispositional knowledge (knowledge how of scientific teams) and shared knowledge by acquiring (knowledge of fundamental similarities and differences in scientific teams) are parts of the real body of scientific knowledge. Scientific common knowledge implies the projection into idealized universal knowledge of all real or potentially rational and well-informed individuals. The respect of scientific knowledge thus presents the norm of rationality. Science as the intensively shared and socially distributed kind of cognition enters into modern society as a special kind of work (general work), as a productive force (social invention drive), and as a form of capital (human, social, cultural, informational, etc.). However any of these categories can adequately represent the socio-economical role of science in modern societies, and we are still waiting for a more adequate general concept that would theoretically cover the 'scientification' of modern societies. My thesis is that the most important general effect of scientification is (and will be) the growth of social processes that structurally resemble parallel distributed shared social cognition of science, and not so much the rise of the use of scientific knowledge and scientifically based information in all kinds of production. Scientification of modern societies inevitably includes a much stronger need for moral and social responsibility of scientists for their work and its effects in their local society and the global world. This need comes in contradiction with the quite large accepted idea (and even the 'norm') on value-free research. I discuss to some degree different arguments for the absence of non-epistemic values and ethical reasoning from science and find them non-convincing. The sound idea of value-free research lies in the need for the absence of categorical value statements from scientific explanations and theoretical deductions. Values and ethical judgment have, and sometimes they even have to play an important role in another part of research and its application in science or outside it. Not the value-loudness of research is a danger for science but its blind, non-reflective use. It leads to the ideologization and politicization of science. A similar danger lies in the purely value-neutrality of research. It leads to the instrumentalization of science to the technical or bureaucratic means for 'any' aims. (shrink)

In this article, logical concepts are defined using the internal syntactic and semantic structure of language. For a first-order language, it has been shown that its logical constants are connectives and a certain type of quantifiers for which the universal and existential quantifiers form a functionally complete set of quantifiers. Neither equality nor cardinal quantifiers belong to the logical constants of a first-order language.

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)

This paper investigates a problem related to quantifiers which has some analogies to that of propositional completeness I give a definition of quantifier in many-valued logics generalizing the cases which already occur in first order many- valued logics. Though other definitions are possible, this particular one, which I call distribution quantifiers, generalizes the classical quantifiers in a very natural way, and occurs in finite numbers in every m-valued logic. We then call the problem of quantificationa2 completeness in (...) m-valued logic the problem of characterizing which are the quantifiers in a given language which can generate all other quantifiers in this language, using the connectives, as is the case, for example, of the universal and exis- tential quantifiers in classical logic, using negation. We are interested, in particular, in those many-valued quantifiers which closely mimic the behavior of existential an universal quantifiers in generating all other quantifiers using negation: these I call perfect quantifiers, as defined below. The main result of this paper is the characterization of all perfect quantifiers in 3-valued logics, which are complete if the logic is functionally complete. As a byproduct, we obtain the same result for the classical logic, which we include mainly for motivation. (shrink)

LetMbe a countably infinite ω-categorical structure. Consider Aut(M) as a complete metric space by definingd(g, h) = Ω{2−n:g(xn) ≠h(xn) org−1(xn) ≠h−1(xn)} where {xn:n∈ ω} is an enumeration ofMAn automorphism α ∈ Aut(M) is generic if its conjugacy class is comeagre. J. Truss has shown in [11] that if the set P of all finite partial isomorphisms contains a co-final subset P1closed under conjugacy and having the amalgamation property and the joint embedding property then there is a generic automorphism. In (...) the present paper we give a weaker condition of this kind which is equivalent to the existence of generic automorphisms. Really we give more: a characterization of the existence of generic expansions (defined in an appropriate way) of an ω-categorical structure. We also show that Truss' condition guarantees the existence of a countable structure consisting of automorphisms ofMwhich can be considered as an atomic model of some theory naturally associated toM. We do it in a general context of weak models for second-order quantifiers.The author thanks Ludomir Newelski for pointing out a mistake in the first version of Theorem 1.2 and for interesting discussions. Also, the author is grateful to the referee for very helpful remarks. (shrink)

This paper presents a simple decidable logic of functional dependence LFD, based on an extension of classical propositional logic with dependence atoms plus dependence quantifiers treated as modalities, within the setting of generalized assignment semantics for first order logic. The expressive strength, complete proof calculus and meta-properties of LFD are explored. Various language extensions are presented as well, up to undecidable modal-style logics for independence and dynamic logics of changing dependence models. Finally, more concrete settings for dependence are (...) discussed: continuous dependence in topological models, linear dependence in vector spaces, and temporal dependence in dynamical systems and games. (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.

We show that the elimination rule for the multiplicative conjunction $\wedge$ is admissible in many important multiplicative substructural logics. These include LL$_m$ and RMI$_m$ An exception is R$_m$. Let SLL$_m$ and SR$_m$ be, respectively, the systems which are obtained from LL$_m$ and R$_m$ by adding this rule as a new rule of inference. The set of theorems of SR$_m$ is a proper extension of that of R$_m$, but a proper subset of the set of theorems of RMI$_m$. Hence it still (...) has the variable-sharing property. SR$_m$ has also the interesting property that classical logic has a strong translation into it. We next introduce general algebraic structures, called strong multiplicative structures, and prove strong soundness and completeness of SLL$_m$ relative to them. We show that in the framework of these structures, the addition of the weakening axiom to SLL$_m$ corresponds to the condition that there will be exactly one designated element, while the addition of the contraction axiom corresponds to the condition that there will be exactly one nondesignated element. Various other systems in which multiplicative conjunction functions as a true conjunction are studied, together with their algebraic counterparts. (shrink)

We show that the elimination rule for the multiplicative (or intensional) conjunction $\wedge$ is admissible in many important multiplicative substructural logics. These include LL m (the multiplicative fragment of Linear Logic) and RMI m (the system obtained from LL m by adding the contraction axiom and its converse, the mingle axiom.) An exception is R m (the intensional fragment of the relevance logic R, which is LL m together with the contraction axiom). Let SLL m and SR m be, respectively, (...) the systems which are obtained from LL m and R m by adding this rule as a new rule of inference. The set of theorems of SR m is a proper extension of that of R m , but a proper subset of the set of theorems of RMI m . Hence it still has the variable-sharing property. SR m has also the interesting property that classical logic has a strong translation into it. We next introduce general algebraic structures, called strong multiplicative structures, and prove strong soundness and completeness of SLL m relative to them. We show that in the framework of these structures, the addition of the weakening axiom to SLL m corresponds to the condition that there will be exactly one designated element, while the addition of the contraction axiom corresponds to the condition that there will be exactly one nondesignated element (in the first case we get the system BCK m , in the second - the system SR m ). Various other systems in which multiplicative conjunction functions as a true conjunction are studied, together with their algebraic counterparts. (shrink)

The author's motivation for constructing the calculi of this paper\nis so that time and tense can be "discussed together in the same\nlanguage" (p. 282). Two types of enriched propositional caluli for\ntense logic are considered, both containing ordinary propositional\nvariables for which any proposition may be substituted. One type\nalso contains "clock-propositional" variables, a,b,c, etc., for\nwhich only clock-propositional variables may be substituted and that\ncorrespond to instants or moments in the semantics. The other type\nalso contains "history-propositional" variables, u,v,w, etc., for\nwhich only history-propositional variables may (...) be substituted and\nthat correspond to maximally linearly ordered subsets of instants.\nThis second type is for tense logics in which time is not linear\nand where at any moment there may exist alternative future courses\nof events. Quantifiers are included in both types but only for clock-\nand history-propositional variables. However, these quantifiers are\ngiven an essentially substitutional interpretation. In addition,\nbesides each clock-propositional variable being true at exactly one\ninstant, the semantics requires that for each instant there be at\nleast one clock-propositional variable true at that instant, which\nin effect amounts to having each instant of the model "denoted" in\nthe formalism by some clock-propositional variable. Thus it is not\nsurprising that quantification over clock-propositional variables\nturns out to have a "well behaved first-order semantics" (p. 284).\n{One axiom relating history- with clock-propositions seems to need\nrevision: CKTauTbuAUabUba. Since the clock-propositional variables\nmight be true at the same instant the consequent should include an\nalternative, namely, LCab.} Besides the standard truth-functional\nand tense operators G,H,F,P, the author includes L for "it is\nalways the case that". The model-theoretic earlier-than relation\nordering instants and the semantic truth-at (an instant) relation\nare formalized by Uab and Taα (for a wff α) which\nare defined respectively as LCbPa (whenever the clock-proposition\nb is the case the clock-proposition a was the case) and LCaα\n(α is the case whenever the clock-proposition a is the case).\nBecause the future tense operators are definable in terms of U\nand T and the latter are definable in terms of L and P, the\ndual of H, the author notes that his completeness proofs can be\nrestricted to L and H as primitive and with the future tense\noperators used for other interpretations, especially Prior, A.'s\nindeterminist "it will be the case that", where time is not linear.\nThe author concludes with some rather technical observations showing\nthat "in ordinary tense logics the instants are nonstandard elements\nof the models. This provides", according to the author, "a semantic\npressure to build the instants into tense logic, so that they will\nappear as standard elements in the models" (p. 283). (shrink)

We can think of functional completeness in systems of propositional logic as a form of expressive completeness: while every logical constant in such system expresses a truth-function of finitely many arguments, functional completeness garantees that every truth-function of finitely many arguments can be expressed with the constants in the system. From this point of view, a functionnaly complete system of propositionnal logic can thus be seen as one where no logical constant is missing. Can a similar question (...) be formulated for quantified first-order logics ? How to make sense of the question whether, e.g., ordinary first-order logic is "functionaly" complete or have no logical constant missing ? In this note, we build on a suggestive proposal made by Bonnay(2006) and shows that it is equivalent to the criterion that a first-order logic L be functionaly complete if and only if every class of structures closed under L-elementary equivalence is L-elementary. Ordinary first-order logic is not complete in this sense. We raise the question whether any logic can be. -/- . (shrink)

The research presented in this paper was motivated by our aim to study a problem due to J. Bourgain [3]. The problem in question concerns the uniform boundedness of the classical separation rank of the elements of a separable compact set of the first Baire class. In the sequel we shall refer to these sets (separable or non-separable) as Rosenthal compacta and we shall denote by ∝(f) the separation rank of a real-valued functionfinB1(X), withXa Polish space. Notice that in [3], (...) Bourgain has provided a positive answer to this problem in the case ofKsatisfyingwithXa compact metric space. The key ingredient in Bourgain's approach is that whenever a sequence of continuous functions pointwise converges to a functionf, then the possible discontinuities of the limit function reflect a local ℓ1-structure to the sequence (fn)n. More precisely the complexity of this ℓ1-structure increases as the complexity of the discontinuities offdoes. This fruitful idea was extensively studied by several authors (c.f. [5], [7], [8]) and for an exposition of the related results we refer to [1]. It is worth mentioning that A.S. Kechris and A. Louveau have invented the rank rND(f) which permits the link between thec0-structure of a sequence (fn)nof uniformly bounded continuous functions and the discontinuities of its pointwise limit. Rosenthal'sc0-theorem [11] and thec0-index theorem [2] are consequences of this interaction.Passing to the case where either (fn)nare not continuous orXis a non-compact Polish space, this nice interaction is completely lost. (shrink)

This paper is dedicated to the memory of Mike Dunn. His untimely death is a loss not only to logic, computer science, and philosophy, but to all of us who knew and loved him. The paper gives an argument for closure under γ in standard systems of relevance logic (first proved by Meyer and Dunn 1969). For definiteness, I chose the example of R. The proof also applies to E and to the quantified systems RQ and EQ. The argument uses (...) semantic tableaux (with one exceptional rule not satisfying the subformula property). It avoids the previous arguments’ use of cutting down inconsistent sets of formulas to consistent sets. Like all tableau arguments, it extends partial valuations to total valuations. (shrink)

A set of propositional connectives is said to be functionally complete if all propositional formulae can be expressed using only connectives from that set. In this paper we give sufficient and necessary conditions for a one-element set of propositional connectives to be functionally complete. These conditions provide a simple and elegant characterization of functionally complete one-element sets of propositional connectives.

The basis of the paper is a logic of analytical consequential implication, CI.0, which is known to be equivalent to the well-known modal system KT thanks to the definition A → B = df A ⥽ B ∧ Ξ (Α, Β), Ξ (Α, Β) being a symbol for what is called here Equimodality Property: (□A ≡ □B) ∧ (◊A ≡ ◊B). Extending CI.0 (=KT) with axioms and rules for the so-called circumstantial operator symbolized by *, one obtains a system CI.0*Eq (...) in whose language one can define an operator ↠ suitable to formalize context-dependent conditionals (so to counterfactual conditionals) via the definition (Def ↠) A ↠ B = df *A ⥽ B ∧ Ξ (Α, Β).. The central problem of the paper is to identify inside CI.0*Eq + Def ↠ a set of axioms yielding the fragment consisting of all and only theorems in which only truth-functional operators and the two operators → and ↠ occur. This system, here called CI.0, is introduced in §3. In view of the intended purpose, it is introduced a complete and tableau-decidable system KTw, which is an extension of KT with axioms for the so-called quasi-variables w(A), w(B)… Three translation functions among the three languages used in the paper are then introduced. The first, Tr, maps every wff of form *A into wffs of form w(A) ∧ A and translates theorems of CI.0*Eq into theorems of KTw. The second, t°, translates theorems of CI.0 into theorems of CI.0*Eq by applying Def↠. A third one, t, translates theorems of CI.0 into theorems of KTw. As a consequence, it follows that t°A = TrtA. In §4 it is proved that CI.0 is complete w.r.t. the class of CI.0-models of form where f is a selection function and R an access relation. It is then proved (i) that CI.0 models may be converted into KTw-models; (ii) that the truth-value of a proposition in a world of a CI.0-model is preserved in the same world of the derived KTw-model; (iii) that A is a CI.0-thesis iff its translation tA is KTw-thesis. It follows that, if A is not a CI.0-thesis, tA is not a KTw-thesis, so also that TrtA is not a CI.0*Eq-thesis. Since t°A = TrtA and t°A is a function built on Def↠, this proves that every t°-translation of a CI.0-wff that is a CI.0*Eq-theorem is a CI.0-theorem. Since the converse proposition is proved in §2, their conjunction establishes the desired result. In the final section, it is proved that CI.0 is tableau-decidable, that ↠ is not a trivial operator and that ↠ enjoys the positive and negative properties required in §1 for context-dependent conditionals. Some final remarks suggest that studying the relations between CI.0 and systems of classical conditional logic may be a promising line of investigation. (shrink)

Both syntactic and semantic solutions are given for the entailment problem of duality theory. The test algebra theorem provides both a syntactic solution to the entailment problem in terms of primitive positive formulae and a new derivation of the corresponding result in clone theory, viz. the syntactic description of $\operatorname{Inv(Pol}(R))$ for a given set R of finitary relations on a finite set. The semantic solution to the entailment problem follows from the syntactic one, or can be given in the form (...) of an algorithm. It shows, in the special case of a purely relational type, that duality-theoretic entailment is describable in terms of five constructs, namely trivial relations, intersection, repetition removal, product, and retractive projection. All except the last are concrete, in the sense that they are described by a quantifier-free formula. It is proved that if the finite algebra $\underline{M}$ generates a congruence-distributive variety and all subalgebras of $\underline{M}$ are subdirectly irreducible, then concrete constructs suffice to describe entailment. The concept of entailment appropriate to strong dualities is also introduced, and described in terms of coordinate projections, restriction of domains, and composition of partial functions. (shrink)

The prime purpose of this paper is, first, to restore to discourse-bound occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of occasion sentences and a mathematical (Boolean) foundation for such a logic, thus preparing the ground for more adequate semantic, logical and mathematical foundations of the study of natural language. Some of the insights elaborated (...) in this paper have appeared in the literature over the past thirty years, and a number of new developments have resulted from them. The present paper aims at providing an integrated conceptual basis for this new development in semantics. In Section 1 it is argued that the reduction by translation of occasion sentences to eternal sentences, as proposed by Russell and Quine, is semantically and thus logically inadequate. Natural language is a system of occasion sentences, eternal sentences being merely boundary cases. The logic has fewer tasks than is standardly assumed, as it excludes semantic calculi, which depend crucially on information supplied by cognition and context and thus belong to cognitive psychology rather than to logic. For sentences to express a proposition and thus be interpretable and informative, they must first be properly anchored in context. A proposition has a truth value when it is, moreover, properly keyed in the world, i.e. is about a situation in the world. Section 2 deals with the logical properties of natural language. It argues that presuppositional phenomena require trivalence and presents the trivalent logic ${\rm PPC}_{3}$ , with two kinds of falsity and two negations. It introduces the notion of Σ-space for a sentence A (or / A /, the set of situations in which A is true) as the basis of logical model theory, and the notion of / ${\bf P}^{A}$ / (the Σ-space of the presuppositions of A), functioning as a 'private' subuniverse for / A /. The trivalent Kleene calculus is reinterpreted as a logical account of vagueness, rather than of presupposition. ${\rm PPC}_{3}$ and the Kleene calculus are refinements of standard bivalent logic and can be combined into one logical system. In Section 3 the adequacy of ${\rm PPC}_{3}$ as a truth-functional model of presupposition is considered more closely and given a Boolean foundation. In a noncompositional extended Boolean algebra, three operators are defined: $1_{a}$ for the conjoined presuppositions of a, ã for the complement of a within $1_{a}$ , and â for the complement of $1_{a}$ within Boolean 1. The logical properties of this extended Boolean algebra are axiomatically defined and proved for all possible models. Proofs are provided of the consistency and the completeness of the system. Section 4 is a provisional exploration of the possibility of using the results obtained for a new discourse-dependent account of the logic of modalities in natural language. The overall result is a modified and refined logical and model-theoretic machinery, which takes into account both the discourse-dependency of natural language sentences and the necessity of selecting a key in the world before a truth value can be assigned. (shrink)

We present an algebraic-style of semantics, which we call a content semantics, for quantified relevant logics based on the weak system BBQ. We show soundness and completeness for all quantificational logics extending BBQ and also treat reduced modelling for all systems containing BB d Q. The key idea of content semantics is that true entailments AB are represented under interpretation I as content containments, i.e. I(A)I(B) (or, the content of A contains that of B). This is opposed to the truth-functional (...) way which represents true entailments as truth-preservations over all set-ups (or worlds), i.e. (VaK) (if I(A, a) = T then I(B, a)= T). (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)

In order to modelize the reasoning of intelligent agents represented by a poset T, H. Rasiowa introduced logic systems called “Approximation Logics”. In these systems the use of a set of constants constitutes a fundamental tool. We have introduced in [8] a logic system called equation image without this kind of constants but limited to the case that T is a finite poset. We have proved a completeness result for this system w.r.t. an algebraic semantics. We introduce in this paper (...) a Kripke-style semantics for a subsystem equation image of equation image for which there existes a deduction theorem. The set of “possible worldsr is enriched by a family of functions indexed by the elements of T and satisfying some conditions. We prove a completeness result for system equation image with respect to this Kripke semantics and define a finite Kripke structure that characterizes the propositional fragment of logic equation image. We introduce a reational semantics which has the advantage to allow an interpretation of the propositionnal logic equation image using only binary relations. We treat also the computational complexity of the satisfiability problem of the propositional fragment of logic equation image. (shrink)

Combining modalities has proven to have interesting applications and many approaches that combine time with other types of modalities have been developed. One of these approaches uses accessibility functions between flows of time to study the basic properties of the functions, such as being total or partial, injective, surjective, etc. The completeness of certain systems expressing many of these properties, with the exception of surjectivity, has been proven. In this paper we propose a language with nominals to denote the initial (...) and final sets of each function in a model in order to obtain a complete system for reasoning about surjective partial functions. (shrink)

We investigate the expressive power of second-order logic over finite structures, when two limitations are imposed. Let SAA ) be the set of second-order formulas such that the arity of the relation variables is bounded by k and the number of alternations of second-order quantification is bounded by n . We show that this imposes a proper hierarchy on second-order logic, i.e. for every k , n there are problems not definable in AA but definable in AA for some c (...) 1 , d 1 . The method to show this is to introduce the set AUTOSAT of formulas in F which satisfy themselves. We study the complexity of this set for various fragments of second-order loeic. For first-order logic FOL with unbounded alternation of quantifiers AUTOSAT is PSpace complete. For first-order logic FOL n with alternation of quantifiers bounded by n , AUTOSAT is definable in AA . AUTOSAT) is definable in AA for some c 1 , d 1. (shrink)

The present paper is a sequel to Robles et al. :349–374, 2020. https://doi.org/10.1007/s10849-019-09306-2). A class of implicative expansions of Kleene’s 3-valued logic functionally including Łukasiewicz’s logic Ł3 is defined. Several properties of this class and/or some of its subclasses are investigated. Properties contemplated include functional completeness for the 3-element set of truth-values, presence of natural conditionals, variable-sharing property and vsp-related properties.

Volume II of Classical Recursion Theory describes the universe from a local (bottom-up or synthetical) point of view, and covers the whole spectrum, from the recursive to the arithmetical sets. The first half of the book provides a detailed picture of the computable sets from the perspective of Theoretical Computer Science. Besides giving a detailed description of the theories of abstract Complexity Theory and of Inductive Inference, it contributes a uniform picture of the most basic complexity classes, ranging from small (...) time and space bounds to the elementary functions, with a particular attention to polynomial time and space computability. It also deals with primitive recursive functions and larger classes, which are of interest to the proof theorist. The second half of the book starts with the classical theory of recursively enumerable sets and degrees, which constitutes the core of Recursion or Computability Theory. Unlike other texts, usually confined to the Turing degrees, the book covers a variety of other strong reducibilities, studying both their individual structures and their mutual relationships. The last chapters extend the theory to limit sets and arithmetical sets. The volume ends with the first textbook treatment of the enumeration degrees, which admit a number of applications from algebra to the Lambda Calculus. The book is a valuable source of information for anyone interested in Complexity and Computability Theory. The student will appreciate the detailed but informal account of a wide variety of basic topics, while the specialist will find a wealth of material sketched in exercises and asides. A massive bibliography of more than a thousand titles completes the treatment on the historical side. (shrink)

A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let [Formula: see text] be the lattice of weak degrees of mass problems associated with nonempty [Formula: see text] subsets of the Cantor (...) space. The lattice [Formula: see text] has been studied in previous publications. The purpose of this paper is to show that [Formula: see text] partakes of hyperarithmeticity. We exhibit a family of specific, natural degrees in [Formula: see text] which are indexed by the ordinal numbers less than [Formula: see text] and which correspond to the hyperarithmetical hierarchy. Namely, for each [Formula: see text], let hα be the weak degree of 0, the αth Turing jump of 0. If p is the weak degree of any mass problem P, let p* be the weak degree of the mass problem P* = {Y | ∃X ⊆ BLR )} where BLR is the set of functions which are boundedly limit recursive in X. Let 1 be the top degree in [Formula: see text]. We prove that all of the weak degrees [Formula: see text], [Formula: see text], are distinct and belong to [Formula: see text]. In addition, we prove that certain index sets associated with [Formula: see text] are [Formula: see text] complete. (shrink)

We give a completeness theorem for a logic with probability quantifiers which is equivalent to the logics described in a recent survey paper of Keisler [K]. This result improves on the completeness theorems in [K] in that it works for languages with function symbols and produces a model whose universe is an analytic subset of the real line, and whose relations and functions are Borel relative to this universe.

This mathematics textbook covers the fundamental ideas used in writing proofs. Proof techniques covered include direct proofs, proofs by contrapositive, proofs by contradiction, proofs in set theory, proofs of existentially or universally quantified predicates, proofs by cases, and mathematical induction. Inductive and deductive reasoning are explored. A straightforward approach is taken throughout. Plenty of examples are included and lots of exercises are provided after each brief exposition on the topics at hand. The text begins with a study of symbolic logic, (...) deductive reasoning, and quantifiers. Inductive reasoning and making conjectures are examined next, and once there are some statements to prove, techniques for proving conditional statements, disjunctions, biconditional statements, and quantified predicates are investigated. Terminology and proof techniques in set theory follow with discussions of the pick-a-point method and the algebra of sets. Cartesian products, equivalence relations, orders, and functions are all incorporated. Particular attention is given to injectivity, surjectivity, and cardinality. The text includes an introduction to topology and abstract algebra, with a comparison of topological properties to algebraic properties. This book can be used by itself for an introduction to proofs course or as a supplemental text for students in proof-based mathematics classes. The contents have been rigorously reviewed and tested by instructors and students in classroom settings. (shrink)

We give an axiomatic characterization for complete elementary extensions, that is, elementary extensions of the first-order structure consisting of all finitary relations and functions on the underlying set. Such axiom systems have been studied using various types of primitive notions . Our system uses the notion of partial functions as primitive. Properties of nonstandard extensions are derived from five axioms in a rather algebraic way, without the use of metamathematical notions such as formulas or satisfaction. For example, when applied (...) to the real number system, it provides a complete framework for developing nonstandard analysis based on hyperreals without having to construct them and without any use of logic. This has possible pedagogical and expository applications as presented in, e.g., [5], [6]. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

The paper describes and proves completeness theorems for a series of proof systems formalizing common sense reasoning about uncertain knowledge in the case where this consists of sets of linear constraints on a probability function.

Contrary to received opinion, the Aristotelian Square of Opposition (square) is logically sound, differing from standard modern predicate logic (SMPL) only in that it restricts the universe U of cognitively constructible situations by banning null predicates, making it less unnatural than SMPL. U-restriction strengthens the logic without making it unsound. It also invites a cognitive approach to logic. Humans are endowed with a cognitive predicate logic (CPL), which checks the process of cognitive modelling (world construal) for consistency. The square is (...) considered a first approximation to CPL, with a cognitive set-theoretic semantics. Not being cognitively real, the null set Ø is eliminated from the semantics of CPL. Still rudimentary in Aristotle’s On Interpretation (Int), the square was implicitly completed in his Prior Analytics (PrAn), thereby introducing U-restriction. Abelard’s reconstruction of the logic of Int is logically and historically correct; the loca (Leaking O-Corner Analysis) interpretation of the square, defended by some modern logicians, is logically faulty and historically untenable. Generally, U-restriction, not redefining the universal quantifier, as in Abelard and loca, is the correct path to a reconstruction of CPL. Valuation Space modelling is used to compute the effects of U-restriction. (shrink)

Here is an account of logical consequence inspired by Bolzano and Tarski. Logical validity is a property of arguments. An argument is a pair of a set of interpreted sentences (the premises) and an interpreted sentence (the conclusion). Whether an argument is logically valid depends only on its logical form. The logical form of an argument is fixed by the syntax of its constituent sentences, the meanings of their logical constituents and the syntactic differences between (...) their non-logical constituents, treated as variables. A constituent of a sentence is logical just if it is formal in meaning, in the sense roughly that its application is invariant under permutations of individuals.1 Thus ‘=’ is a logical constant because no permutation maps two individuals to one or one to two; ‘∈’ is not a logical constant because some permutations interchange the null set and its singleton. Truth functions, the usual quantifiers and bound variables also count as logical constants. An argument is logically valid if and only if the conclusion is true under every assignment of semantic values to variables (including all non-logical expressions) under which all its premises are true. A sentence is logically true if and only if the argument with no premises of which it is the conclusion is logically valid, that is, if and only if the sentence is true under every assignment of semantic values to variables. An interpretation assigns values to all variables. (shrink)

The quantified relevant logic RQ is given a new semantics in which a formula for all xA is true when there is some true proposition that implies all x-instantiations of A. Formulae are modelled as functions from variable-assignments to propositions, where a proposition is a set of worlds in a relevant model structure. A completeness proof is given for a basic quantificational system QR from which RQ is obtained by adding the axiom EC of 'extensional confinement': for all x(A V (...) B) -> (A V for all xB). with x not free in A. Validity of EC requires an additional model condition involving the boolean difference of propositions. A QR-model falsifying EC is constructed by forming the disjoint union of two natural arithmetical structures in which negation is interpreted by the minus operation. (shrink)

The paper describes and proves completeness theorems for a series of proof systems formalizing common sense reasoning about uncertain knowledge in the case where this consists of sets of linear constraints on a probability function.

§1. Introduction. Natural sets that can be enumerated by a computable function always seem to be either actually computable or of the same complexity as the Halting Problem, the complete r.e. set K. The obvious question, first posed in Post [1944] and since then called Post's Problem is then just whether there are r.e. sets which are neither computable nor complete, i.e., neither recursive nor of the same Turing degree as K?Let be the r.e. degrees, i.e., the r.e. (...) sets modulo the equivalence relation of equicomputable with the partial order induced by Turing computability. This structure is a partial order with least element 0, the degree of the computable sets, and greatest element 1 or 0′, the degree of K. Post's problem then asks if there are any other elements of.The solution of Post's problem by Friedberg [1957] and Muchnik [1956] was followed by various algebraic or order theoretic results that were interpreted as saying that the structure was in some way well behaved:Theorem 1.1. Every countable partial ordering or even uppersemilattice can be embedded into.Theorem 1.2. For every nonrecursive r.e. degreeathere are r.e. degreesb, c < asuch thatb ∨ c = a.Theorem 1.3. For every pair of nonrecursive r.e. degreesa < bthere is an r.e. degreecsuch thata < c < b. (shrink)

The aim of this article is to give an introduction to functional interpretations of set theory given by the authorin Burr (2000a). The first part starts with some general remarks on Gödel's functional interpretation with a focus on aspects related to problems that arise in the context of set theory. The second part gives an insight in the techniques needed to perform a functional interpretation of systems of set theory. However, the first part of this article is not intended to (...) be a complete survey of functional interpretations and here we recommend, for example, Avigad and Feferman (1998),Troelstra (1990) and Troelstra (1973). (shrink)

This textbook is designed for an Introduction to Proofs course organized around the themes of number and space. Concepts are illustrated using both geometric and number examples, while frequent analogies and applications help build intuition and context in the humanities, arts, and sciences. Sophisticated mathematical ideas are introduced early and then revisited several times in a spiral structure, allowing students to progressively develop rigorous thinking. Throughout, the presentation is enlivened with whimsical illustrations, apt quotations, and glimpses of mathematical history and (...) culture. Early chapters integrate an introduction to sets, logic, and beginning proof techniques with a first exposure to more advanced mathematical structures. The middle chapters focus on equivalence relations, functions, and induction. Carefully chosen examples elucidate familiar topics, such as natural and rational numbers and angle measurements, as well as new mathematics, such as modular arithmetic and beginning graph theory. The book concludes with a thorough exploration of the cardinalities of finite and infinite sets and, in two optional chapters, brings all the topics together by constructing the real numbers and other complete metric spaces. Designed to foster the mental flexibility and rigorous thinking needed for advanced mathematics, Introduction to Mathematics suits either a lecture-based or flipped classroom. A year of mathematics, statistics, or computer science at the university level is assumed, but the main prerequisite is the willingness to engage in a new challenge. (shrink)

The paper is dedicated to the 80th birthday of the outstanding Russian logician A.V. Kuznetsov. It is addressing a history of the ideas and research conducted by him in non-classical and intermediate logics.

Fibring has been shown to be useful for combining logics endowed withtruth-functional semantics. However, the techniques used so far are unableto cope with fibring of logics endowed with non-truth-functional semanticsas, for example, paraconsistent logics. The first main contribution of thepaper is the development of a suitable abstract notion of logic, that mayalso encompass systems with non-truth-functional connectives, and wherefibring can still be dealt with. Furthermore, it is shown that thisextended notion of fibring preserves completeness under certain reasonableconditions. This completeness transfer (...) result, the second main contributionof the paper, generalizes the one established in Zanardo et al. (2001) butis obtained using new techniques that explore the properties of a suitablemeta-logic (conditional equational logic) where the (possibly)non-truth-functional valuations are specified. The modal paraconsistentlogic of da Costa and Carnielli (1988) is studied in the context of this novel notionof fibring and its completeness is so established. (shrink)

Kooi and Tamminga's correspondence analysis is a technique for designing proof systems, mostly, natural deduction and sequent systems. In this paper it is used to generate sequent calculi with invertible rules, whose only branching rule is the rule of cut. The calculi pertain to classical propositional logic and any of its fragments that may be obtained from adding a set of rules characterizing a two-argument Boolean function to the negation fragment of classical propositional logic. The properties of soundness and completeness (...) of the calculi are demonstrated. The proof of completeness is conducted by Kalmár's method. Most of the presented sequent-calculus rules have been obtained automatically, by a rule-generating algorithm implemented in Python. Correctness of the algorithm is demonstrated. This automated approach allowed us to analyse thousands of possible rules' schemes, hundreds of rules corresponding to Boolean functions, and to nd dozens of those invertible. Interestingly, the analysis revealed that the presented proof-theoretic framework provides a syntactic characteristics of such an important semantic property as functional completeness. (shrink)

Gelfond and Lifschitz proposed the notion of a stable model of a logic program. We establish that the set of all stable models in a Herbrand universe of a recursive logic program is, up to recursive renaming, the set of all infinite paths of a recursive, countably branching tree, and conversely. As a consequence, the problem, given a recursive logic program, of determining whether it has at least one stable model, is Σ11-complete. Due to the equivalences established in the (...) authors' previous nonmonotone rule systems papers ), this applies equally to truth maintenance systems and default logics. (shrink)

We present a set-theoretic model of the mental representation of classically quantified sentences (All P are Q, Some P are Q, Some P are not Q, and No P are Q). We take inclusion, exclusion, and their negations to be primitive concepts. We show that although these sentences are known to have a diagrammatic expres- sion (in the form of the Gergonne circles) that constitutes a semantic representation, these concepts can also be expressed syntactically in the form of algebraic formulas. (...) We hypothesized that the quantified sen- tences have an abstract underlying representation common to the formulas and their associated sets of dia- grams (models). We derived 9 predictions (3 semantic, 2 pragmatic, and 4 mixed) regarding people’s as- sessment of how well each of the 5 diagrams expresses the meaning of each of the quantified sentences. We report the results from 3 experiments using Gergonne’s (1817) circles or an adaptation of Leibniz (1903/ 1988) lines as external representations and show them to support the predictions. (shrink)

In the paper we explore the idea of describing Pawlak’s rough sets using three-valued logic, whereby the value t corresponds to the positive region of a set, the value f — to the negative region, and the undefined value u — to the border of the set. Due to the properties of the above regions in rough set theory, the semantics of the logic is described using a non-deterministic matrix (Nmatrix). With the strong semantics, where only the value t is (...) treated as designated, the above logic is a “common denominator” for Kleene and Łukasiewicz 3-valued logics, which represent its two different “determinizations”. In turn, the weak semantics—where both t and u are treated as designated—represents such a “common denominator” for two major 3-valued paraconsistent logics. We give sound and complete, cut-free sequent calculi for both versions of the logic generated by the rough set Nmatrix. Then we derive from these calculi sequent calculi with the same properties for the various “determinizations” of those two versions of the logic (including Łukasiewicz 3-valued logic). Finally, we show how to embed the four above-mentioned determinizations in extensions of the basic rough set logics obtained by adding to those logics a special two-valued “definedness” or “crispness” operator. (shrink)

In many-valued logic the decision of functional completeness is a basic and important problem, and the thorough solution to this problem depends on determining all maximal closed sets in the set of many-valued logic functions. It includes three famous problems, i.e., to determine all maximal closed sets in the set of the total, of the partial and of the unary many-valued logic functions, respectively. The first two problems have been completely solved , and the solution to the third problem boils (...) down to determining all maximal subgroups in the k-degree symmetric group Sk, which is an open problem in the finite group theory. In this paper, all maximal closed sets in the set of unary p-valued logic functions are determined, where p is a prime. (shrink)