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)

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 (...) class='Hi'>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)

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)

Working in a fixed Grothendieck topos Sh(C, J C ) we generalize \({\mathcal{L}_{{\infty},\omega}}\) to allow our languages and formulas to make explicit reference to Sh(C, J C ). We likewise generalize the notion of model. We then show how to encode these generalized structures by models of a related sentence of \({\mathcal{L}_{{\infty},\omega}}\) in the category of sets and functions. Using this encoding we prove analogs of several results concerning \({\mathcal{L}_{{\infty},\omega}}\) , such as the downward Löwenheim–Skolem theorem, the completeness theorem (...) and Barwise compactness. (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)

This is an exposition of Lambek's strengthening and generalization of the deduction theorem in categories related to intuitionistic propositional logic. Essential notions of category theory are introduced so as to yield a simple reformulation of Lambek's Functional Completeness Theorem, from which its main consequences can be readily drawn. The connections of the theorem with combinatory logic, and with modal and substructural logics, are briefly considered at the end.

The relevance of the research is defined by the need to create a set of linguistic means, which would contribute to effective communication with the general public, and the need to study different functional-semantic categories, including appeals, for the competent formation of public opinion in the political context. The research aims to comprehend the functioning of linguistic means used as appeals in the example of political propaganda texts in the English media field. The methodology is based on the (...) theoretical study of the works of modern linguists, linguistic, structural, and communicative analysis of appeal linguistic units and contexts. The research considered the linguistic means that form the functional-semantic category of the appeal, examples of political contexts from the British and American media were presented, the functions of appeal were presented using specific examples, an idea of the communicative side of political propaganda texts and audience participation in this process was formed, emphasis was placed on different types of propaganda (white, black, gray), the following groups of appeals were characterized: imperatives (volitional and involuntary). The materials presented in this research can be used to form an idea of the functional-semantic category of appeal, the choice of linguistic means for the purpose of information promotion in the media or social networks, the study of communicative strategies in linguistics and their successful implementation, consideration of political propaganda texts, increasing efficiency when influencing the audience, further implementation of means of appeal in machine learning. (shrink)

The aim of this paper is to offer two novel solutions to two perennial problems of categorial ontology, namely, the problem of the categorial structure: how are the categories related to one another? And the problem of categorial completeness: how is the completeness of a proposed list of categories justified? First, I argue that a system of categories should have a structure such that there is a most basic category that is a bearer of all (...) other categories and that has what I shall call “combinatorial conditions”. To do so, I compare this approach to categorial structure with the approaches given by substantialist, mereological, factualist, and geo ontologies. Second, I argue that the problem of categorial completeness is only a problem for certain ontologies. In this connection, I explore views on categorial completeness proposed by substantialists and geo-ontologists. Lastly, I conclude that factualism does a better job of accounting for categorial structure and categorial completeness than other categorial ontologies. (shrink)

The aim of this investigation is to discuss the merits of the three radically divergent views as to the nature and function of the categories held by Kant, Ward and Alexander.

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.

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)

This study aims to find out how high the level and trends of student misconceptions experienced by high school students in Indonesia. The subject of research that is a class XI student of Natural Science (IPA) SMA Negeri 1 Anjatan with the subject matter limit function. Forms of research used in this study is a qualitative research, with a strategy that is descriptive qualitative research. The data analysis focused on the results of the students' answers on the test essay subject (...) matter limit function with the number of students by 16 (sixteen). Data collection techniques used are shaped test methods essay, interview method to students who have misconceptions, and methods of documentation of the test answers. Examination of the validity of the data using a triangulation technique that compares the data written test results with data from interviews. The results of this study can be described as follows; (1) The level of misconceptions experienced by students belonging to the category of low, amounting to 12.18%. However, students who do not understand the concept quite high at 40.38%, and the others are students who understand the concept that is equal to 47.44%. (2) The misconception most experienced students lie in subconcepts explain the meaning of limit function at one point through the calculation of values around that point, that is equal to 20.51%. The tendency misconceptions experienced by students is located on errors and operating concepts that misconceptions students that there should be no limit in the completion of the writing of the emblem and misconceptions about the limit students to conclude if the limit value of 0 is no limit to the value of the function. (shrink)

In focusing on the systematic deduction of the categories from a principle, Schulting takes up anew the controversial project of the eminent German Kant scholar Klaus Reich, whose monograph “The Completeness of Kant's Table of Judgments” made the case that the logical functions of judgement can all be derived from the objective unity of apperception and can be shown to link up with one another systematically. -/- Common opinion among Kantians today has it that Kant did not mean (...) to derive the functions of judgement, and accordingly the categories, from the principle of apperception. Schulting challenges this standard view and aims to resuscitate the main motivation behind Reich’s project. He argues, in agreement with Reich’s main thesis about the derivability of the functions of judgement, that Kant indeed does mean to derive, in full a priori fashion, the categories from the principle of apperception. -/- Schulting also shows that, given the general assumptions of the Critical philosophy, Kant's derivation is successful and that absent an account of the derivation of the categories from apperception, the B-Deduction cannot really be understood. -/- New edition. First published 2012 as „Kant’s Deduction and Apperception. Explaining the Categories" (Palgrave Macmillan). (shrink)

ObjectiveBulimia nervosa and binge eating disorder are eating disorders characterized by recurrent binge eating episodes. Overlap exists between ED diagnostic groups, with BE episodes presenting one clinical feature that occurs transdiagnostically. Neuroimaging of the responses of those with BN and BED to disorder-specific stimuli, such as food, is not extensively investigated. Furthermore, to our knowledge, there have been no previous published studies examining the neural response of individuals currently experiencing binge eating, to low energy foods. Our objective was to examine (...) the neural responses to both low energy and high energy food images in three emotive categories in BN and BED participants.MethodsNineteen females with BN or BED, comprising the binge eating group, and 19 age-matched healthy control ’s completed thorough clinical assessment prior to functional MRI. Neural response to low energy and high energy foods and non-food images was compared between groups using whole-brain exploratory analyses, from which six regions of interest were then selected: frontal, occipital, temporal, and parietal lobes; insula and cingulate.ResultsIn response to low energy food images, the BEG demonstrated differential neural responses to all three low energy foods categories compared to HCs. Correlational analyses found a significant association between frequency of binge episodes and diminished temporal lobe and greater occipital lobe response. In response to high energy food images, compared to HC’s, the BEG demonstrated significantly decreased neural activity in response to all high energy food images. The HC’s had significantly greater neural activity in the limbic system, occipital lobe, temporal lobe, frontal lobe, and limbic system in response to high energy food images.ConclusionResults in the low energy food condition indicate that binge frequency may be related to increased aberrant neural responding. Furthermore, differences were found between groups in all ROI’s except the insula. The neural response seen in the BEG to disgust food images may indicate disengagement with this particular stimuli. In the high energy food condition, results demonstrate that neural activity in BN and BED patients may decrease in response to high energy foods, suggesting disengagement with foods that may be more consistent with those consumed during a binge eating episode. (shrink)

Background: Although performance on the category fluency test is influenced by many cognitive functions, item-level scoring methods of CFT performance might be a promising way to capture aspects of semantic memory that are less influenced by intervenient abilities. One such approach is based on the calculation of correlation coefficients that quantify the association between item-level features and the serial order with which words are recalled.Methods: We explored the neural underpinnings of 10 of these correlational indices in a sample of 40 (...) healthy adults who completed a classic 1-min CFT and an MRI protocol inclusive of T1-weighted and resting-state fMRI sequences for the evaluation of the default-mode network. Two sets of linear models were defined to test the association between neural maps and each correlational index: a first set in which major demographic and clinical descriptors were controlled for and a second set in which, additionally, all other 9 correlational indices were regressed out.Results: In the analysis of the DMN, ‘SRO-frequency’, ‘SRO-dominance’ and ‘SRO-body-object interaction’ correlational indices were all negatively associated with the anterior portion of the right temporoparietal junction. The ‘SRO-frequency’ correlational index was also negatively associated with the right dorsal anterior cingulate and the ‘SRO-dominance’ correlational index with the right lateral prefrontal cortex. From the second set of models, the ‘SRO-typicality’ correlational index was positively associated with the left entorhinal cortex. No association was found in relation to grey matter maps.Conclusion: The ability to retrieve more difficult words during CFT performance as measured by the correlational indices between SRO and item-level descriptors is associated with DMN expression in regions deputed to attentional reorienting and processing of salience of infrequent stimuli and dominance status. Of all item-level features, typicality appears to be that most closely linked with entorhinal functioning and may thus play a relevant role in assessing its value in testing procedures for early detection of subtle cognitive difficulties in people with suspected Alzheimer’s degeneration. Although exploratory, these findings warrant further investigations in larger cohorts. (shrink)

Diagnostics of a mental disorder completely bases on an estimation of patient’s behaviour, verbal behaviour being the most important. The behaviour, in turn, is ruled by a situation expressed as a system of signs. Perception of a situation could be seen as a function, which depends on the context resulting from the previous situations, structuring personal world. So the world is not given — it is being formed while the person is in action. We argue that distinctive features of behaviour, (...) including its abnormal variants, can be explained not in categories of characters and diseases but in terms of situations taking place in individual worlds. The situation in which a person perceives himself is not simply a site in a three-dimensional space at a certain moment, but a part of the world and an episode of his life. Like a text composed of words, individual world is composed of situations. Each of them needs certain context to cope with ambiguity. This context is induced by the world as a whole. And the world, in turn, is presented as a chain of situations. If the context cannot help to interpret a situation adequately, uncertainty can be eliminated by actions clarifying a situation, which is changed in a predictable way. Thus, purposeful activity, skills to make predictions and corrections of one’s own actions are crucial. Weakness of any of them inevitably leads to the distortion of the presentation of the world, to wrongevaluation of situations and, as a result, to inadequate actions that finally reduce the activity as being ineffective. Thus, the lack of activity becomes the key factor in the development of disorder, being simultaneously its cause and effect. In periods of insufficient activity conditions for violated (and violating) sign processing arise. Possible variants of sign malfunction are: oligosemia (reduction of the number of perceivable signs), hyposemia (decrease of significance of signs), hypersemia (increase of significance of some signs at the expense of others), ambisemia (uncertainty of sign, when situation remains unclear), cryptosemia (recognition of signs not obvious for other observers), and parasemia (perverted interpretation of signs influenced by a false context). (shrink)

Tarski 1968 makes a move in the course of providing an account of ?definitionally equivalent? classes of algebras with a businesslike lack of fanfare and commentary, the significance of which may accordingly be lost on the casual reader. In ?1 we present this move as a response to a certain difficulty in the received account of what it is to define a function symbol (or ?operation symbol?). This difficulty, which presents itself as a minor technicality needing to be got around (...) especially for the case of symbols for zero-place functions (for ?distinguished elements?), has repercussions?not widely recognised?for the account of functional completeness in sentential logic. A similarly stark comment in Church 1956 reveals an appreciation of this difficulty, though not every subsequent author on the topic has taken the point. We fill out this side of the picture in ?2. The discussion of functional completeness in ?2 is supplemented by some remarks on what is involved in defining a connective, which have been included in an Appendix. The emphasis throughout is on conceptual clarification rather than on proving theorems, and the main body of the paper may be regarded as an elaboration on the remarks just mentioned by Tarski and Church. The Appendix (?3) is intended to be similarly clarificatory, though this time with some corrective intent, of remarks made in and about Makinson 1973. (shrink)

A hybrid logic is obtained by adding to an ordinary modal logic further expressive power in the form of a second sort of propositional symbols called nominals and by adding so-called satisfaction operators. In this paper we consider hybridized versions of S5 (“the logic of everywhere”) and the modal logic of inequality (“the logic of elsewhere”). We give natural deduction systems for the logics and we prove functional completeness results.

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.

Equational hybrid propositional type theory ) is a combination of propositional type theory, equational logic and hybrid modal logic. The structures used to interpret the language contain a hierarchy of propositional types, an algebra and a Kripke frame. The main result in this paper is the proof of completeness of a calculus specifically defined for this logic. The completeness proof is based on the three proofs Henkin published last century: Completeness in type theory, The completeness of (...) the first-order functional calculus and Completeness in propositional type theory. More precisely, from and we take the idea of building the model described by the maximal consistent set; in our case the maximal consistent set has to be named, \-saturated and extensionally algebraic-saturated due to the hybrid and equational nature of \. From, we use the result that any element in the hierarchy has a name. The challenge was to deal with all the heterogeneous components in an integrated system. (shrink)

Equational hybrid propositional type theory ) is a combination of propositional type theory, equational logic and hybrid modal logic. The structures used to interpret the language contain a hierarchy of propositional types, an algebra and a Kripke frame. The main result in this paper is the proof of completeness of a calculus specifically defined for this logic. The completeness proof is based on the three proofs Henkin published last century: Completeness in type theory, The completeness of (...) the first-order functional calculus and Completeness in propositional type theory. More precisely, from and we take the idea of building the model described by the maximal consistent set; in our case the maximal consistent set has to be named, \-saturated and extensionally algebraic-saturated due to the hybrid and equational nature of \. From, we use the result that any element in the hierarchy has a name. The challenge was to deal with all the heterogeneous components in an integrated system. (shrink)

Diagnostics of a mental disorder completely bases on an estimation of patient’s behaviour, verbal behaviour being the most important. The behaviour, in turn, is ruled by a situation expressed as a system of signs. Perception of a situation could be seen as a function, which depends on the context resulting from the previous situations, structuring personal world. So the world is not given — it is being formed while the person is in action. We argue that distinctive features of behaviour, (...) including its abnormal variants, can be explained not in categories of characters and diseases but in terms of situations taking place in individual worlds. The situation in which a person perceives himself is not simply a site in a three-dimensional space at a certain moment, but a part of the world and an episode of his life. Like a text composed of words, individual world is composed of situations. Each of them needs certain context to cope with ambiguity. This context is induced by the world as a whole. And the world, in turn, is presented as a chain of situations. If the context cannot help to interpret a situation adequately, uncertainty can be eliminated by actions clarifying a situation, which is changed in a predictable way. Thus, purposeful activity, skills to make predictions and corrections of one’s own actions are crucial. Weakness of any of them inevitably leads to the distortion of the presentation of the world, to wrongevaluation of situations and, as a result, to inadequate actions that finally reduce the activity as being ineffective. Thus, the lack of activity becomes the key factor in the development of disorder, being simultaneously its cause and effect. In periods of insufficient activity conditions for violated (and violating) sign processing arise. Possible variants of sign malfunction are: oligosemia (reduction of the number of perceivable signs), hyposemia (decrease of significance of signs), hypersemia (increase of significance of some signs at the expense of others), ambisemia (uncertainty of sign, when situation remains unclear), cryptosemia (recognition of signs not obvious for other observers), and parasemia (perverted interpretation of signs influenced by a false context). (shrink)

THE PURE INTUITIONS OF SPACE AND TIME and the pure concepts of understanding are the two basic elements in Kant's critical philosophy. Whereas his account of pure intuitions is relatively straightforward, his theory of categories is quite complicated. When he presents space and time as two forms of intuition, he never sees the need to prove that there are no other forms of intuition than these two. But when he presents his table of categories, he tries to prove (...) its completeness in one of the most obscure chapters of the first Critique, known as the "Metaphysical Deduction." Kant complicates his picture of categories by claiming two different functions for them: the logical and the real. The obscurity surrounding the relation of these two functions is the chief obstacle for understanding the "Transcendental Deduction," perhaps the most controversial chapter of the Critique. He further aggravates the problem in the "Schematism," perhaps the most oracular chapter of the Critique, by producing a set of categories quite different from the original set given in the "Metaphysical Deduction.". (shrink)

The logics of the modal operators and of the quantifiers show striking analogies. The analogies are so extensive that, when a special class of entities (possible worlds) is postulated, natural and non-arbitrary translation procedures can be defined from the language with the modal operators into a purely quantificational one, under which the necessity and possibility operators translate into universal and existential quantifiers. In view of this I would be willing to classify the modal operators as ‘disguised’ quantifiers, and I think (...) that wholehearted acceptance of modal language should be considered to carry ontological commitment to something like possible worldsConsidered as two languages for describing the same subject matter, modal and purely quantificational languages show interesting differences. The operator variables of the purely quantificational languages give them more power than the modal languages, but at least some of the functions performed by the apparatus of operator variables are also performed, in a more primitive and less versatile way, by actuality operators in modal languages.A final note. Quine has written much on the inter-relations of quantifiers, identity, and the concept of existence. These, he holds, form a tightly knit conceptual system which has been evolved to a high point of perfection, but which might conceivably change yet further.29 He has also dropped hints about the possibility of a simpler, primitive or defective version of the system, in which the quantifiers are not backed up in their accustomed way by the concept of identity. He has dubbed the resulting concept a ‘pre-individuative’ concept of existence, or a concept of ‘entity without identity.’ What would a pre-individuative concept of existence be like? Quine has sometimes suggested that one might be embodied in the use of mass nouns, but the identity concept is used in connection with stuff as well as with things: “is that the same coffee that was in the cup last night?” I would submit that modality provides a better case. In view of the comparative weakness of modal languages, compared to the explicitly quantificational ones Quine takes as canonical, there is surely a sense in which the concept of existence embodied in that disguised existential quantifier, the possibility operator, is a defective one. And as we have seen, one of the differences between modal operators and explicit quantifiers is that modal operators cannot be joined with the identity predicate in the way quantifiers with operator variables can. Surely, then, there is a sense in which ordinary speech, as opposed to the metaphysical theorizing of a Leibniz or a David Lewis, conceives of possible worlds as entities without identity. (shrink)

Well-known results due to David Makinson show that there are exactly two Post complete normal modal logics, that in both of them, the modal operator is truth-functional, and that every consistent normal modal logic can be extended to at least one of them. Lloyd Humberstone has recently shown that a natural analog of this result in congruential modal logics fails, by showing that not every congruential modal logic can be extended to one in which the modal operator is truth- (...) class='Hi'>functional. As Humberstone notes, the issue of Post completeness in congruential modal logics is not well understood. The present article shows that in contrast to normal modal logics, the extent of the property of Post completeness among congruential modal logics depends on the background set of logics. Some basic results on the corresponding properties of Post completeness are established, in particular that although a congruential modal logic is Post complete among all modal logics if and only if its modality is truth-functional, there are continuum many modal logics Post complete among congruential modal logics. (shrink)

Categorical-theoretic semantics for the relevance logic is proposed which is based on the construction of the topos of functors from a relevant algebra (considered as a preorder category endowed with the special endofunctors) in the category of sets Set. The completeness of the relevant system R of entailment is proved in respect to the semantic considered.

A categorical, higher dimensional algebra and generalized topos framework for Łukasiewicz–Moisil Algebraic–Logic models of non-linear dynamics in complex functional genomes and cell interactomes is proposed. Łukasiewicz–Moisil Algebraic–Logic models of neural, genetic and neoplastic cell networks, as well as signaling pathways in cells are formulated in terms of non-linear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable ‘next-state functions’ is extended to a (...) Łukasiewicz–Moisil Topos with an n-valued Łukasiewicz–Moisil Algebraic Logic subobject classifier description that represents non-random and non-linear network activities as well as their transformations in developmental processes and carcinogenesis. The unification of the theories of organismic sets, molecular sets and Robert Rosen’s (M,R)-systems is also considered here in terms of natural transformations of organismal structures which generate higher dimensional algebras based on consistent axioms, thus avoiding well known logical paradoxes occurring with sets. Quantum bionetworks, such as quantum neural nets and quantum genetic networks, are also discussed and their underlying, non-commutative quantum logics are considered in the context of an emerging Quantum Relational Biology. (shrink)

Discontinuity refers to the character of many natural language constructions wherein signs differ markedly in their prosodic and semantic forms. As such it presents interesting demands on monostratal computational formalisms which aspire to descriptive adequacy. Pied piping, in particular, is argued by Pollard (1988) to motivate phrase structure-style feature percolation. In the context of categorial grammar, Bach (1981, 1984), Moortgat (1988, 1990, 1991) and others have sought to provide categorial operators suited to discontinuity. These attempts encounter certain difficulties with respect (...) to model theory and/or proof theory, difficulties which the current proposals are intended to resolve.Lambek calculus is complete for interpretation byresiduation with respect to the adjunction operation of groupoid algebras (Buszkowski 1986). In Moortgat and Morrill (1991) it is shown how to give calculi for families of categorial operators, each defined by residuation with respect to an operation of prosodic adjunction (associative, non-associative, or with interactive axioms). The present paper treats discontinuity in this way, by residuation with respect to three adjunctions: + (associative), (.,.) (split-point marking), andW (wrapping) related by the equations 1+s 2+s 3=(s 1,s 3)Ws 2. We show how the resulting methods apply to discontinuous functors, quantifier scope and quantifier scope ambiguity, pied piping, and subject and object antecedent reflexivisation. (shrink)

The most difficult problem that Leniewski came across in constructing his system of the foundations of mathematics was the problem of defining definitions, as he used to put it. He solved it to his satisfaction only when he had completed the formalization of his protothetic and ontology. By formalization of a deductive system one ought to understand in this context the statement, as precise and unambiguous as possible, of the conditions an expression has to satisfy if it is added to (...) the system as a new thesis. Now, some protothetical theses, and some ontological ones, included in the respective systems, happen to be definitions. In the present essay I employ Leniewski's method of terminological explanations for the purpose of formalizing ukasiewicz's system of implicational calculus of propositions, which system, without having recourse to quantification, I first extended some time ago into a functionally complete system. This I achieved by allowing for a rule of implicational definitions, which enabled me to define any propositionforming functor for any finite number of propositional arguments. (shrink)

The article studies properties of functional dependencies between strategies of players in Nash equilibria of multiplayer strategic games. The main focus is on the properties of functional dependencies in the context of a fixed dependency graph for payoff functions. A logical system describing properties of functional dependence for any given graph is proposed and is proven to be complete.

Peirce’s anti-psychologism hinges on two main assumptions. First, logic and psychology belong to two separate disciplines – respectively, the normative sciences and the experimental sciences. Second, externalism must be understood as a crucial and inescapable epistemological criterion. The introspectionist illusion, according to which individuals have direct and epistemologically flawless access to their own internal states, should be dismissed. As Colapietro (2003) and Calcaterra (2006) observe, Peirce’s standpoint is far different from the Kantian classical account of anti-psychologism. This original take on (...) anti-psychologism leaves room for a functional distinction between logic and psychology, emerging from a semiotic and communicative continuity. This means that psychology, unlike logic as a formal doctrine of signs, will be epistemologically appropriate for dealing with internal psychological states, on the condition that this inquiry be focused on the communicative processes through which these internal states are expressed and conveyed. Such a Peircean account of anti-psychologism forms the epistemological background of this paper. My goal is to show how Peirce’s approach to communication and semiosis can be applied in order to discuss a specific psychological theory, in this case, attachment theory. Specifically, I propose employing Giovanni Maddalena’s Peircean distinction between complete and incomplete gestures (Maddalena 2015) to account for the distinction between secure, dismissing and preoccupied attachment patterns. To this end, I will be discussing three different measurements of attachment: the Adult Attachment Interview (AAI, George, Kaplan & Main 1985), the Adult Attachment Projection (AAP, George & West 2006), and the Patient Attachment Coding System (PACS, Talia, Miller-Bottome & Daniel 2015). Throughout this discussion, I will examine the connection between the semiotic and phenomenological category of completeness, and the psychological category of security. This connection involves an interesting normative import, which I briefly discuss in the conclusions. (shrink)

A categorical, higher dimensional algebra and generalized topos framework for Łukasiewicz–Moisil Algebraic–Logic models of non-linear dynamics in complex functional genomes and cell interactomes is proposed. Łukasiewicz–Moisil Algebraic–Logic models of neural, genetic and neoplastic cell networks, as well as signaling pathways in cells are formulated in terms of non-linear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable ‘next-state functions’ is extended to a (...) Łukasiewicz–Moisil Topos with an n-valued Łukasiewicz–Moisil Algebraic Logic subobject classifier description that represents non-random and non-linear network activities as well as their transformations in developmental processes and carcinogenesis. The unification of the theories of organismic sets, molecular sets and Robert Rosen’s (M,R)-systems is also considered here in terms of natural transformations of organismal structures which generate higher dimensional algebras based on consistent axioms, thus avoiding well known logical paradoxes occurring with sets. Quantum bionetworks, such as quantum neural nets and quantum genetic networks, are also discussed and their underlying, non-commutative quantum logics are considered in the context of an emerging Quantum Relational Biology. (shrink)

In this paper functional completeness results are obtained for certain positive and constructive propositional logics associated with a Tarski-type structured consequence relation as defined by Gabbay.

The goal of this paper is to develop a systematic taxonomy of cognitive artifacts, i.e., human-made, physical objects that functionally contribute to performing a cognitive task. First, I identify the target domain by conceptualizing the category of cognitive artifacts as a functional kind: a kind of artifact that is defined purely by its function. Next, on the basis of their informational properties, I develop a set of related subcategories in which cognitive artifacts with similar properties can be grouped. In (...) this taxonomy, I distinguish between three taxa, those of family, genus, and species. The family includes all cognitive artifacts without further specifying their informational properties. Two genera are then distinguished: representational and non-representational (or ecological) cognitive artifacts. These genera are further divided into species. In case of representational artifacts, these species are iconic, indexical, or symbolic. In case of ecological artifacts, these species are spatial or structural. Within species, token artifacts are identified. The proposed taxonomy is an important first step towards a better understanding of the range and variety of cognitive artifacts and is a helpful point of departure, both for conceptualizing how different artifacts augment or impair cognitive performance and how they transform and are integrated into our cognitive system and practices. (shrink)

It is a classic result in modal logic, often referred to as Jónsson-Tarski duality, that the category of modal algebras is dually equivalent to the category of descriptive frames. The latter are Kripke frames equipped with a Stone topology such that the binary relation is continuous. This duality generalizes the celebrated Stone duality for boolean algebras. Our goal is to generalize descriptive frames so that the topology is an arbitrary compact Hausdorff topology. For this, instead of working with the boolean (...) algebra of clopen subsets of a Stone space, we work with the ring of continuous real-valued functions on a compact Hausdorff space. The main novelty is to define a modal operator on such a ring utilizing a continuous relation on a compact Hausdorff space.Our starting point is the well-known Gelfand duality between the category ${\sf KHaus}$ of compact Hausdorff spaces and the category $\boldsymbol {\mathit {uba}\ell }$ of uniformly complete bounded archimedean $\ell $ -algebras. We endow a bounded archimedean $\ell $ -algebra with a modal operator, which results in the category $\boldsymbol {\mathit {mba}\ell }$ of modal bounded archimedean $\ell $ -algebras. Our main result establishes a dual adjunction between $\boldsymbol {\mathit {mba}\ell }$ and the category ${\sf KHF}$ of what we call compact Hausdorff frames; that is, Kripke frames equipped with a compact Hausdorff topology such that the binary relation is continuous. This dual adjunction restricts to a dual equivalence between ${\sf KHF}$ and the reflective subcategory $\boldsymbol {\mathit {muba}\ell }$ of $\boldsymbol {\mathit {mba}\ell }$ consisting of uniformly complete objects of $\boldsymbol {\mathit {mba}\ell }$. This generalizes both Gelfand duality and Jónsson-Tarski duality. (shrink)

We enrich intuitionistic logic with a lax modal operator and define a corresponding intensional enrichment of Kripke models M = (W, , V) by a function T giving an effort measure T(w, u) {} for each -related pair (w, u). We show that embodies the abstraction involved in passing from true up to bounded effort to true outright. We then introduce a refined notion of intensional validity M |= p : and present a corresponding intensional calculus iLC-h which gives a (...) natural extension by lax modality of the well-known G: odel/Dummett logic LC of (finite) linear Kripke models. Our main results are that for finite linear intensional models L the intensional theory iTh(L) = {p : | L |= p : } characterises L and that iLC-h generates complete information about iTh(L).Our paper thus shows that the quantitative intensional information contained in the effort measure T can be abstracted away by the use of and completely recovered by a suitable semantic interpretation of proofs. (shrink)

We enrich intuitionistic logic with a lax modal operator ○ and define a corresponding intensional enrichment of Kripke models M = (W, ⊑, V) by a function T giving an effort measure T(w, u) ∈ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\mathbb{N}} \cup$$ \end{document} {∞} for each ⊑-related pair (w, u). We show that ○ embodies the abstraction involved in passing from “ϕ true up to bounded effort” to “ϕ true outright”. We then introduce a refined (...) notion of intensional validityM |= p : ϕ and present a corresponding intensional calculus iLC-h which gives a natural extension by lax modality of the well-known G: odel/dummett logic LC of (finite) linear Kripke models. Our main results are that for finite linear intensional models L the intensional theory iTh(L) = {p : ϕ | L |= p : ϕ} characterises L and that iLC-h generates complete information about iTh(L).Our paper thus shows that the quantitative intensional information contained in the effort measure T can be abstracted away by the use of ○ and completely recovered by a suitable semantic interpretation of proofs. (shrink)