In 2001, W. Carnielli and Marcos considered a 3-valued logic in order to prove that the schema ϕ ∨ (ϕ → ψ) is not a theorem of da Costa’s logic Cω. In 2006, this logic was studied (and baptized) as G'3 by Osorio et al. as a tool to define semantics of logic programming. It is known that the truth-tables of G'3 have the same expressive power than the one of Łukasiewicz 3-valued logic (...) as well as the one of Gödel 3-valued logic G3. From this, the three logics coincide up-to language, taking into acccount that 1 is the only designated truth-value in these logics. -/- From the algebraic point of view, Canals-Frau and Figallo have studied the 3-valued modal implicative semilattices, where the modal operator is the well-known Moisil-Monteiro-Baaz Δ operator, and the supremum is definable from this. We prove that the subvariety obtained from this by adding a bottom element 0 is term-equivalent to the variety generated by the 3-valued algebra of G'3. The algebras of that variety are called G'3-algebras. From this result, we obtain the equations which axiomatize the variety of G'3-algebras. Moreover, we prove that this variety is semisimple, and the 3-element and the 2-element chains are the unique simple algebras of the variety. Finally an extension of G'3 to first-order languages is presented, with an algebraic semantics based on complete G'3-algebras. The corresponding soundness and completeness theorems are obtained. (shrink)

Fred shows how problems with Slater's restriction of the classical propositional logic can be solved.

In a recent paper we have defined an analytic tableau calculus PL_16 for a functionally complete extension of Shramko and Wansing's logic based on the trilattice SIXTEEN_3. This calculus makes it possible to define syntactic entailment relations that capture central semantic relations of the logic---such as the relations |=_t, |=_f, and |=_i that each correspond to a lattice order in SIXTEEN_3; and |=, the intersection of |=_t and |=_f,. -/- It turns out that our method of characterising these (...) semantic relations---as intersections of auxiliary relations that can be captured with the help of a single calculus---lends itself well to proving interpolation. All entailment relations just mentioned have the interpolation property, not only when they are defined with respect to a functionally complete language, but also in a range of cases where less expressive languages are considered. For example, we will show that |=, when restricted to L_{tf}, the language originally considered by Shramko and Wansing, enjoys interpolation. This answers a question that was recently posed by M. Takano. (shrink)

One of the most prominent myths in analytic philosophy is the so- called “Fregean Axiom”, according to which the reference of a sentence is a truth value. In contrast to this referential semantics, a use-based formal semantics will be constructed in which the logical value of a sentence is not its putative referent but the information it conveys. Let us call by “Question Answer Semantics” (thereafter: QAS) the corresponding formal semantics: a non-Fregean many-valued logic, where the meaning of (...) any sentence is an ordered n-tupled of yes-no answers to corresponding questions. A sample of philosophical problems will be approached in order to justify the relevance of QAS. These include: (1) illocutionary forces, and the logical analysis of speech-acts; (2) the variety of logical negations, and their characterization in terms of restricted ranges of logical values; (3) change in meaning, and the use of dynamic oppositions for belief sets. (shrink)

Boolean-valued models of set theory were independently introduced by Scott, Solovay and Vopěnka in 1965, offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, Löwe and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific 3-valued model called PS3 which satisfies all (...) the axioms of ZF, and can be expanded with a paraconsistent negation *, thus obtaining a paraconsistent model of ZF. The logic (PS3 ,*) coincides (up to language) with da Costa and D'Ottaviano logic J3, a 3-valued paraconsistent logic that have been proposed independently in the literature by several authors and with different motivations such as CluNs, LFI1 and MPT. We propose in this paper a family of algebraic models of ZFC based on LPT0, another linguistic variant of J3 introduced by us in 2016. The semantics of LPT0, as well as of its first-order version QLPT0, is given by twist structures defined over Boolean agebras. From this, it is possible to adapt the standard Boolean-valued models of (classical) ZFC to twist-valued models of an expansion of ZFC by adding a paraconsistent negation. We argue that the implication operator of LPT0 is more suitable for a paraconsistent set theory than the implication of PS3, since it allows for genuinely inconsistent sets w such that [(w = w)] = 1/2 . This implication is not a 'reasonable implication' as defined by Löwe and Tarafder. This suggests that 'reasonable implication algebras' are just one way to define a paraconsistent set theory. Our twist-valued models are adapted to provide a class of twist-valued models for (PS3,*), thus generalizing Löwe and Tarafder result. It is shown that they are in fact models of ZFC (not only of ZF). (shrink)

(1) This paper is about how to build an account of the normativity of logic around the claim that logic is constitutive of thinking. I take the claim that logic is constitutive of thinking to mean that representational activity must tend to conform to logic to count as thinking. (2) I develop a natural line of thought about how to develop the constitutive position into an account of logical normativity by drawing on constitutivism in metaethics. (3) (...) I argue that, while this line of thought provides some insights, it is importantly incomplete, as it is unable to explain why we should think. I consider two attempts at rescuing the line of thought. The first, unsuccessful response is that it is self-defeating to ask why we ought to think. The second response is that we need to think. But this response secures normativity only if thinking has some connection to human flourishing. (4) I argue that thinking is necessary for human flourishing. Logic is normative because it is constitutive of this good. (5) I show that the resulting account deals nicely with problems that vex other accounts of logical normativity. (shrink)

After some generalities about connections between functions and relations in Sections 1 and 2 recalls the possibility of taking the semantic values of ‐ary Boolean connectives as ‐ary relations among truth‐values rather than as ‐ary truth functions. Section 3, the bulk of the paper, looks at correlates of these truth‐value relations as applied to formulas, and explores in a preliminary way how their properties are related to the properties of “logical relations” among formulas such as equivalence, implication (entailment) and contrariety (...) (logical incompatibility), concentrating for illustrative purposes on binary logical relations such as those just listed. To avoid an excess of footnotes, some points have been deferred to an Appendix as “Longer Notes”. (shrink)

Multialgebras have been much studied in mathematics and in computer science. In 2016 Carnielli and Coniglio introduced a class of multialgebras called swap structures, as a semantic framework for dealing with several Logics of Formal Inconsistency that cannot be semantically characterized by a single finite matrix. In particular, these LFIs are not algebraizable by the standard tools of abstract algebraic logic. In this paper, the first steps towards a theory of non-deterministic algebraization of logics by swap structures are given. (...) Specifically, a formal study of swap structures for LFIs is developed, by adapting concepts of universal algebra to multialgebras in a suitable way. A decomposition theorem similar to Birkhoff’s representation theorem is obtained for each class of swap structures. Moreover, when applied to the 3-valued algebraizable logics J3 and Ciore, their classes of algebraic models are retrieved, and the swap structures semantics become twist structures semantics. This fact, together with the existence of a functor from the category of Boolean algebras to the category of swap structures for each LFI, suggests that swap structures can be seen as non-deterministic twist structures. This opens new avenues for dealing with non-algebraizable logics by the more general methodology of multialgebraic semantics. (shrink)

How to say no less, no more about conditional than what is needed? From a logical analysis of necessary and sufficient conditions (Section 1), we argue that a stronger account of conditional can be obtained in two steps: firstly, by reminding its historical roots inside modal logic and set-theory (Section 2); secondly, by revising the meaning of logical values, thereby getting rid of the paradoxes of material implication whilst showing the bivalent roots of conditional as a speech-act based on (...) affirmations and rejections (Section 3). Finally, the two main inference rules for conditional, viz. Modus Ponens and Modus Tollens, are reassessed through a broader definition of logical consequence that encompasses both a normal relation of truth propagation and a weaker relation of falsity non-propagation from premises to conclusion (Section 3). (shrink)

Paraconsistent logics are logical systems that reject the classical principle, usually dubbed Explosion, that a contradiction implies everything. However, the received view about paraconsistency focuses only the inferential version of Explosion, which is concerned with formulae, thereby overlooking other possible accounts. In this paper, we propose to focus, additionally, on a meta-inferential version of Explosion, i.e. which is concerned with inferences or sequents. In doing so, we will offer a new characterization of paraconsistency by means of which a logic (...) is paraconsistent if it invalidates either the inferential or the meta-inferential notion of Explosion. We show the non-triviality of this criterion by discussing a number of logics. On the one hand, logics which validate and invalidate both versions of Explosion, such as classical logic and Asenjo–Priest’s 3-valued logic LP. On the other hand, logics which validate one version of Explosion but not the other, such as the substructural logics TS and ST, introduced by Malinowski and Cobreros, Egré, Ripley and van Rooij, which are obtained via Malinowski’s and Frankowski’s q- and p-matrices, respectively. (shrink)

Free logics aim at freeing logic from existence assumptions by making them explicit, e.g., by adding an existence premisse to the antecedence of the classical axiom-schema of Universal Instantiation. Their historical development was motivated by the problem of empty singular terms, and that one of simple statements containing at least one such singular term: what is the referential status of such singular terms and what truth-value, if any, do such statemants have? Free logics can be classified with regard to (...) their respective answers to these problems. Negative free logics assume that non-existent objects cannot have any properties at all; hence, in particular, they cannot be self-identical or rotate. Positive free logics believe that non-existents can be self-identical according to the Leibnizian concept of identity. Neutral free logics think that statements of self-identity are truth-valueless because of the Fregean principle of compositionality. Since only in negative free logic, but not in positive free logic, two statements of the forms "a = a" and "E!a" are logically equivalent, one also can define only for NFL, but not for PFL, existence by self-identity. (shrink)

We here make preliminary investigations into the model theory of DeMorgan logics. We demonstrate that Łoś's Theorem holds with respect to these logics and make some remarks about standard model-theoretic properties in such contexts. More concretely, as a case study we examine the fate of Cantor's Theorem that the classical theory of dense linear orderings without endpoints is $\aleph_{0}$-categorical, and we show that the taking of ultraproducts commutes with respect to previously established methods of constructing nonclassical structures, namely, Priest's Collapsing (...) Lemma and Dunn's Theorem in 3-Valued Logic. (shrink)

The depth-bounded approach seeks to provide realistic models of reasoners. Recognizing that most useful logics are idealizations in that they are either undecidable or likely to be intractable, the approach accounts for how they can be approximated in practice by resource-bounded agents. The approach has been applied to Classical Propositional Logic (CPL), yielding a hierarchy of tractable depth-bounded approximations to that logic, which in turn has been based on a KE/KI system. -/- This Thesis shows that the approach (...) can be naturally extended to useful nonclassical logics such as First-Degree Entailment (FDE), the Logic of Paradox (LP), Strong Kleene Logic (K3 ) and Intuitionistic Propositional Logic (IPL). To do this, we introduce a KE/KI-style system for each of those logics such that: is formulated via signed formulae, consist of linear operational rules and branching structural rule(s), can be used as a direct-proof and a refutation method, and is interesting independently of the approach in that it has an exponential speed-up on its tableau system counterpart. The latter given that we introduce a new class of examples which we prove to be hard for all tableau systems sharing the V/& rules with the classical one, but easy for their analogous KE-style systems. Then we focus on showing that each of our KE/KI-style systems naturally yields a hierarchy of tractable depth-bounded approximations to the respective logic, in terms of the maximum number of allowed nested applications of the branching rule(s). The rule(s) express(es) a generalized rule of bivalence, is (are) essentially cut rule(s) and govern(s) the manipulation of virtual information, which is information that an agent does not hold but she temporarily assumes as if she held it. Intuitively, the more virtual information needs to be invoked via the branching rule(s), the harder the inference is for the agent. So, the nested application the branching rule(s) provides a sensible measure of inferential depth. We also show that each hierarchy approximating FDE, LP, and K3 , admits of a 5-valued non-deterministic semantics; whereas, paving the way for a semantical characterization of the hierarchy approximating IPL, we provide a 3-valued non-deterministic semantics for the full logic that fixes the meaning of the connectives without appealing to “structural” conditions. -/- Moreover, we show a super-polynomial lower bound for the strongest possible version of clausal tableaux on the well-known class of “truly fat” expressions (which are easy for KE), settling a problem left open in the literature. Further, we investigate a hierarchy of tractable depth-bounded approximations to CPL based only on KE. Finally, we propose a refinement of the p-simulation relation which is adequate to establish positive results about the superiority of a system over another with respect to proof-search. (shrink)

Description Logics (DLs) are a family of well-known terminological knowledge representation formalisms in modern semantics-based systems. This research focuses on analysing how our developed Occurrence Logic (OccL) can conceptually and logically support the development of a description logic. OccL is integrated into the alternative theory of natural language syntax in `Deviational Syntactic Structures' under the label `EFA(X)3' (or the third version of Epi-Formal Analysis in Syntax, EFA(X), which is a radical linguistic theory). From the logical point of view, (...) OccL is a formal logic that mainly deals with the occurrences of symbols as well as with their priorities within linguistic descriptions, i.e. natural language syntax, semantics and phonology. In this article---based on our OccL-based definitions of the concepts of `strong implication' and `occurrence value' as well as of the logical concept `identical occurrence constructor (IDOC)' that is the most fundamental logical concept in our formalism---we will model Occurrence Description Logic ($\mathcal{ODL}$). Accordingly, we will formally-logically analyse `occurrence(s) of symbol(s)' within descriptions of the world in $\mathcal{ODL}$. In addition, we will analyse and assess the logical concepts of `occurrence' and `occurrence priority' in $\mathcal{ODL}$. This research can make a strong logical background for our future research in the development of a Modal Occurrence Description Logic. (shrink)

The logics of formal inconsistency (LFIs, for short) are paraconsistent logics (that is, logics containing contradictory but non-trivial theories) having a consistency connective which allows to recover the ex falso quodlibet principle in a controlled way. The aim of this paper is considering a novel semantical approach to first-order LFIs based on Tarskian structures defined over swap structures, a special class of multialgebras. The proposed semantical framework generalizes previous aproaches to quantified LFIs presented in the literature. The case of QmbC, (...) the simpler quantified LFI expanding classical logic, will be analyzed in detail. An axiomatic extension of QmbC called QLFI1o is also studied, which is equivalent to the quantified version of da Costa and D'Ottaviano 3-valued logic J3. The semantical structures for this logic turn out to be Tarkian structures based on twist structures. The expansion of QmbC and QLFI1o with a standard equality predicate is also considered. (shrink)

Modal logic is one of philosophy’s many children. As a mature adult it has moved out of the parental home and is nowadays straying far from its parent. But the ties are still there: philosophy is important to modal logic, modal logic is important for philosophy. Or, at least, this is a thesis we try to defend in this chapter. Limitations of space have ruled out any attempt at writing a survey of all the work going on (...) in our field—a book would be needed for that. Instead, we have tried to select material that is of interest in its own right or exemplifies noteworthy features in interesting ways. Here are some themes that have guided us throughout the writing: • The back-and-forth between philosophy and modal logic. There has been a good deal of give-and-take in the past. Carnap tried to use his modal logic to throw light on old philosophical questions, thereby inspiring others to continue his work and still others to criticise it. He certainly provoked Quine, who in his turn provided—and continues to provide—a healthy challenge to modal logicians. And Kripke’s and David Lewis’s philosophies are connected, in interesting ways, with their modal logic. Analytic philosophy would have been a lot different without modal logic! • The interpretation problem. The problem of providing a certain modal logic with an intuitive interpretation should not be conflated with the problem of providing a formal system with a model-theoretic semantics. An intuitively appealing model-theoretic semantics may be an important step towards solving the interpretation problem, but only a step. One may compare this situation with that in probability theory, where definitions of concepts like ‘outcome space’ and ‘random variable’ are orthogonal to questions about “interpretations” of the concept of probability. • The value of formalisation. Modal logic sets standards of precision, which are a challenge to—and sometimes a model for—philosophy. Classical philosophical questions can be sharpened and seen from a new perspective when formulated in a framework of modal logic. On the other hand, representing old questions in a formal garb has its dangers, such as simplification and distortion. • Why modal logic rather than classical (first or higher order) logic? The idioms of modal logic—today there are many!—seem better to correspond to human ways of thinking than ordinary extensional logic. (Cf. Chomsky’s conjecture that the NP + VP pattern is wired into the human brain.) In his An Essay in Modal Logic (1951) von Wright distinguished between four kinds of modalities: alethic (modes of truth: necessity, possibility and impossibility), epistemic (modes of being known: known to be true, known to be false, undecided), deontic (modes of obligation: obligatory, permitted, forbidden) and existential (modes of existence: universality, existence, emptiness). The existential modalities are not usually counted as modalities, but the other three categories are exemplified in three sections into which this chapter is divided. Section 1 is devoted to alethic modal logic and reviews some main themes at the heart of philosophical modal logic. Sections 2 and 3 deal with topics in epistemic logic and deontic logic, respectively, and are meant to illustrate two different uses that modal logic or indeed any logic can have: it may be applied to already existing (non-logical) theory, or it can be used to develop new theory. (shrink)

Trivalence is quite natural for deontic action logic, where actions are treated as good, neutral or bad.We present the ideas of trivalent deontic logic after J. Kalinowski and its realisation in a 3-valued logic of M. Fisher and two systems designed by the authors of the paper: a 4-valued logic inspired by N. Belnap’s logic of truth and information and a 3-valued logic based on nondeterministic matrices. Moreover, we combine Kalinowski’s idea (...) of trivalence with deontic action logic based on boolean algebra. (shrink)

The endeavour to locate value in moral progress faces various substantive as well as more formal challenges. This paper focuses on challenges of the latter kind. After some preliminaries, Section 3 introduces two general kinds of “evaluative moral progress-claims”, and outlines a possible novel analysis of a descriptive notion of moral progress. While Section 4 discusses certain logical features of betterness in light of recent work in value theory which are pertinent to the notion of moral progress, Sections 5 and (...) 6 outline the ambiguous character of “making moral progress”. (shrink)

Chapter 3 surveys objectivity in the natural sciences. Thomas Kuhn problematized the logicist understanding of the objectivity or rationality of scientific change, providing a very different picture than that of the cumulative or step-wise progress of theoretical science. Theories often compete, and when consensus builds around one competitor it may be for a variety of reasons other than just the direct logical implications of experimental successes and failures. Kuhn pitted the study of the actual history of science against what Hans (...) Reichenbach referred to as the “logical substitutes” logicists used to reconstruct the rationality of theory change. Such substitutes were promoted to show that theory confirmation is a purely epistemic affair, where purely epistemic implies purely inferential. But where the positivists saw logical compulsion as the reason for theory change, Kuhn saw psychology in values as relevant to how theories are actually confirmed or accepted by the scientific community. (shrink)

We examine the set of formula-to-formula valid inferences of Classical Logic, where the premise and the conclusion share at least a propositional variable in common. We review the fact, already proved in the literature, that such a system is identical to the first-degree entailment fragment of R. Epstein's Relatedness Logic, and that it is a non-transitive logic of the sort investigated by S. Frankowski and others. Furthermore, we provide a semantics and a calculus for this logic. (...) The semantics is defined in terms of a \-matrix built on top of a 5-valued extension of the 3-element weak Kleene algebra, whereas the calculus is defined in terms of a Gentzen-style sequent system where the left and right negation rules are subject to linguistic constraints. (shrink)

1971. Discourse Grammars and the Structure of Mathematical Reasoning II: The Nature of a Correct Theory of Proof and Its Value, Journal of Structural Learning 3, #2, 1–16. REPRINTED 1976. Structural Learning II Issues and Approaches, ed. J. Scandura, Gordon & Breach Science Publishers, New York, MR56#15263. -/- This is the second of a series of three articles dealing with application of linguistics and logic to the study of mathematical reasoning, especially in the setting of a concern for improvement (...) of mathematical education. The present article presupposes the previous one. Herein we develop our ideas of the purposes of a theory of proof and the criterion of success to be applied to such theories. In addition we speculate at length concerning the specific kinds of uses to which a successful theory of proof may be put vis-a-vis improvement of various aspects of mathematical education. The final article will deal with the construction of such a theory. The 1st is the 1971. Discourse Grammars and the Structure of Mathematical Reasoning I: Mathematical Reasoning and Stratification of Language, Journal of Structural Learning 3, #1, 55–74. https://www.academia.edu/s/fb081b1886?source=link . (shrink)

The passionate and staunch defence of logic of the controversial thinker Ibn Ḥazm, Abū Muḥammad ʿAlī b. Aḥmad b. Saʿīd of Córdoba (384-456/994-1064), had lasting consequences in the Islamic world. Indeed, his book Facilitating the Understanding of the Rules of Logic and Introduction Thereto, with Common Expressions and Juristic Examples (Kitāb al-Taqrīb li-ḥadd al-manṭiq wa-l-mudkhal ilayhi bi-l-alfāẓ al-ʿāmmiyya wa-l-amthila al-fiqhiyya), composed in 1025-1029, was well known and discussed during and after his time; and it paved the way for (...) the studies of his compatriots Ibn Bājjah (d. 1138), Ibn Ṭufayl (1185), and Ibn Rushd (1198), who each gave demonstrative reasoning a privileged place within the methods of attaining knowledge. Unfortunately, as too often in the history of science, Ibn Ḥazm’s innovative perspectives and contributions in logic have been overlooked or considered with an attitude of contempt. On the one hand, his work has been seen, at best, as promulgating the benefits of studying Aristotle’s logic, so that his contribution is assessed as more didactical than conceptual. And on the other hand, those who do examine his innovations often consider them to be mistaken. As indicated by Chejne (1984, p. 2) contempt towards the logical work of Ibn Ḥazm was also present in its reception by the Eastern philosophers who accused him of deviating from Aristotelian logic and of dabbling in things beyond his capability. However, a reassessment of his work on logic has since begun, by delving into the ways the thinker of Córdoba studied the links between deontic and modal qualification of propositions. In this context Lameer’s (2013) paper on the logical sources of Ibn Ḥazm is worth mentioning; the author (p. 417, footnote 1) observes that, although, strictly speaking, it was al-Fārābī who first drew the parallelism between deontic and modal concepts, it was Ibn Ḥazm who developed it and worked it out in a more precise manner. A primary aim of this paper is to help fill some of these gaps by stressing the role of the work of Ibn Ḥazm in developing a notion of deontic necessity deeply rooted in legal normativity, and in explicitly discussing the transference between deontic and modal concepts. According to our view; the basic units of Islamic deontic logic are what we might call, indulging in terminological anachronism, heteronomous imperatives. As it turns out, the heteronomy of imperatives within Islamic legal systems contrasts with those of the purely moral realm, which seem to be closer to an autonomous conception of moral law. There it concurs again with Leibniz’s proposal to define obligatory as “what is necessary for a good person to do”. In the present paper, we will focus on the heteronomous imperatives of legal systems rather than on the imperatives of the purely moral realm. In this context, the work of Ibn Ḥazm extends the parallelism between the necessity of events and that of human actions stressed by his predecessors. According to our understanding, Ibn Ḥazm’s parallelism can be rendered explicit formally by means of a conditional (or hypothetical) structure shared by both deontic and modal propositions. Moreover, this structure makes apparent that this parallelism displays an underlying system of “degrees”. Indeed, while in the domain of events, given some conditions, it makes sense to distinguish between an event that is more likely to happen than another. Ibn Ḥazm speaks of near and distant possibility (such as the higher degree of likelihood of rain, given the condensation of clouds in December, and its lower degree, when the condensation occurs in summer); and, in the domain of actions, the Islamic notion of weighting actions determines degrees of virtue (or of legal and moral value). So whereas the degree of virtue of performing a forbidden act as determined by the distribution of sanction and reward is 0, we obtain the same value by pondering the likelihood of an impossible event to take place. Furthermore, as discussed in section II.3.2, the system of values at work in the parallelism can be seen as opening the way to a more direct correspondence. Thus, while in the realm of nature, likelihood of occurrence of an event is dependent upon the conditions specific to the occurrence of that event; while, in the realm of jurisprudence, likelihood of performing an action is dependent upon the distribution of reward and sanction specific to that action (being that, given the choice to perform or not perform a given action, those that will be rewarded are more likely to be performed than those that are not). According to this perspective, the point of the parallelism is that both deontic and modal qualifications measure the degree of an action or event to become actual; that is – indulging once more in anachronism (but this time from the Leibnizian background) – the degree measures how feasible (facile) an action or event may be. This suggests that Ibn Ḥazm’s perspectives already herald the links to probability and possibility explored by Leibniz six centuries later – see for example Leibniz’s (1671, A VI, I, p. 424-26) use of probability in the context of conditional right. Nevertheless, it also shows a crucial difference to the approach developed in the Elementa Jura Naturalis: whereas Leibniz’s studies seek to define what is to be just or virtuous, the logical system of deontic imperatives within Islamic jurisprudence presupposes that what is to be virtuous has already been settled. Determining what is to be virtuous is not achieved by logical reasoning within the system of legal jurisprudence, but by delving into the higher objectives of the Sharīʿa. While developing our point, we will delve into the logical structure of the heteronomous imperatives. This distinguishes our contribution from the existing literature, such as the papers of Chejne (1984), Lameer (2014), and Guerrero (1997, 2010, 2014), which do not provide a logical analysis of the deontic concepts put into work by Ibn Ḥazm. The true antecedent to the present paper is the work of Farid Zidani (2007, 2015), who, so far as we know, was the first to undertake such a task. Some of our own developments and general epistemological thoughts go beyond Ibn Ḥazm’s framework and motivations; mainly those contained in sections II.3.2 and IV. However, according to our view, these reflections suggest that Ibn Ḥazm’s approach has the substance for a broader and deeper exploration of the logic of norms. The paper is structured as follows:I.Ibn Ḥazm’s Logic of Heteronomous Imperatives. After presenting some extracts of the relevant text, we proceed by providing a formal reconstruction of the five forms of deontic modalities. II. A Landmark in the History of the Logical Analysis of Norms. Duties and Modalities. In this section, we study the transferences from deontic to modal necessity and possibility. We briefly compare the deontic system of Islamic Jurisprudence with that of Leibniz.III. Leibniz and Hypothetical Imperatives in Law.IV. Beyond Ibn Ḥazm: Conclusions and the Work Ahead. We will conclude the paper by discussing briefly some conceptual points that distinguish the logic of heteronomous imperatives from contemporary deontic logic. Our final words will discuss deontic-modal parallelism in the context of Ibn Ḥazm’s rejection of reasoning by conjecture and what we call “the internalization of nature.”. (shrink)

Abstract Introduction Logically valid deductive arguments are clear examples of abstract recursive computational proce‐ dures on propositions or on probabilities. However, it is not known if the cortical time‐consuming inferential pro‐ cesses in which logical arguments are eventually realized in the brain are in fact physically different from other kinds of inferential processes. Methods In order to determine whether an electrical EEG discernible pattern of logical deduction exists or not, a new experimental paradigm is proposed contrasting logically valid and invalid (...) inferences with exactly the same content (same premises and same relational variables) and distinct logical complexity (propositional truth‐functional operators). Electroencephalographic signals from 19 subjects (24.2 ± 3.3 years) were acquired in a two‐condition para‐ digm (100 trials for each condition). After the initial general analysis, a trial‐by‐trial approach in beta‐2 band allowed to uncover not only evoked but also phase asynchronous activity between trials. Results showed that (i) deductive inferences with the same content evoked the same response pattern in logically valid and invalid conditions, (ii) mean response time in logically valid inferences is 61.54% higher, (iii) logically valid inferences are subjected to an early (400 ms) and a late reprocessing (600 ms) verified by two distinct beta‐2 activa‐ tions (p‐value < 0,01, Wilcoxon signed rank test). Conclusion We found evidence of a subtle but measurable electrical trait of logical validity. Results put forward the hypothesis that some logically valid deductions are recursive or computational cortical events. Keywords Beta‐2 band, Evoked potentials, Induced potentials, Deductive inference, Logical validity, Cortical bases of logical reasoning. (shrink)

Desires matter. How are we to understand the intentionality of desire? According to the two classical views, desire is either a positive evaluation or a disposition to act: to desire a state is to positively evaluate it or to be disposed to act to realize it. This Ph.D. Dissertation examines these conceptions of desire and proposes a deontic alternative inspired by Meinong. On this view, desiring is representing a state of affairs as what ought to be or, if one prefers, (...) as what should be. Desire involves a deontic manner of representing: a norm of the ought-to-be type features in desire’s intentional mode, as opposed to content. The dissertation is structured in three parts. In order to defend this conception, I formulate three main desiderata for a promising theory of the intentionality of desire in the introduction (§0). The first concerns desire’s direction of fit, i.e. the intuition that the world should conform to our desires. The second concerns the death of desire principle, i.e. the intuition that one cannot desire what one represents as actual. The last pertains to desire’s role in psychological explanations, i.e. the intuition that desires can explain some mental states and be explained by other mental states. The first part examines the main conceptions of desire in light of these desiderata. I argue that the classical pictures of desire do not adequately meet our desiderata. The first chapter is devoted to the evaluative conception (§1), while the second examines the motivational approach (§2). Following these criticisms, I then present the deontic view of desire (§3). In the second part, I defend this conception with the help of three arguments. The main idea is that appealing to norms of the ought-to-be type can satisfy our chief desiderata: the world should conform to norms (world-to-mind direction of fit, §4), norms are grounded on values and in turn ground obligations (explanation, §5), and norms are about non-actual states of affairs (death of desire principle, §6). In the last part, I develop the deontic view to draw a cartography of the various types of desire. Some desires are correct, while others are inappropriate. This distinction is explained by the deontic conception, as it matches that between states of affairs that ought to obtain and states that should not obtain (§7). Two study cases are examined: caprice and the impermissibility of desire aggregation. Intuitively, hopes, wishes, or urges are types of desire. The next chapter presents a typology inspired by the deontic view and the type of norms there are (§8). The last chapter discusses the main objections to the deontic approach (§9). In conclusion, I show the relevance of the deontic view for several debates in philosophy of mind and ethics. Desires are crucial because they are the ‘eye’ of what should be. (shrink)

Knowledge is good, ignorance is bad. So it seems, anyway. But in this dissertation, I argue that some ignorance is epistemically valuable. Sometimes, we should suspend judgment even though by believing we would achieve knowledge. In this apology for ignorance (ignorance, that is, of a certain kind), I defend the following four theses: 1) Sometimes, we should continue inquiry in ignorance, even though we are in a position to know the answer, in order to achieve more than mere knowledge (e.g. (...) understanding) while minimizing the effects of confirmation bias. 2) It’s false that we should believe every proposition such that we are guaranteed to be right about it (and even such that we are guaranteed to know it) if we believe it. 3) Being in a position to know is the norm of assertion: importantly, this does not require belief or (thereby) knowledge, and so proper assertion can survive speaker-ignorance. 4) It can be permissible and conversationally useful to tell audiences things that it is logically impossible for them to come to know: Proper assertion can survive (necessary) audience-side ignorance. Cumulatively, this project suggests that, properly understood, ignorance has an important role to play in the good epistemic life. (shrink)

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

The main thesis of this paper is directed against the traditional (cognitivetheoretical) definition of the concept which claims that the concept is the '' thought about the essence of the object being thought'', i.e. that it is “a set of essential features or essential characteristics of an object''. But the '' set of essential features or essential characteristics of an object of thought'' is a '' content’’ of the thought. The thought about the essence of an object is definition and (...) the concept is not definition but the part of definition! Besides as the part of formal structure of thought, the concept possesses calculative logical properties that in formal logic (be it syllogistics, or the logic of propositions, or the logic of predicates) come to the front place of formal logical computation. Without the calculative properties of the concept, there would be no calculative properties of propositions which express the thought (thought structures). The calculative properties of a concept include the (1) degree of its logical generality (degree of variability), the (2) logical relations it can establish within the whole of the conceptual content, the (3) operability of the concept in structure of affirmation and negation, the (4) deducibility of either axiomatic or probabilistic systems. Therefore, I believe that, from the logical point of view, the definition of a concept should be applied in favor of its calculative properties that it possesses. (shrink)

The main goal of the article is a further development of the usage of income and comparable approaches to company valuation aimed at defining market capitalization sensitivity to value changing in the conditions of dynamization of internal and external business parameters. The relevance of the researched topic is determined by the importance of establishing the factors influencing the change in company market capitalization based on the synthesis of approaches to company valuation. To obtain the results of the study, the following (...) methods were used: structural and logical analysis – to improve the process approach to company valuation in the context of determining the impact of its change on market capitalization; discounted cash flow method (DCF-model) – to determine the value of the enterprise based on the income approach; comparable companies method – to determine the value of the company on the basis of a market approach; factor analysis method – to assess the impact of changes in value on company market capitalization; generalization and logical analysis – in conclusions building. The results of critical analysis of publications of domestic and foreign scientists on the application of company valuation methods identified the need of improvement the process approach to company valuation in the context of determining the impact of its change on the company market capitalization. To adequately assess company value, the use of a synthesis of income and market approaches is proposed. The implementation of the DCF-model and comparable companies valuation method while valuation of Private Joint-Stock Company “SKF Ukraine” allowed to establish that the greatest impact on the target metric (company market capitalization) has an average industry multiplier EV/EBITDA, terminal growth rate and weighted average cost of capital – 5.9 %, 3.2% and 2.7% respectively. The practical value of the results of the study is determined by the importance of forming a full analytical basis for establishing the attractiveness of the performance of joint-stock companies in the market and consolidating management decisionsin order to increase market capitalization. (shrink)

Plato’s attitude towards drunkenness (μέθη) is surprisingly positive in the Laws, especially as compared to his negative treatment of intoxication in the Republic. In the Republic, Plato maintains that intoxication causes cowardice and intemperance (3.398e-399e, 3.403e, and 9.571c-573b), while in the Laws, Plato holds that it can produce courage and temperance (1.635b, 1.645d-650a, and 2.665c-672d). This raises the question: Did Plato change his mind, and if he did, why? Ultimately, this paper answers affirmatively and argues that this marks a substantive (...) shift in Plato’s attitude towards anti-rational desires. More precisely, this paper argues that in the Republic, Plato holds that anti-rational desires are always detrimental to health and virtue, while in the Laws, Plato maintains that anti-rational desires can be instrumental to health and virtue. (shrink)

In this paper we consider the logics L(i,n) obtained from the (n+1)-valued Lukasiewicz logics L(n+1) by taking the order filter generated by i/n as the set of designated elements. In particular, the conditions of maximality and strong maximality among them are analyzed. We present a very general theorem that provides sufficient conditions for maximality between logics. As a consequence of this theorem, it is shown that L(i,n) is maximal w.r.t. CPL whenever n is prime. Concerning strong maximality (i.e. maximality (...) w.r.t. rules instead of only axioms), we provide algebraic arguments in order to show that the logics L(i,n) are not strongly maximal w.r.t. CPL, even for n prime. Indeed, in such case, we show that there is just one extension between L(i,n) and CPL obtained by adding to L(i,n) a kind of graded explosion rule. Finally, using these results, we show that the logics L(i,n) with n prime and i/n < 1/2 are ideal paraconsistent logics. (shrink)

In this dissertation I offer a global and detailed reconstruction of the logic developed by Gottlob Frege throughout his career. Even though Frege's logic suffered profound modifications from his initial formulation in Begriffsschrift to its revised version in Grundgesetze, the significant differences between these two works have been rarely taken at face value. I not only argue that these differences exist, but I also explain how they should be understood in the light of the evolution of Frege's thought. (...) First, I suggest a new reconstruction of Begriffsschrift's logic, which amounts to a completely novel reading of its formal system—one that contradicts the core of modern historical studies. In particular, I defend that this logic is not—as it has been repeatedly said—a second-order logic and provide the following reasons. (1) The language is not properly a formal language. (2) In Begriffsschrift there is only one sort of quantification: quantification over arguments. (3) Begriffsschrift's logic does not have a semantics in the modern sense. Second, I offer an explanation of the reasons that drive the evolution of Frege's logic. The transition from Begriffsschrift to Grundgesetze has been seldom addressed and never fully explained. According to my historical analysis, the switch from Frege's position concerning logic in Begriffsschrift to his later conception—finally established in Grundgesetze—can be articulated through the adoption of the distinction between concept and object as the basic element of the formal system. This leads to a formalisation of the notion of concept, which in the end drives to Grundgesetze's notion of function. Finally, I put forward a global analysis of Grundgesetze's logic. In this work, Frege develops a formal system that resembles in many relevant ways a second-order one. I suggest a reconstruction of this formal system that allows us to compare it with Begriffsschrift's. In particular, I formulate precisely every rule of inference proposed by Frege and especially focus on the rules of substitution. Moreover, I reflect on several meta-logical results that can be drawn from this reconstruction. (shrink)

Equality and identity. Bulletin of Symbolic Logic. 19 (2013) 255-6. (Coauthor: Anthony Ramnauth) Also see https://www.academia.edu/s/a6bf02aaab This article uses ‘equals’ [‘is equal to’] and ‘is’ [‘is identical to’, ‘is one and the same as’] as they are used in ordinary exact English. In a logically perfect language the oxymoron ‘the numbers 3 and 2+1 are the same number’ could not be said. Likewise, ‘the number 3 and the number 2+1 are one number’ is just as bad from a logical (...) point of view. In normal English these two sentences are idiomatically taken to express the true proposition that ‘the number 3 is the number 2+1’. Another idiomatic convention that interferes with clarity about equality and identity occurs in discussion of numbers: it is usual to write ‘3 equals 2+1’ when “3 is 2+1” is meant. When ‘3 equals 2+1’ is written there is a suggestion that 3 is not exactly the same number as 2+1 but that they merely have the same value. This becomes clear when we say that two of the sides of a triangle are equal if the two angles they subtend are equal or have the same measure. -/- Acknowledgements: Robert Barnes, Mark Brown, Jack Foran, Ivor Grattan-Guinness, Forest Hansen, David Hitchcock, Spaulding Hoffman, Calvin Jongsma, Justin Legault, Joaquin Miller, Tania Miller, and Wyman Park. -/- ► JOHN CORCORAN AND ANTHONY RAMNAUTH, Equality and identity. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA E-mail: [email protected] The two halves of one line are equal but not identical [one and the same]. Otherwise the line would have only one half! Every line equals infinitely many other lines, but no line is [identical to] any other line—taking ‘identical’ strictly here and below. Knowing that two lines equaling a third are equal is useful; the condition “two lines equaling a third” often holds. In fact any two sides of an equilateral triangle is equal to the remaining side! But could knowing that two lines being [identical to] a third are identical be useful? The antecedent condition “two things identical to a third” never holds, nor does the consequent condition “two things being identical”. If two things were identical to a third, they would be the third and thus not be two things but only one. The plural predicate ‘are equal’ as in ‘All diameters of a given circle are equal’ is useful and natural. ‘Are identical’ as in ‘All centers of a given circle are identical’ is awkward or worse; it suggests that a circle has multiple centers. Substituting equals for equals [replacing one of two equals by the other] makes sense. Substituting identicals for identicals is empty—a thing is identical only to itself; substituting one thing for itself leaves that thing alone, does nothing. There are as many types of equality as magnitudes: angles, lines, planes, solids, times, etc. Each admits unit magnitudes. And each such equality analyzes as identity of magnitude: two lines are equal [in length] if the one’s length is identical to the other’s. Tarski [1] hardly mentioned equality-identity distinctions (pp. 54-63). His discussion begins: -/- Among the logical concepts […], the concept of IDENTITY or EQUALITY […] has the greatest importance. -/- Not until page 62 is there an equality-identity distinction. His only “notion of equality”, if such it is, is geometrical congruence—having the same size and shape—an equivalence relation not admitting any unit. Does anyone but Tarski ever say ‘this triangle is equal to that’ to mean that the first is congruent to that? What would motivate him to say such a thing? This lecture treats the history and philosophy of equality-identity distinctions. [1] ALFRED TARSKI, Introduction to Logic, Dover, New York, 1995. [This is expanded from the printed abstract.] . (shrink)

This chapter is concerned with plural discourse in the grammatical sense. The goal of the chapter is to urge the value of the event analysis of the matrix of action sentences in thinking about logical form in plural discourse about action. Among the claims advanced are that: -/- 1. The ambiguity between distributive and collective readings of plural action sentences is not lexical ambiguity, either in the noun phrase (NP) or in the verb phrase (VP), but an ambiguity tracing to (...) the scope of the event quantifier introduced by the action verb. 2. This allows us to analyze collective action sentences in a way that commits us only to individual agents acting when we say that groups act, without denying that there are groups as such or that we talk about them as such. 3. Intermediate readings, that seem to be neither purely distributive nor purely collective, can be explained in terms of the same apparatus. (shrink)

In this paper, I consider a family of three-valued regular logics: the well-known strong and weak S.C. Kleene’s logics and two intermedi- ate logics, where one was discovered by M. Fitting and the other one by E. Komendantskaya. All these systems were originally presented in the semantical way and based on the theory of recursion. However, the proof theory of them still is not fully developed. Thus, natural deduction sys- tems are built only for strong Kleene’s logic both (...) with one (A. Urquhart, G. Priest, A. Tamminga) and two designated values (G. Priest, B. Kooi, A. Tamminga). The purpose of this paper is to provide natural deduction systems for weak and intermediate regular logics both with one and two designated values. (shrink)

Infectious logics are systems that have a truth-value that is assigned to a compound formula whenever it is assigned to one of its components. This paper studies four-valued infectious logics as the basis of transparent theories of truth. This take is motivated as a way to treat different pathological sentences differently, namely, by allowing some of them to be truth-value gluts and some others to be truth-value gaps and as a way to treat the semantic pathology suffered by at (...) least some of these sentences as infectious. This leads us to consider four distinct four-valued logics: one where truth-value gaps are infectious, but gluts are not; one where truth-value gluts are infectious, but gaps are not; and two logics where both gluts and gaps are infectious, in some sense. Additionally, we focus on the proof theory of these systems, by offering a discussion of two related topics. On the one hand, we prove some limitations regarding the possibility of providing standard Gentzen sequent calculi for these systems, by dualizing and extending some recent results for infectious logics. On the other hand, we provide sound and complete four-sided sequent calculi, arguing that the most important technical and philosophical features taken into account to usually prefer standard calculi are, indeed, enjoyed by the four-sided systems. (shrink)

The problem addressed in this paper is “the main epistemic problem concerning science”, viz. “the explication of how we compare and evaluate theories [...] in the light of the available evidence” (van Fraassen, BC, 1983, Theory comparison and relevant Evidence. In J. Earman (Ed.), Testing scientific theories (pp. 27–42). Minneapolis: University of Minnesota Press). Sections 1– 3 contain the general plausibility-informativeness theory of theory assessment. In a nutshell, the message is (1) that there are two values a theory should exhibit: (...) truth and informativeness—measured respectively by a truth indicator and a strength indicator; (2) that these two values are conflicting in the sense that the former is a decreasing and the latter an increasing function of the logical strength of the theory to be assessed; and (3) that in assessing a given theory by the available data one should weigh between these two conflicting aspects in such a way that any surplus in informativeness succeeds, if the shortfall in plausibility is small enough. Particular accounts of this general theory arise by inserting particular strength indicators and truth indicators. In Section 4 the theory is spelt out for the Bayesian paradigm of subjective probabilities. It is then compared to incremental Bayesian confirmation theory. Section 4 closes by asking whether it is likely to be lovely. Section 5 discusses a few problems of confirmation theory in the light of the present approach. In particular, it is briefly indicated how the present account gives rise to a new analysis of Hempel’s conditions of adequacy for any relation of confirmation (Hempel, CG, 1945, Studies in the logic of comfirmation. Mind, 54, 1–26, 97–121.), differing from the one Carnap gave in § 87 of his Logical foundations of probability (1962, Chicago: University of Chicago Press). Section 6 adresses the question of justification any theory of theory assessment has to face: why should one stick to theories given high assessment values rather than to any other theories? The answer given by the Bayesian version of the account presented in section 4 is that one should accept theories given high assessment values, because, in the medium run, theory assessment almost surely takes one to the most informative among all true theories when presented separating data. The concluding section 7 continues the comparison between the present account and incremental Bayesian confirmation theory. (shrink)

Logics based on weak Kleene algebra (WKA) and related structures have been recently proposed as a tool for reasoning about flaws in computer programs. The key element of this proposal is the presence, in WKA and related structures, of a non-classical truth-value that is “contaminating” in the sense that whenever the value is assigned to a formula ϕ, any complex formula in which ϕ appears is assigned that value as well. Under such interpretations, the contaminating states represent occurrences of a (...) flaw. However, since different programs and machines can interact with (or be nested into) one another, we need to account for different kind of errors, and this calls for an evaluation of systems with multiple contaminating values. In this paper, we make steps toward these evaluation systems by considering two logics, HYB1 and HYB2, whose semantic interpretations account for two contaminating values beside classical values 0 and 1. In particular, we provide two main formal contributions. First, we give a characterization of their relations of (multiple-conclusion) logical consequence—that is, necessary and sufficient conditions for a set Δ of formulas to logically follow from a set Γ of formulas in HYB1 or HYB2 . Second, we provide sound and complete sequent calculi for the two logics. (shrink)

This document presents a Gentzen-style deductive calculus and proves that it is complete with respect to a 3-valued semantics for a language with quantifiers. The semantics resembles the strong Kleene semantics with respect to conjunction, disjunction and negation. The completeness proof for the sentential fragment fills in the details of a proof sketched in Arnon Avron (2003). The extension to quantifiers is original but uses standard techniques.

Logical Probability (LP) is strictly distinguished from Statistical Probability (SP). To measure semantic information or confirm hypotheses, we need to use sampling distribution (conditional SP function) to test or confirm fuzzy truth function (conditional LP function). The Semantic Information Measure (SIM) proposed is compatible with Shannon’s information theory and Fisher’s likelihood method. It can ensure that the less the LP of a predicate is and the larger the true value of the proposition is, the more information there is. So the (...) SIM can be used as Popper's information criterion for falsification or test. The SIM also allows us to optimize the true-value of counterexamples or degrees of disbelief in a hypothesis to get the optimized degree of belief, i. e. Degree of Confirmation (DOC). To explain confirmation, this paper 1) provides the calculation method of the DOC of universal hypotheses; 2) discusses how to resolve Raven Paradox with new DOC and its increment; 3) derives the DOC of rapid HIV tests: DOC of “+” =1-(1-specificity)/sensitivity, which is similar to Likelihood Ratio (=sensitivity/(1-specificity)) but has the upper limit 1; 4) discusses negative DOC for excessive affirmations, wrong hypotheses, or lies; and 5) discusses the DOC of general hypotheses with GPS as example. (shrink)

By pure calculus of names we mean a quantifier-free theory, based on the classical propositional calculus, which defines predicates known from Aristotle’s syllogistic and Leśniewski’s Ontology. For a large fragment of the theory decision procedures, defined by a combination of simple syntactic operations and models in two-membered domains, can be used. We compare the system which employs `ε’ as the only specific term with the system enriched with functors of Syllogistic. In the former, we do not need an empty name (...) in the model, so we are able to construct a 3-valued matrix, while for the latter, for which an empty name is necessary, the respective matrices are 4-valued. (shrink)

The purpose of this paper is to demonstrate how Bertrand Russell depicts the relation of mysticism to three other areas of human activity: philosophy, science, and religion, and thus: its special role. In his essay Mysticism and Logic (1914), Russell defines mystical thinking as beliefs in (1) the existence of special insight, (2) the unity of all things, (3) the unreality of time, and (4) the effacement of the boundaries between good and evil. Although he considers full mysticism – (...) as a belief about the ontic structure of the world – to be erroneous – as a life attitude he attributes to it an element of wisdom that is lacking in other areas of human intellectual activity. Mysticism proves wrong also at the epistemological level, i.e. as a certain test of truth. But in its spirit of inquiry, it contains something that science also benefits from, and from which, therefore, scientific philosophy should take its cue. What additionally gives mysticism its value is the claim of impartial contemplation which gives rise to an attitude of love towards the whole world. Significantly, it was the combination of the best features of mystical thinking with scientific thinking and method which gave rise to Russell’s advocacy of scientific philosophy. In light of this, I argue that according to Russell mysticism is not an exclusively religious phenomenon, which is demonstrated, among other things, by the fact that he attributes two aspects of mysticism to the field of mathematics. (shrink)

Formal systems are standardly envisaged in terms of a grammar specifying well-formed formulae together with a set of axioms and rules. Derivations are ordered lists of formulae each of which is either an axiom or is generated from earlier items on the list by means of the rules of the system; the theorems of a formal system are simply those formulae for which there are derivations. Here we outline a set of alternative and explicitly visual ways of envisaging and analyzing (...) at least simple formal systems using fractal patterns of infinite depth. Progressively deeper dimensions of such a fractal can be used to map increasingly complex wffs or increasingly complex 'value spaces', with tautologies, contradictions, and various forms of contingency coded in terms of color. This and related approaches, it turns out, offer not only visually immediate and geometrically intriguing representations of formal systems as a whole but also promising formal links (1) between standard systems and classical patterns in fractal geometry, (2) between quite different kinds of value spaces in classical and infinite-valued logics, and (3) between cellular automata and logic. It is hoped that pattern analysis of this kind may open possibilities for a geometrical approach to further questions within logic and metalogic. (shrink)

This paper outlines the fundamental of a new Pragmatics, formulated in a new conceptual framework, including a new normative system - Conversational Etiquette. It's claimed that it does better than the Gricean system or its offshoots, and covers a much broader domain. It's main new concepts are: Steering-Thrust; Posting; and Pragmatic Stances. Its main applications are: 1. Assertion is a Pragmatic construction, which invokes a raise in the degree of Steering Thrust. 2. 'presupposition' as used in Logic/Linguistics (or what's (...) in the 'Common Ground') is analyzed in terms of the new Pragmatics: The main phenomenon involves is: Posting-without Steering. 3. There is no Pragmatic Encroachment (into the semantics of 'know'). Fluctuations in knowledge-ascriptions in High/Low Stakes convey a DIFFERENT dominant Pragmatic content that drives our intuitions, which track so-called 'Sayability' rather than Truth-Values. 4. Pragmatic Inconsistencies amount to Steering-Thrusts in opposite directions (polarity). Moorean sentences invoke Pragmatic Inconsistencies. 5. Misleading is a Pragmatic phenomenon. (shrink)

2nd edition. Many-valued logics are those logics that have more than the two classical truth values, to wit, true and false; in fact, they can have from three to infinitely many truth values. This property, together with truth-functionality, provides a powerful formalism to reason in settings where classical logic—as well as other non-classical logics—is of no avail. Indeed, originally motivated by philosophical concerns, these logics soon proved relevant for a plethora of applications ranging from switching theory to cognitive (...) modeling, and they are today in more demand than ever, due to the realization that inconsistency and vagueness in knowledge bases and information processes are not only inevitable and acceptable, but also perhaps welcome. The main modern applications of (any) logic are to be found in the digital computer, and we thus require the practical knowledge how to computerize—which also means automate—decisions (i.e. reasoning) in many-valued logics. This, in turn, necessitates a mathematical foundation for these logics. This book provides both these mathematical foundation and practical knowledge in a rigorous, yet accessible, text, while at the same time situating these logics in the context of the satisfiability problem (SAT) and automated deduction. The main text is complemented with a large selection of exercises, a plus for the reader wishing to not only learn about, but also do something with, many-valued logics. (shrink)

I have three main objectives in this essay. First, in chapter 2, I shall put forward and justify what I call worldlessness, by which I mean the following: All truths (as well as falsehoods) are wholly independent of any circumstances, not only time and place but also possible worlds. It follows from this view that whatever is actually true must be taken as true with respect to every possible world, which means that all truths are (in a sense) necessary. However, (...) the account I shall propound is different from what is known in the trade as necessitarianism, i.e. the view that there is only one possible world, viz. the actual one, for the doctrine of the worldlessness of truth values, despite its commitment to the necessity of truths and falsehoods, is quite compatible with the idea of there being other possible worlds. Another important issue in chapter 2, explored in particular in section 2.12, is the claim that there is no real change in the world. Secondly, in chapter 3 I consider the eminent traditional argument for determinism, deriving from Aristotle, namely, logical determinism, i.e. determinism justified by an appeal to the logical principle of bivalence (that all proper statements, including those concerning the future, are either true or false). In this connection I try to show that, (i), the formulation of the conclusion of this argument as "Whatever will happen will happen of necessity" is implausible, at least from the modern point of view, (ii), the formulation as "Whatever will happen will happen inevitably" is more to the point, and (iii), on the basis of the worldless and timeless aspect advocated in chapter 2, this latter formulation is quite harmless, essentially amounting to the trivial statement, "Whatever will happen will happen". Thirdly, in chapter 4 I study theological determinism, or determinism that arises from God's supposed providential control over everything that happens. In this connection, I shall survey some historical accounts of the relation between human free will and determinism (not only theological but also causal determinism); the philosophers the views of whom I shall attend to include Chrysippus, St. Augustine, Boethius and Aquinas. I shall in particular consider G.W. Leibniz' theodicean aspirations, viz. his solution to the problem of evil and, especially, his compatibilist attempts to reconcile human free will with the strictly deterministic flow of actual events. I think it is important to try to explicate Leibniz' ingenious account of these matters, since it seems that it has not been fully appreciated in the literature, not even by contemporary Leibniz scholars (such as B. Mates, R.C. Sleigh, C. Wilson, R.M. Adams and D. Rutherford). In providing the Leibnizian compatibilist solution of the problem of determinism and freedom in chapter 4, I shall utilize the approach of chapter 2. (shrink)

Enterprise Risk Management involves the process of identification, evaluation, treatment, and communication regarding risks throughout the enterprise. To support the tasks associated with this process, several frameworks and modeling languages have been proposed, such as the Risk and Security Overlay (RSO) of ArchiMate. An ontological investigation of this artifact would reveal its adequacy, capabilities, and limitations w.r.t. the domain of risk and security. Based on that, a language redesign can be proposed as a refinement. Such analysis and redesign have been (...) executed for the risk elements of the RSO grounded in the Common Ontology of Value and Risk. The next step along this line of research is to address the following research problems: What would be the outcome of an ontological analysis of security-related elements of the RSO? That is, can we identify other semantic deficiencies in the RSO through an ontological analysis? Once such an analysis is provided, can we redesign the security elements of the RSO accordingly, in order to produce an improved artifact? Here, with the aid of the Reference Ontology for Security Engineering (ROSE) and the ontological theory of prevention behind it, we address the remaining gap by proceeding with an ontological analysis of the security-related constructs of the RSO. The outcome of this assessment is an ontology-based redesign of the ArchiMate language regarding security modeling. In a nutshell, we report the following contributions: (1) an ontological analysis of the RSO that identifies six limitations concerning security modeling; (2) because of the key role of the notion of prevention in security modeling, the introduction of the ontological theory of prevention in ArchiMate; (3) a well-founded redesign of security elements of ArchiMate; and (4) ontology-based security modeling patterns that are logical consequences of our proposal of redesign due to its underlying ontology of security. As a form of evaluation, we show that our proposal can describe risk treatment options, according to ISO 31000. Finally, besides presenting multiple examples, we proceed with a real-world illustrative application taken from the cybersecurity domain. (shrink)

This paper analyzes selected examples of uses of argumentation tactics that exploit emotive language, many of them criticized as deceptive and even fallacious by classical and recent sources, including current informal logic textbooks. The analysis is based on six argumentation schemes, and an account of the dialectical setting in which these schemes are used. The three conclusions are (1) that such uses of emotive language are often reasonable and necessary in argumentation based on values, (2) but that they are (...) defeasible, and hence need to be seen as open to critical questioning (3) and that when they are used fallaciously, it is because they interfere with critical questioning or conceal the need for it. The analysis furnishes criteria for distinguishing between arguments based on the use of emotive words that are reasonable tools of persuasion, and those that are fallacious tactics used to conceal and distort information. (shrink)

W. V. Quine’s criterion of ontological commitment is commonly presented as the slogan: “to be is to be the value of a variable”. More specifically, to be is to be in the domain of values of a bound variable attached to a existential quantifier in the body of a theory. If a sentence of our best available theory quantifies over bachelors, so we conclude that, according to it, there are bachelors. Simplicity and the use of logical apparatus for determining this (...) criterion are responsible for many philosophers to adhere to this standpoint within analytical metaphysics. I will argue that despite this strong influence of the Quinean criterion of ontological commitment in many debates, we find at least three interlinked problems in its original formulation: (1) The criterion depends of a clear definition of theory for the determination of the quantifications counting as entailers of ontological commitments. However, Quine does not present such clear concept of what would be a theory and of how to formalize it. Quine develops this notion under just the metaphor of the knowledge fabric, in which we have a group of sentences, which does not suffice for the required account of ontological commitment. (2) There are distinct possibilities of quantifying over given sentences, which leaves room for different ontological commitments associated with the same statement. In that sense, someone could argue that the Quinean solution would be to endorse just the first order predicate logic. Nonetheless, the Quinean should explain why we ought to be restricted to this logic. (3) The previous problems lead to a possible explosion of the Quinean criterion of ontological commitment, making it trivial, because it permits the quantification over any supposed entities and, hence, that anything can exist. Thus, I will discuss Quine’s criterion of ontological commitment in the light of the abovementioned problems, showing that, unless the Quinean presents a more specific definition of theory beyond just a group of sentences, he leads to an explosion of commitments with different entities. I will conclude that, if you understand ‘theory’ as Alonzo Church’s concept of ‘formalized language’, this knot can be undone. However, Church’s concept is much more akin to Carnap’s linguistic framework than to a notion of knowledge fabric “without borders” as the one proposed by Quine. (shrink)

I intend to: a) clarify the origins and de facto meanings of the term relativism; b) reconstruct the reasons for the birth of the thesis named “cultural relativism”; d) reconstruct ethical implications of the above thesis; c) revisit the recent discussion between universalists and particularists in the light of the idea of cultural relativism.. -/- 1.Prescriptive Moral Relativism: “everybody is justified in acting in the way imposed by criteria accepted by the group he belongs to”. Universalism: there are at least (...) some judgments which are valid inter-culturally Absolutism: there are at least some particular prescriptions which are valid without exception everywhere and always -/- 2. The traditional proof of prescriptive moral relativism: the argument from variability: Judgments, rules, and shared values are de facto variable in time and space. The traditional counter-proof: examples of variability do not prove what skeptics contend. -/- 3. Pre-history of the doctrine -Ancient sophists: either immoralist or contractualist -Modern moral scepticism (xvii c.): variability as an historical and ethnographic fact supports a sceptical conclusion more moderate than sheer immoralism. - Voltaire, Kant, Reid counter-attack pointing at a universally shared moral sense - Romantics and idealists stage an even more moderate reformulation: instead of universally shared moral sense they point at the Spirit of a People which is: a)alternative to abstract and universal philosophical systems as far as it is lived ‘culture’; b) indivisible unity with an inner harmony and a source of normative standards; c) dynamic, in so far as it is a manifestation of the Spirit through the becoming of National cultures. -/- 4. The birth of Cultural Relativism and its ethical implications 4.1. The 18th c. doctrine was the noble savage (a non-historical doctrine: state of nature vs. social state) 4.2 Edward Tylor (1832-1817) and ethnocentric historicism Savage moral standards are real enough, but they are far and weaker than ours. 4.3 Boas and Malinowski and an holistic reaction to ethnocentric historicism -/- Franz Boas (1858-1942): a) Development of civilizations is not ruled by technical progress nor does it follow a one-way path; instead there are parallel developments (for ex. Agriculture does not follow stock-raising); b) racial characters have no relevance in development of civilization; c) we are not yet in a position to compare externally identical kinds of behaviour till we have not yet understood beliefs and intentions laying at their roots (for ex.: “From an ethnological point of view murder cannot be considered as a single phenomenon”; d) we should distinguish among different practices which are only superficially similar (fro ex. practices traditionally classified under the label “tabù”); e) there is as a fact just one normative ethic, constant in its contents but varying in its extension; f) the implication is not that we cannot judge behavior by members of other groups; it is only a recommendation of caution. -/- Bronislaw Malinowski (1884-1942): a) against Tylor’s and Frazer’s “magpie” methodology, field-work is required, a culture as a whole should be observed from inside; individual elements are incomprehensible; b) a culture is an organic whole; c) its elements are accounted for by their function (economy), avoiding non-observables (empio-criticism). -/- Ruth Benedict and Melville Herskovitz identify Boas’s approach with “cultural relativism”. Benedict: what is normal and abnormal is to be judged on a culture’s own standards, not on our own (“Anthropology and the Abnormal”). Herskovits: “Boas adumbrates what we have come to call cultural relativism” (The Mind, p. 10); “Judgements are based on experience, and experience is interpreted by each individual in terms of his own enculturation” (Man and his Works). -/- 4. How analytic philosophy understood and misunderstood the discussion 4.1. At the beginning of the 20th c., the new view in ethics was non-cognitivism (emotivist and subjectivist). Eric Westermark combines this view with an old-style ethnographic approach in support of relativity of moralities. Moralities are codes, or systems of emotive ‘disinterested’ reactions selected by evolution on their usefulness in terms of survival value for the society that is the carrier of such systems or codes. The moral relativity thesis: there are cases of disagreement that cannot be settled even after agreement about facts. 4.2 Anti-realists Brandt, Mackie, Gilbert, Harman adopt Westermark’s approach in a more sophisticated version: a) moralities are codes with an overall function and may be appraised only as wholes; b) variability is an argument for moral subjectivism; c) apparent legitimacy of deriving shift from ought is legitimized only within one institution d) morality should not be described but instead made, and existing moralities may be improved. Is it ‘real’ relativism? It is clearly subjectivism (a metaethical thesis). The normative thesis is that there better and worse codes, and survival values is the normative standard. -/- 4.3 Particularists MacIntyre, Sandel, Taylor, Wiggins, McDowell ‘Wittgensteinian’ prospectivist arguments bent to support weak-relativist claims MacIntyre: there is ‘incommensurability’ between different theoretical systems in both science and ethics. No argument is possible through different systems Different traditions may coexist for a long time without being able to bring their conflicts to a rational solution. -/- 4.4 Kantian universalists Baier, Gewirth, Rawls, Apel, Habermas Shared claim: justice concerns the right and is universal in so far as it may be based on minimal assumptions Other virtues are relative to context in so far as they are related to comprehensive views of the good - O’Neill criticism: a) it is an assumption shared by both alignments; b) after an alleged crisis brought about by alleged loss of metaphysical certainties, theories of justice have dropped demanding assumptions and kept universalism, virtue theories have kept demanding assumptions and dropped universalism; c) the opposition of virtue and justice has arisen in an unjustified way. O’Neill’s positive proposal: ‘constructive’ procedures may be adopted both (i) concerning all the range of virtues and (ii) across cultures once we abandon idealization and confine ourselves to abstraction from real-world cases. -/- 4.5 A metaethical relativist and anti-relativist normative ethicists: Bernard Williams Williams: vulgar relativism may be assumed to claim that: a) 'just' means 'just in a given society'; b) 'just in a given society' is to be understood in functionalist sense; c) it is wrong for one society’s members to condemn another society’s values. It is inconsistent since in (c) uses ‘just’ in a non-relative way that has been excluded in (a). William’s positive proposal: i) keep a number of substantive or thick ethical concepts that will be different in space and time; ii) admit that public choices are to be legitimized through recourse to more abstract procedures and relying on more thin ethical concepts. -/- 5. Critical remarks 5.1 The only real relativism available is ‘vulgar’ relativism (Westermark?) 5.2. Descriptive universalism (or absolutism) has a long pedigree, from Cicero on, reaching Boas himself but it is useless as an answer to normative questions 5.3. Twentieth-century philosophical discussion seems to discuss an ad hoc doctrine reconstructed by assembling obsolete philosophical ideas but ignoring the real theory of cultural relativism as formulated by anthropologists. -/- 6. A distinction between ethoi and ethical theories as a way out of confusions a)There are systems of conventions de facto existing. These may be studies from outside as phenomena or facts. b)There is moral argument and this, when studies from outside, is a fact, but this does not influence in any degree the possible validity of claims advanced. c) the difference between the above claims and Mackie’s criticism to Searle’s argument of the promising game is that promises, arguments etc. are also phenomena, but they are also communicative phenomena with a logical and pragmatic structure. -/- 7.Conclusions: a) cultural relativism, as a name for Boas’s methodology is a valuable discovery, and in this sense we are all relativists; b) ethical relativism, as an alleged implication of cultural relativism, has been argued in a philosophically quite unsophisticated way by Benedict and Herskovits; philosophers apparently discussed ethical relativism in the basis of a rather faint impression of what cultural relativism had been. c) a full-fledged ethical relativism has hardly been defended by anybody among philosophers; virtually no modern philosopher really argued a prescriptive version of the thesis; d) we may accept the grain of truth in ethical relativism by including relativist critique to ethical absolutism into a universalist normative doctrine that be careful in separating open-textured formulations of universal claims from culturally conditioned particular prescriptions. -/- . 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