Does it make sense to employ modern logical tools for ancient philosophy? This well-known debate2 has been re-launched by the indologist Piotr Balcerowicz, questioning those who want to look at the Eastern school of Jainism with Western glasses. While plainly acknowledging the legitimacy of Balcerowicz's mistrust, the present paper wants to propose a formal reconstruction of one of the well-known parts of the Jaina philosophy, namely: the saptabhangi, i.e. the theory of sevenfold predication. Before arguing for this formalist approach to (...) philosophy, let us return to the reasons to be reluctant at it. (shrink)

In this paper we re-assess the philosophical foundation of Exactly True Logic ($$\mathcal {ET\!L}$$ ET L ), a competing variant of First Degree Entailment ($$\mathcal {FDE}$$ FDE ). In order to do this, we first rebut an argument against it. As the argument appears in an interview with Nuel Belnap himself, one of the fathers of $$\mathcal {FDE}$$ FDE, we believe its provenance to be such that it needs to be taken seriously. We submit, however, that the argument ultimately (...) fails, and that $$\mathcal {ET\!L}$$ ET L cannot easily be dismissed. We then proceed to give an overview of the research that was inspired by this logic over the last decade, thus providing further motivation for the study of $$\mathcal {ET\!L}$$ ET L and, more generally, of $$\mathcal {FDE}$$ FDE -related logics that result from semantical analyses alternative to Belnap’s canonical one. We focus, in particular, on philosophical questions that these developments raise. (shrink)

The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory (...) of finite-valued first order logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics. (shrink)

The aim of this paper is to emphasize the fact that for all finitely-many-valued logics there is a completely systematic relation between sequent calculi and tableau systems. More importantly, we show that for both of these systems there are al- ways two dual proof sytems (not just only two ways to interpret the calculi). This phenomenon may easily escape one’s attention since in the classical (two-valued) case the two systems coincide. (In two-valued logic the assignment of (...) a truth value and the exclusion of the opposite truth value describe the same situation.). (shrink)

The problem of approximating a propositional calculus is to find many-valued logics which are sound for the calculus (i.e., all theorems of the calculus are tautologies) with as few tautologies as possible. This has potential applications for representing (computationally complex) logics used in AI by (computationally easy) many-valued logics. It is investigated how far this method can be carried using (1) one or (2) an infinite sequence of many-valued logics. It is shown that the optimal candidate matrices (...) for (1) can be computed from the calculus. (shrink)

A wide family of many-valued logics—for instance, those based on the weak Kleene algebra—includes 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. In such systems, the contaminating value enjoys a wide range of interpretations, suggesting scenarios in which more than one of these interpretations are called for. This calls for an evaluation of systems with multiple (...) contaminating values. In this paper, we consider the countably infinite family of multiple-conclusion consequence relations in which classical logic is enriched with one or more contaminating values whose behavior is determined by a linear ordering between them. We consider some motivations and applications for such systems and provide general characterizations for all consequence relations in this family. Finally, we provide sequent calculi for a pair of four-valued logics including two linearly ordered contaminating values before defining two-sided sequent calculi corresponding to each of the infinite family of many-valued logics studied in this paper. (shrink)

The thesis that the two-valued system of classical logic is insufficient to explanation the various intermediate situations in the entity, has led to the development of many-valued and fuzzy logic systems. These systems suggest that this limitation is incorrect. They oppose the law of excluded middle (tertium non datur) which is one of the basic principles of classical logic, and even principle of non-contradiction and argue that is not an obstacle for things both to exist (...) and to not exist at the same time. However, contrary to these claims, there is no inadequacy in the two-valued system of classical logic in explanation the intermediate situations in existence. The law of exclusion and the intermediate situations in the external world are separate things. The law of excluded middle has been inevitably accepted by other logic systems which are considered to reject this principle. The many-valued and the fuzzy logic systems do not transcend the classical logic. Misconceptions from incomplete information and incomplete research are effective in these criticisms. In addition, it is also effective to move the discussion about the intellectual conception (tasawwur) into the field of judgmental assent (tasdiq) and confusion of the mawhum (imaginable) with the ma‘kûl (intellegible). (shrink)

Given a three-valued definition of validity, which choice of three-valued truth tables for the connectives can ensure that the resulting logic coincides exactly with classical logic? We give an answer to this question for the five monotonic consequence relations $st$, $ss$, $tt$, $ss\cap tt$, and $ts$, when the connectives are negation, conjunction, and disjunction. For $ts$ and $ss\cap tt$ the answer is trivial (no scheme works), and for $ss$ and $tt$ it is straightforward (they are the (...) collapsible schemes, in which the middle value acts like one of the classical values). For $st$, the schemes in question are the Boolean normal schemes that are either monotonic or collapsible. (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)

Plural expressions found in natural languages allow us to talk about many objects simultaneously. Plural logic — a logical system that takes plurals at face value — has seen a surge of interest in recent years. This book explores its broader significance for philosophy, logic, and linguistics. What can plural logic do for us? Are the bold claims made on its behalf correct? After introducing plural logic and its main applications, the book provides a systematic analysis (...) of the relation between this logic and other theoretical frameworks such as set theory, mereology, higher-order logic, and modal logic. The applications of plural logic rely on two assumptions, namely that this logic is ontologically innocent and has great expressive power. These assumptions are shown to be problematic. The result is a more nuanced picture of plural logic's applications than has been given thus far. Questions about the correct logic of plurals play a central role in the final chapters, where traditional plural logic is rejected in favor of a "critical" alternative. The most striking feature of this alternative is that there is no universal plurality. This leads to a novel approach to the relation between the many and the one. In particular, critical plural logic paves the way for an account of sets capable of solving the set-theoretic paradoxes. (shrink)

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)

This article explores the question of how the members of the Lvov-Warsaw School promoted values that can be regarded as components of so-called logical culture. The author argues that these values are strictly connected with science. With references to Łukasiewicz, Czeżowski, and Kotarbiński,the article explores how values shape the logical culture and determines society as directed towards values. The article connects the meta-philosophical perspective with the philosophical one.

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)

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)

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)

This book treats ancient logic: the logic that originated in Greece by Aristotle and the Stoics, mainly in the hundred year period beginning about 350 BCE. Ancient logic was never completely ignored by modern logic from its Boolean origin in the middle 1800s: it was prominent in Boole’s writings and it was mentioned by Frege and by Hilbert. Nevertheless, the first century of mathematical logic did not take it seriously enough to study the ancient (...) class='Hi'>logic texts. A renaissance in ancient logic studies occurred in the early 1950s with the publication of the landmark Aristotle’s Syllogistic by Jan Łukasiewicz, Oxford UP 1951, 2nd ed. 1957. Despite its title, it treats the logic of the Stoics as well as that of Aristotle. Łukasiewicz was a distinguished mathematical logician. He had created many-valued logic and the parenthesis-free prefix notation known as Polish notation. He co-authored with Alfred Tarski’s an important paper on metatheory of propositional logic and he was one of Tarski’s the three main teachers at the University of Warsaw. Łukasiewicz’s stature was just short of that of the giants: Aristotle, Boole, Frege, Tarski and Gödel. No mathematical logician of his caliber had ever before quoted the actual teachings of ancient logicians. -/- Not only did Łukasiewicz inject fresh hypotheses, new concepts, and imaginative modern perspectives into the field, his enormous prestige and that of the Warsaw School of Logic reflected on the whole field of ancient logic studies. Suddenly, this previously somewhat dormant and obscure field became active and gained in respectability and importance in the eyes of logicians, mathematicians, linguists, analytic philosophers, and historians. Next to Aristotle himself and perhaps the Stoic logician Chrysippus, Łukasiewicz is the most prominent figure in ancient logic studies. A huge literature traces its origins to Łukasiewicz. -/- This Ancient Logic and Its Modern Interpretations, is based on the 1973 Buffalo Symposium on Modernist Interpretations of Ancient Logic, the first conference devoted entirely to critical assessment of the state of ancient logic studies. (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)

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)

According to one tradition, the virtues and vices should be understood in terms of their relation to value. But inside this tradition, there are three distinct proposals: virtues are intrinsically valuable; virtues are instrumentally valuable; or a hybrid proposal on which virtues are either intrinsically or instrumentally valuable. In this paper, I offer an alternative proposal inside this tradition. I propose that virtues and vices should be understood in terms of the degreed properties of being virtuous and being vicious, which (...) I analyze in terms of the value and disvalue of actions and attitudes. I defend my proposal as the best inside this tradition by showing how it is immune from standard problems the other three proposals face. (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)

The thesis defended, the “guise of the ought”, is that the formal objects of desires are norms (oughts to be or oughts to do) rather than values (as the “guise of the good” thesis has it). It is impossible, in virtue of the nature of desire, to desire something without it being presented as something that ought to be or that one ought to do. This view is defended by pointing to a key distinction between values and norms: positive and (...) negative norms (obligation and interdiction) are interdefinable through negation; positive and negative values aren’t. This contrast between the norms and values, it is argued, is mirrored, within the psychological realm, by the contrast between the desires and emotions. Positive and negative desires are interdefinable through negation, but positive and negative emotions aren’t. The overall, Meinongian picture suggested is that norms are to desires what values are to emotions. (shrink)

We analyze the logical form of the domain knowledge that grounds analogical inferences and generalizations from a single instance. The form of the assumptions which justify analogies is given schematically as the "determination rule", so called because it expresses the relation of one set of variables determining the values of another set. The determination relation is a logical generalization of the different types of dependency relations defined in database theory. Specifically, we define determination as a relation between schemata of first (...) order logic that have two kinds of free variables: (1) object variables and (2) what we call "polar" variables, which hold the place of truth values. Determination rules facilitate sound rule inference and valid conclusions projected by analogy from single instances, without implying what the conclusion should be prior to an inspection of the instance. They also provide a way to specify what information is sufficiently relevant to decide a question, prior to knowledge of the answer to the question. (shrink)

Quantum information is discussed as the universal substance of the world. It is interpreted as that generalization of classical information, which includes both finite and transfinite ordinal numbers. On the other hand, any wave function and thus any state of any quantum system is just one value of quantum information. Information and its generalization as quantum information are considered as quantities of elementary choices. Their units are correspondingly a bit and a qubit. The course of time is what generates choices (...) by itself, thus quantum information and any item in the world in final analysis. The course of time generates necessarily choices so: The future is absolutely unorderable in principle while the past is always well-ordered and thus unchangeable. The present as the mediation between them needs the well-ordered theorem equivalent to the axiom of choice. The latter guarantees the choice even among the elements of an infinite set, which is the case of quantum information. The concrete and abstract objects share information as their common base, which is quantum as to the formers and classical as to the latters. The general quantities of matter in physics, mass and energy can be considered as particular cases of quantum information. The link between choice and abstraction in set theory allows of “Hume’s principle” to be interpreted in terms of quantum mechanics as equivalence of “many” and “much” underlying quantum information. Quantum information as the universal substance of the world calls for the unity of physics and mathematics rather than that of the concrete and abstract objects and thus for a form of quantum neo-Pythagoreanism in final analysis. (shrink)

There is some consensus on the claim that imagination as suppositional thinking can have epistemic value insofar as it’s constrained by a principle of minimal alteration of how we know or believe reality to be – compatibly with the need to accommodate the supposition initiating the imaginative exercise. But in the philosophy of imagination there is no formally precise account of how exactly such minimal alteration is to work. I propose one. I focus on counterfactual imagination, arguing that this can (...) be modeled as simulated belief revision governed by Laplacian imaging. So understood, it can be rationally justified by accuracy considerations: it minimizes expected belief inaccuracy, as measured by the Brier score. (shrink)

This paper presents a semantical analysis of the Weak Kleene Logics Kw3 and PWK from the tradition of Bochvar and Halldén. These are three-valued logics in which a formula takes the third value if at least one of its components does. The paper establishes two main results: a characterisation result for the relation of logical con- sequence in PWK – that is, we individuate necessary and sufficient conditions for a set.

One of the primary, if not most critical, difficulties in the design and implementation of autonomous systems is the black-boxed nature of the decision-making structures and logical pathways. How human values are embodied and actualised in situ may ultimately prove to be harmful if not outright recalcitrant. For this reason, the values of stakeholders become of particular significance given the risks posed by opaque structures of intelligent agents (IAs). This paper explores how decision matrix algorithms, via the belief-desire-intention model for (...) autonomous vehicles, can be designed to minimize the risks of opaque architectures. Primarily through an explicit orientation towards designing for the values of explainability and verifiability. In doing so, this research adopts the Value Sensitive Design (VSD) approach as a principled framework for the incorporation of such values within design. VSD is recognized as a potential starting point that offers a systematic way for engineering teams to formally incorporate existing technical solutions within ethical design, while simultaneously remaining pliable to emerging issues and needs. It is concluded that the VSD methodology offers at least a strong enough foundation from which designers can begin to anticipate design needs and formulate salient design flows that can be adapted to the changing ethical landscapes required for utilisation in autonomous vehicles. (shrink)

Intuitionistic logic provides an elegant solution to the Sorites Paradox. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages. Second, switching from classical to intuitionistic logic, while it may help with the Sorites, does not appear to offer any advantages (...) when dealing with the so-called paradoxes of higher-order vagueness. We offer a proposal that makes strides on both issues. We argue that the intuitionist’s characteristic rejection of any third alethic value alongside true and false is best elaborated by taking the normal modal system S4M to be the sentential logic of the operator ‘it is clearly the case that’. S4M opens the way to an account of higher-order vagueness which avoids the paradoxes that have been thought to infect the notion. S4M is one of the modal counterparts of the intuitionistic sentential calculus and we use this fact to explain why IPC is the correct sentential logic to use when reasoning with vague statements. We also show that our key results go through in an intuitionistic version of S4M. Finally, we deploy our analysis to reply to Timothy Williamson’s objections to intuitionistic treatments of vagueness. (shrink)

In this paper we discuss the extent to which conjunction and disjunction can be rightfully regarded as such, in the context of infectious logics. Infectious logics are peculiar many-valued logics whose underlying algebra has an absorbing or infectious element, which is assigned to a compound formula whenever it is assigned to one of its components. To discuss these matters, we review the philosophical motivations for infectious logics due to Bochvar, Halldén, Fitting, Ferguson and Beall, noticing that none of them (...) discusses our main question. This is why we finally turn to the analysis of the truth-conditions for conjunction and disjunction in infectious logics, employing the framework of plurivalent logics, as discussed by Priest. In doing so, we arrive at the interesting conclusion that —in the context of infectious logics— conjunction is conjunction, whereas disjunction is not disjunction. (shrink)

The principle of universal instantiation plays a pivotal role both in the derivation of intensional paradoxes such as Prior’s paradox and Kaplan’s paradox and the debate between necessitism and contingentism. We outline a distinctively free logical approach to the intensional paradoxes and note how the free logical outlook allows one to distinguish two different, though allied themes in higher-order necessitism. We examine the costs of this solution and compare it with the more familiar ramificationist approaches to higher-order logic. Our (...) assessment of both approaches is largely pessimistic, and we remain reluctantly inclined to take Prior’s and Kaplan’s derivations at face value. (shrink)

In this paper, we make distinctions between Classical Logic (where the propositions are 100% true, or 100 false) and the Neutrosophic Logic (where one deals with partially true, partially indeterminate and partially false propositions) in order to respond to K. Georgiev’s criticism [1]. We recall that if an axiom is true in a classical logic system, it is not necessarily that the axiom be valid in a modern (fuzzy, intuitionistic fuzzy, neutrosophic etc.) logic system.

To hold an environmental ethic is to hold that non-human beings and states of affairs in the natural world have intrinsic value. This seemingly straightforward claim has been the focus of much recent philosophical discussion of environmental issues. Its clarity is, however, illusory. The term ‘intrinsic value’ has a variety of senses and many arguments on environmental ethics suffer from a conflation of these different senses: specimen hunters for the fallacy of equivocation will find rich pickings in the area. This (...) paper is largely the work of the underlabourer. I distinguish different senses of the concept of intrinsic value, and, relatedly, of the claim that non-human beings in the natural world have intrinsic value; I exhibit the logical relations between these claims and examine the distinct motivations for holding them. The paper is not however merely an exercise in conceptual underlabouring. It also defends one substantive thesis: that while it is the case that natural entities have intrinsic value in the strongest sense of the term, i.e., in the sense of value that exists independently of human valuations, such value does not as such entail any obligations on the part of human beings. The defender of nature’s intrinsic value still needs to show that such value contributes to the well-being of human agents. (shrink)

By and large, the conditional connective in three-valued logic has two different functions. First, by means of a deduction theorem, it can express a specific relation of logical consequence in the logical language itself. Second, it can represent natural language structures such as "if/then'' or "implies''. This chapter surveys both approaches, shows why none of them will typically end up with a three-valued material conditional, and elaborates on connections to probabilistic reasoning.

Entailment in propositional Gödel logics can be defined in a natural way. While all infinite sets of truth values yield the same sets of tautologies, the entailment relations differ. It is shown that there is a rich structure of infinite-valued Gödel logics, only one of which is compact. It is also shown that the compact infinite-valued Gödel logic is the only one which interpolates, and the only one with an r.e. entailment relation.

Sören Halldén’s logic of nonsense is one of the most well-known many-valued logics available in the literature. In this paper, we discuss Peter Woodruff’s as yet rather unexplored attempt to advance a version of such a logic built on the top of a constructive logical basis. We start by recalling the basics of Woodruff’s system and by bringing to light some of its notable features. We then go on to elaborate on some of the difficulties attached to (...) it; on our way to offer a possible solution to such difficulties, we discuss the relation between Woodruff’s system and two-dimensional semantics for many-valued logics, as developed by Hans Herzberger. (shrink)

We define consciousness as the category of all conscious experiences. This immediately raises the question: What is the essence in which every conscious experience in the category of conscious experiences partakes? We consider various abstract essences of conscious experiences as theories of consciousness. They are: (i) conscious experience is an action of memory on sensation, (ii) conscious experience is experiencing a particular as an exemplar of a general, (iii) conscious experience is an interpretation of sensation, (iv) conscious experience is referring (...) sensation to an object as its cause, and (v) conscious experience is a model of stimulus. Corresponding to each one of these theories we obtain a category of models of conscious experiences: (i) category of actions, (ii) category of idempotents, (iii) category of two sequential maps, (iv) category of brain-generalized figures, and (v) functor categories with intuition as base and conceptual repertoire as exponent, respectively. For each theory of consciousness we also calculate its truth value object and characterize the objective logic intrinsic to the corresponding category of models of consciousness experiences. (shrink)

This paper contributes to explaining the rise of logical empiricism in mid-twentieth century (North) America and to a better understanding of American philosophy of science before the dominance of logical empiricism. We show that, contrary to a number of existing histories, philosophy of science was already a distinct subfield of philosophy, one with its own approaches and issues, even before logical empiricists arrived in America. It was a form of speculative philosophy with a concern for speculative metaphysics, normative issues relating (...) to science and society and issues which later were associated with logical empiricist philosophy of science, issues such as confirmation, scientific explanation, reductionism and laws of nature. Further, philosophy of science was not primarily pragmatist in orientation. We also show, with the help of our historical characterization, that a recent account of the emergence of analytic philosophy applies to the rise of logical empiricism. It has been argued that the emergence of American analytic philosophy is partly explained by analytic philosophers’ use of key institutions, including of journals, to marginalize speculative philosophy and promote analytic philosophy. We argue that this use of institutions included the marginalization of speculative and value-laden philosophy of science and the promotion of logical empiricism. (shrink)

Kuhn's alleged taxonomic interpretation of incommensurability is grounded on an ill defined notion of untranslatability and is hence radically incomplete. To supplement it, I reconstruct Kuhn's taxonomic interpretation on the basis of a logical-semantic theory of taxonomy, a semantic theory of truth-value, and a truth-value conditional theory of cross-language communication. According to the reconstruction, two scientific languages are incommensurable when core sentences of one language, which have truth values when considered within its own context, lack truth values when considered within (...) the context of the other due to the unmatchable taxonomic structures underlying them. So constructed, Kuhn's mature interpretation of incommensurability does not depend upon the notion of truth-preserving translatability, but rather depends on the notion of truth-value-status-preserving cross-language communication. The reconstruction makes Kuhn's notion of incommensurability a well grounded, tenable and integrated notion.Author Keywords: Incommensurability; Thomas Kuhn; Taxonomic structures; Lexicons; Truth-value; Untranslatability; Cross-language communication. (shrink)

This book is written for those who wish to learn some basic principles of formal logic but more importantly learn some easy methods to unpick arguments and assess their value for truth and validity. -/- The first section explains the ideas behind traditional logic which was formed well over two thousand years ago by the ancient Greeks. Terms such as ‘categorical syllogism’, ‘premise’, ‘deduction’ and ‘validity’ may appear at first sight to be inscrutable but will easily be understood (...) with examples bringing the subjects to life. Traditionally, Venn diagrams have been employed to test arguments. These are very useful but their application is limited and they are not open to quantification. The mid-section of this book introduces a methodology that makes the analysis of arguments accessible with the use of a new form of diagram, modified from those of the mathematician Leonhard Euler. These new diagrammatic methods will be employed to demonstrate an addition to the basic form of syllogism. This includes a refined definition of the terms ‘most’ and ‘some’ within propositions. This may seem a little obscure at the moment but one will readily apprehend these new methods and principles of a more modern logic. (shrink)

Among recent objections to Pascal's Wager, two are especially compelling. The first is that decision theory, and specifically the requirement of maximizing expected utility, is incompatible with infinite utility values. The second is that even if infinite utility values are admitted, the argument of the Wager is invalid provided that we allow mixed strategies. Furthermore, Hájek has shown that reformulations of Pascal's Wager that address these criticisms inevitably lead to arguments that are philosophically unsatisfying and historically unfaithful. Both the objections (...) and Hájek's philosophical worries disappear, however, if we represent our preferences using relative utilities rather than a one-place utility function. Relative utilities provide a conservative way to make sense of infinite value that preserves the familiar equation of rationality with the maximization of expected utility. They also provide a means of investigating a broader class of problems related to the Wager. (shrink)

One can construct a mapping between Hilbert space and the class of all logic if the latter is defined as the set of all well-orderings of some relevant set (or class). That mapping can be further interpreted as a mapping of all states of all quantum systems, on the one hand, and all logic, on the other hand. The collection of all states of all quantum systems is equivalent to the world (the universe) as a whole. Thus that (...) mapping establishes a fundamentally philosophical correspondence between the physical world and universal logic by the meditation of a special and fundamental structure, that of Hilbert space, and therefore, between quantum mechanics and logic by mathematics. Furthermore, Hilbert space can be interpreted as the free variable of "quantum information" and any point in it, as a value of the same variable as "bound" already axiom of choice. (shrink)

According to Cantor (Mathematische Annalen 21:545–586, 1883 ; Cantor’s letter to Dedekind, 1899 ) a set is any multitude which can be thought of as one (“jedes Viele, welches sich als Eines denken läßt”) without contradiction—a consistent multitude. Other multitudes are inconsistent or paradoxical. Set theoretical paradoxes have common root—lack of understanding why some multitudes are not sets. Why some multitudes of objects of thought cannot themselves be objects of thought? Moreover, it is a logical truth that such multitudes do (...) exist. However we do not understand this logical truth so well as we understand, for example, the logical truth $${\forall x \, x = x}$$ . In this paper we formulate a logical truth which we call the productivity principle. Rusell (Proc Lond Math Soc 4(2):29–53, 1906 ) was the first one to formulate this principle, but in a restricted form and with a different purpose. The principle explicates a logical mechanism that lies behind paradoxical multitudes, and is understandable as well as any simple logical truth. However, it does not explain the concept of set. It only sets logical bounds of the concept within the framework of the classical two valued $${\in}$$ -language. The principle behaves as a logical regulator of any theory we formulate to explain and describe sets. It provides tools to identify paradoxical classes inside the theory. We show how the known paradoxical classes follow from the productivity principle and how the principle gives us a uniform way to generate new paradoxical classes. In the case of ZFC set theory the productivity principle shows that the limitation of size principles are of a restrictive nature and that they do not explain which classes are sets. The productivity principle, as a logical regulator, can have a definite heuristic role in the development of a consistent set theory. We sketch such a theory—the cumulative cardinal theory of sets. The theory is based on the idea of cardinality of collecting objects into sets. Its development is guided by means of the productivity principle in such a way that its consistency seems plausible. Moreover, the theory inherits good properties from cardinal conception and from cumulative conception of sets. Because of the cardinality principle it can easily justify the replacement axiom, and because of the cumulative property it can easily justify the power set axiom and the union axiom. It would be possible to prove that the cumulative cardinal theory of sets is equivalent to the Morse–Kelley set theory. In this way we provide a natural and plausibly consistent axiomatization for the Morse–Kelley set theory. (shrink)

On one standard philosophical position adopted by evolutionary naturalists, human ethical systems are nothing more than evolutionary adaptations that facilitate social behavior. Belief in an absolute moral foundation is therefore in error. But evolutionary naturalism, by its commitment to the basic valutional concept of fitness, reveals another, logical error: standard conceptions of value in terms of simple predication and properties are mistaken. Valuation has instead, a relational structure that makes reference to respects, subjects and environments. This relational nature is illustrated (...) by the analogy commonly drawn between value and color. Color perception, as recognized by the ecological concept, is relational and dependent on subject and environment. In a similar way, value is relational and dependent on subject and environment. This makes value subjective, but also objective in that it is grounded on facts about mattering. At bottom, values are complex relational facts. The view presented here, unlike other prominent relational and naturalistic conceptions of value, recognizes the full range of valuation in nature. The advantages of this relational conception are first, that it gets valuation right; second, it provides a framework to better explain and understand valuation in all its varieties and patterns. (shrink)

ABSTRACT: In its strongest unqualified form, the principle of wholistic reference is that in any given discourse, each proposition refers to the whole universe of that discourse, regardless of how limited the referents of its non-logical or content terms. According to this principle every proposition of number theory, even an equation such as "5 + 7 = 12", refers not only to the individual numbers that it happens to mention but to the whole universe of numbers. This principle, its history, (...) and its relevance to some of Oswaldo Chateaubriand's work are discussed in my 2004 paper "The Principle of Wholistic Reference" in Essays on Chateaubriand's "Logical Forms". In Chateaubriand's réplica (reply), which is printed with my paper, he raised several important additional issues including the three I focus on in this tréplica (reply to his reply): truth-values, universes of discourse, and formal ontology. This paper is self-contained: it is not necessary to have read the above-mentioned works. The principle of wholistic reference (PWR) was first put forth by George Boole in 1847 when he espoused a monistic fixed-universe viewpoint similar to the one Frege and Russell espoused throughout their careers. Later, Boole elaborated PWR in 1854 from the pluralistic multiple-universes perspective. (shrink)

Supererogatory acts are those that lie “beyond the call of duty.” There are two standard ways to define this idea more precisely. Although the definitions are often seen as equivalent, I argue that they can diverge when options are infinite, or when there are cycles of better options; moreover, each definition is acceptable in only one case. I consider two ways out of this dilemma.

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)

The truth functional account of conditional statements ‘if A then B’ is not only inadequate; it eliminates the very conditionality expressed by ‘if’. Focusing only on the truth-values of the statements ‘A’ and ‘B’ and different combinations of these, one is bound to miss out on the conditional relation expressed between them. All approaches that treat conditionals as functions of their antecedents and consequents will end up in some sort of logical atomism where causal matters simply are reduced to the (...) joint occurrence of A and B. (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)

The phenomenon of self-love elicits conflicting reactions. Some believe it is the key to happiness, while others are skeptical. This essay defines self-love as wholehearted concern for one's well-being, argues that it does not imply selfishness, arrogance, or vanity, discusses reasons to value self-love, and describes ways Logic-Based Therapy can be used to help people love themselves.

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)

This essay is part of larger project in which I attempt to show that Western formal logic, from its inception in Aristotle onward, has both been partially constituted by, and partially constitutive of, what has become known as racism. More specifically, (a) racist/quasi-racist/proto-racist political forces were part of the impetus for logic’s attempt to classify the world into mutually exclusive, hierarchically-valued categories in the first place; and (b) these classifications, in turn, have been deployed throughout history to (...) justify and empower racism/quasi-racism/proto-racism. In other words, (a) an important part of the chaos and messiness that so troubles logic has been the natural bio-cultural diversity of human beings as described by concepts such as race and ethnicity; and (b) once these concepts were historically in place, taken for granted, buttressed with pseudo-science, and had their origins forgotten, they became tools for the oppression of diverse human groups. The central thesis of this essay is that close readings of a variety of Leibniz’s writings (including several essays and one letter) will reveal a pattern of mutual dependency and creation in regard to logic and racism in Leibniz. My conclusion will be that there is a meaningful connection between logic and racism (or at least between logic and politics in general) in Leibniz such that one should be skeptical of his logic’s claim to offer transparent and neutral descriptions of the world as well as imperatives for action. Put differently, I want to ask the following specific question: how do Leibniz’s own racially-informed political goals, along with the emergence of race-centric European colonialism, both invest and disguise themselves within Leibniz’s mathematizing of logic? (shrink)