This paper develops a trivalent semantics for indicative conditionals and extends it to a probabilistic theory of valid inference and inductive learning with conditionals. On this account, (i) all complex conditionals can be rephrased as simple conditionals, connecting our account to Adams's theory of p-valid inference; (ii) we obtain Stalnaker's Thesis as a theorem while avoiding the well-known triviality results; (iii) we generalize Bayesian conditionalization to an updating principle for conditional sentences. The final result is a unified semantic and probabilistic (...) theory of conditionals with attractive results and predictions. (shrink)

I survey the role of trivalence in theories of anaphora.

The claim that is examined in this chapter is that, as is bivalent logic, trivalent logic occupies a place in the field of logic. A trivalent logic is a three-value logical system, and a bivalent logic is a two-value logical system. As part of advancing this claim, the chapter uses the examples of trivalent logic in Charles Sanders Peirce’s thought, the trivalent logic of Janus, the Aymará trivalent logical system, and African trivalent logic. Using the example of ancestorhood, where characteristically (...) an ancestor, as a living dead, is both a spiritual and physical entity or is considered neither a spiritual nor physical being, the overarching view or thesis that is defended and advanced is that African trivalent logic mirrors a trivalent African metaphysics or ontology. (shrink)

Michael Tye responds to the problem of higher-order vagueness for his trivalent semantics by maintaining that truth-value predicates are “vaguely vague”: it’s indeterminate, on his view, whether they have borderline cases and therefore indeterminate whether every sentence is true, false, or indefinite. Rosanna Keefe objects (1) that Tye’s argument for this claim tacitly assumes that every sentence is true, false, or indefinite, and (2) that the conclusion is any case not viable. I argue – contra (1) – that Tye’s argument (...) needn’t make that assumption. A version of her objection is in fact better directed against other arguments Tye advances, though Tye can absorb this criticism without abandoning his position’s core. On the other hand, Keefe’s second objection does hit the mark: embracing ‘vaguely vague’ truth-value predicates undermines Tye’s ability to support validity claims needed to defend his position. To see this, however, we must develop Keefe’s remarks further than she does. (shrink)

We consider language models for the Lambek calculus that allow empty antecedents and enrich them with constants for the empty language and for the language containing only the empty word. No complete calculi are known with respect to these semantics, and in this paper we consider several trivalent systems that arise as fragments of these models? logics.

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)

This paper explores trivalent truth conditions for indicative conditionals, examining the “defective” truth table proposed by de Finetti and Reichenbach. On their approach, a conditional takes the value of its consequent whenever its antecedent is true, and the value Indeterminate otherwise. Here we deal with the problem of selecting an adequate notion of validity for this conditional. We show that all standard validity schemes based on de Finetti’s table come with some problems, and highlight two ways out of the predicament: (...) one pairs de Finetti’s conditional with validity as the preservation of non-false values, but at the expense of Modus Ponens; the other modifies de Finetti’s table to restore Modus Ponens. In Part I of this paper, we present both alternatives, with specific attention to a variant of de Finetti’s table proposed by Cooper and Cantwell. In Part II, we give an in-depth treatment of the proof theory of the resulting logics, DF/TT and CC/TT: both are connexive logics, but with significantly different algebraic properties. (shrink)

This paper discusses some formal properties of trivalent approaches to presupposition projection, and in particular of the middle Kleene system of Peters (1977) and Krahmer (1998). After exploring the relationship between trivalent truth-functional accounts and dynamic accounts in the tradition of Heim (1983), I show how the middle Kleene trivalent account can be formulated in a way which shows that it meets the explanatory challenge of Schlenker (2006, 2008a,b), and provide some results relating to the application of the middle Kleene (...) approach to generalised quantifiers. (shrink)

ABSTRACT A trivalent theory of indicative conditionals automatically enforces Stalnaker's thesis – the equation between probabilities of conditionals and conditional probabilities. This result holds because the trivalent semantics requires, for principled reasons, a modification of the ratio definition of conditional probability in order to accommodate the possibility of undefinedness. I explain how this modification is motivated and how it allows the trivalent semantics to avoid a number of well-known triviality results, in the process clarifying why these results hold for many (...) bivalent theories. In short, the slew of triviality results published in the last 40-odd years need not be viewed as an argument against Stalnaker's thesis: it can be construed instead as an argument for abandoning the bivalent requirement that indicative conditionals somehow be assigned a truth-value in worlds in which their antecedents are false. (shrink)

ABSTRACTThe paper deals with a problem posed by Mathieu Vidal to provide a formal representation for defective conditional in mathematics Vidal, M. [. The defective conditional in mathematics. Journal of Applied Non-Classical Logics, 24, 169–179]. The key feature of defective conditional is that its truth-value is indeterminate if its antecedent is false. In particular, we are interested in two explanations given by Vidal with the use of trivalent logics. By analysing a simple argument from plane geometry, where defective conditional is (...) in use, he gives two trivalent formal explanations for it. For both explanations, Vidal rigorously shows that trivalent logics cannot adequately represent defective conditional. Preserving Vidal's criteria of defective conditional ad max, we indicate some arguable points in his explanations and present an alternative explanation containing the original conjunction and disjunction in order to show that there are trivalent logics that might be an adequate formal explanation for defective conditional. (shrink)

(2014). Capturing the relationship between conditionals and conditional probability with a trivalent semantics. Journal of Applied Non-Classical Logics: Vol. 24, Three-Valued Logics and their Applications, pp. 144-152. doi: 10.1080/11663081.2014.911535.

A quasi-canonical Gentzen-type system is a Gentzen-type system in which each logical rule introduces either a formula of the form , or of the form , and all the active formulas of its premises belong to the set . In this paper we investigate quasi-canonical systems in which exactly one of the two classical rules for negation is included, turning the induced logic into either a paraconsistent logic or a paracomplete logic, but not both. We provide a constructive coherence criterion (...) for such systems, and show that a quasi-canonical system of the type we investigate is coherent iff it is strongly paraconsistent or strongly paracomplete (in a sense defined in the paper), iff it has a trivalent, non-deterministic semantics of a special type (also defined in the paper) for which it is sound and complete. Finally, we determine when a system of this sort admits cut-elimination, and provide a simple procedure for transforming one which does not into one which does. (shrink)

Both the phenomenon of presupposition and that of vagueness have motivated the use of one form or another of trivalent logic, in which a declarative sentence can not only receive the standard values true and false , but also a third, non-standard truth-value which is usually understood as ‘undefined’ . The goal of this paper is to propose a multivalent framework which can deal simultaneously with presupposition and vagueness, and, more specifically, capture their projection properties as well as their different (...) roles in language. Now, there is a prima facie simple way of doing this, which simply consists in assimilating the two phenomena, and using an appropriate type of trivalent logic. On this view, we just need a compositional system that deals with the ‘undefined’ truth-value, and does not care about whether the source of undefinedness is ‘presuppositional’ or related to vagueness. I will argue that such a simple solution cannot succeed, and point out a number of desiderata that any successful approach must meet. I will then present and discuss two seven-valued semantics, inspired, respectively, by the Strong Kleene semantics and by supervaluationism, which meet these desiderata. (shrink)

Regarding the famous Sea Battle Argument, which Aristotle presents in De Interpretatione 9, there has never been a general agreement not only about its correctness but also, and mainly, about what the argument really is. According to the most natural reading of the chapter, the argument appeals to a temporal concept of truth and concludes that not every statement is always either true or false. However, many of Aristotle’s followers and commentators have not adopted this reading. I believe that it (...) has faced so much resistance for reasons of hermeneutic charity: denying the law of universal bivalence seems to be overly disruptive to logical orthodoxy – the kind of logical orthodoxy represented by what we now call classical propositional logic, much of which Aristotle clearly supports in many texts. I intend to show that the logical-semantic theses that the traditional reading finds in De Interpretatione 9 are much more conservative than they may seem to be at first glance. First, I will show that they complement, and do not contradict in any way, the orthodox definitions of the concepts of truth and statement that Aristotle advances in other texts. Second, by resorting in an anachronistic vein to concepts and methods peculiar to contemporary logic, I will show that a trivalent modal semantics conforming to those theses can be built for a standard formal language of the classical propositional calculus. It is remarkable that reasonable concepts of logical truth and logical consequence that may be defined on the basis of this trivalent modal semantics are coextensive with their orthodox counterparts, the concepts of tautology and tautological consequence of classical bivalent and extensional semantics. (shrink)

This paper develops a trivalent semantics for the truth conditions and the probability of the natural language indicative conditional. Our framework rests on trivalent truth conditions first proposed by Cooper (1968) and Belnap (1973) and it yields two logics of conditional reasoning: (i) a logic C of certainty-preserving inference; and (ii) a logic U for uncertain reasoning that preserves the probability of the premises. We show systematic correspondences between trivalent and probabilistic representations of inferences in either framework, and we use (...) the distinction between the two systems to cast light on the validity of inferences such as Modus Ponens, Or-To-If, and Conditional Excluded Middle. Specifically, the conditional behaves monotonically in C, but non-monotonically in U; Modus Ponens is valid in C, but valid in U only for non-nested conditionals. The result is a unified account of the semantics and epistemology of indicative conditionals that can be fruitfully applied to analyzing the validity of conditional inferences. (shrink)

Citrate, an organic trivalent anion, is a major substrate for generation of energy in most cells. It is produced in mitochondria and used either in the Krebs' cycle or released into cytoplasm through a specific mitochondrial carriers. Citrate can also be taken up from blood through different plasma membrane transporters. In the cytoplasm, citrate can be used ultimately for fatty acid synthesis, which is increased in cancer cells. Here, we review the ways in which citrate can be transported and discuss (...) the changes in transport and metabolism that occur in cancer cells. The primary focus is on the prostate gland, which is known to produce and release large amounts of citrate during its normal secretory function. The significant changes that occur in citrate‐related metabolism and transport in prostate cancer are the second focus. This review strives to relate these mechanisms to molecular biology on the one hand and to clinical applications on the other. (shrink)

Psychological research on people’s understanding of natural language connectives has traditionally used truth table tasks, in which participants evaluate the truth or falsity of a compound sentence given the truth or falsity of its components in the framework of propositional logic. One perplexing result concerned the indicative conditional if A then C which was often evaluated as true when A and C are true, false when A is true and C is false but irrelevant“ (devoid of value) when A is (...) false (whatever the value of C). This was called the “psychological defective table of the conditional.” Here we show that far from being anomalous the “defective” table pattern reveals a coherent semantics for the basic connectives of natural language in a trivalent framework. This was done by establishing participants’ truth tables for negation, conjunction, disjunction, conditional, and biconditional, when they were presented with statements that could be certainly true, certainly false, or neither. We review systems of three-valued tables from logic, linguistics, foundations of quantum mechanics, philosophical logic, and artificial intelligence, to see whether one of these systems adequately describes people’s interpretations of natural language connectives. We find that de Finetti’s (1936/1995) three-valued system is the best approximation to participants’ truth tables. (shrink)

Different inferences in probabilistic logics of conditionals 'preserve' the probabilities of their premisses to different degrees. Some preserve certainty, some high probability, some positive probability, and some minimum probability. In the first case conclusions must have probability I when premisses have probability 1, though they might have probability 0 when their premisses have any lower probability. In the second case, roughly speaking, if premisses are highly probable though not certain then conclusions must also be highly probable. In the third case (...) conclusions must have positive probability when premisses do, and in the last case conclusions must be at least as probable as their least probable premisses. Precise definitions and well known examples are given for each of these properties, characteristic principles are shown to be valid and complete for deriving conclusions of each of these kinds, and simple trivalent truthtable 'tests' are described for determining which properties are possessed by any given inference. Brief comments are made on the application of these results to certain modal inferences such as "Jones may own a car, and if he does he will have a driver's license. Therefore, he may have a driver's license.". (shrink)

In this paper we investigate a semantics for first-order logic originally proposed by R. van Rooij to account for the idea that vague predicates are tolerant, that is, for the principle that if x is P, then y should be P whenever y is similar enough to x. The semantics, which makes use of indifference relations to model similarity, rests on the interaction of three notions of truth: the classical notion, and two dual notions simultaneously defined in terms of it, (...) which we call tolerant truth and strict truth. We characterize the space of consequence relations definable in terms of those and discuss the kind of solution this gives to the sorites paradox. We discuss some applications of the framework to the pragmatics and psycholinguistics of vague predicates, in particular regarding judgments about borderline cases. (shrink)

Suszko's Thesis maintains that many-valued logics do not exist at all. In order to support it, R. Suszko offered a method for providing any structural abstract logic with a complete set of bivaluations. G. Malinowski challenged Suszko's Thesis by constructing a new class of logics (called q-logics by him) for which Suszko's method fails. He argued that the key for logical two-valuedness was the "bivalent" partition of the Lindenbaum bundle associated with all structural abstract logics, while his q-logics were generated (...) by "trivalent" matrices. This paper will show that contrary to these intuitions, logical two-valuedness has more to do with the geometrical properties of the deduction relation of a logical structure than with the algebraic properties embedded on it. (shrink)

We investigate the interactions between scalar implicatures and presuppositions in sentences containing both a scalar item and presupposition trigger. We first critically discuss Gajewski and Sharvit’s previous approach. We then closely examine two ways of integrating an exhaustivity-based theory of scalar implicatures with a trivalent approach to presuppositions. The empirical side of our discussion focuses on two novel observations: the interactions between prosody and monotonicity, and what we call presupposed ignorance. In order to account for these observations, our final proposal (...) relies on two mechanisms of scalar strengthening, the Presupposed Ignorance Principle and an exhaustivity operator which lets the presuppositions of negated alternatives project. (shrink)

the world of phenomena is immensely large this means we can perceive only part of the world. We see, feel and hear parts of reality, not the whole of it, and it seems that a sentence containing a verb of perception like 'John sees a house burn' is most naturally treated as saying that the subject sees an incomplete world in which the embedded sentence is true (see Barwise (1981) for this analysis). But if we want to analyse perception verbs (...) thus, we must introduce some form of incompleteness into our formal system, the.. (shrink)

In this paper, we use antecedent-final conditionals to formulate two problems for parsing-based theories of presupposition projection and triviality of the kind given in Schlenker 2009. We show that, when it comes to antecedent-final conditionals, parsing-based theories predict filtering of presuppositions where there is in fact projection, and triviality judgments for sentences which are in fact felicitous. More concretely, these theories predict that presuppositions triggered in the antecedent of antecedent-final conditionals will be filtered (i.e. will not project) if the negation (...) of the consequent entails the presupposition. But this is wrong: John isn’t in Paris, if he regrets being in France intuitively presupposes that John is in France, contrary to this prediction. Likewise, parsing-based approaches to triviality predict that material entailed by the negation of the consequent will be redundant in the antecedent of the conditional; but John isn’t in Paris, if he’s in France and Mary is with him is intuitively felicitous, contrary to these predictions. Importantly, given that the trigger appears in sentence-final position, both incremental (left-to-right) and symmetric versions of such theories make the same predictions. These data constitute a challenge to the idea that presupposition projection and triviality should be computed on the basis of parsing. This issue is important because it relates to the more general question as to whether presupposition and triviality calculation should be thought of as a pragmatic post-compositional phenomenon or as part of compositional semantics (as in the more traditional dynamic approaches). We discuss a solution which allows us to maintain the parsing-based pragmatic approach; it is based on an analysis of conditionals which incorporates a presupposition that their antecedent is compatible with the context, together with a modification to Schlenker’s (2009) algorithm for calculating local contexts so that it takes into account presupposed material. As we will discuss, this solution works within a framework broadly similar to that of Schlenker’s (2009), but it doesn’t extend in an obvious way to other parsing-based accounts (e.g. parsing-based trivalent approaches). We conclude that a parsing-based theory can be maintained, but only if we adopt a substantial change of perspective on the framework. (shrink)

This article focuses on defective conditionals ? namely indicative conditionals whose antecedents are false and whose truth-values therefore cannot be determined. The problem is to decide which formal connective can adequately represent this usage. Classical logic renders defective conditionals true whereas traditional mathematics dismisses them as irrelevant. This difference in treatment entails that, at the propositional level, classical logic validates some sentences that are intuitively false in plane geometry. With two proofs, I show that the same flaw is shared by (...) a family of trivalent logics. I go on to examine the strict conditional and its derivatives. This family is the only one to avoid the faulty inference but it does so without addressing the status of the truth-value assigned to defective conditionals. (shrink)

In Part I of this paper, we identified and compared various schemes for trivalent truth conditions for indicative conditionals, most notably the proposals by de Finetti and Reichenbach on the one hand, and by Cooper and Cantwell on the other. Here we provide the proof theory for the resulting logics DF/TT and CC/TT, using tableau calculi and sequent calculi, and proving soundness and completeness results. Then we turn to the algebraic semantics, where both logics have substantive limitations: DF/TT allows for (...) algebraic completeness, but not for the construction of a canonical model, while CC/TT fails the construction of a Lindenbaum-Tarski algebra. With these results in mind, we draw up the balance and sketch future research projects. (shrink)

This article tries to present a possible way of proliferating bivalent, trivalent, etc. calculation. Beyond the controversies as to the effects that involve the presence versus the absence of the temporal factor, no significant attempts of adapting a bivalent logic to other pairs of values have been mentioned. Should the transformation of the alethiologic modalities into logic values be possible, five more bivalent calculations may be added to the classic bivalent calculation. All these new calculations are compatible with the traditional (...) one, to the extent that they submit to the conditions of non-contradictoriness and exhaustibility, and the connectors receive, even if partially, the corresponding value interpretations. (shrink)

Fictionalists maintain that possible worlds, numbers or composite objects exist only according to theories which are useful but false. Hale, Divers and Woodward have provided arguments which threaten to show that fictionalists must be prepared to regard the theories in question as contingently, rather than necessarily, false. If warranted, this conclusion would significantly limit the appeal of the fictionalist strategy rendering it unavailable to anyone antecedently convinced that mathematics and metaphysics concern non-contingent matters. I try to show that their arguments (...) can be resisted by developing and defending a strategy suggested by Rosen, Nolan and Dorr, according to which the fiction-operator is to be analysed in terms of a counterfactual that admits of non-trival truth-values even when the antecedent is impossible. (shrink)

Republican freedom is freedom from domination juxtaposed to negative freedom as freedom from interference. Proponents argue that republican freedom is superior since it highlights that individuals lose freedoms even when they are not subject to interference, and claim republican freedom is more ‘resilient’. Republican freedom is trivalent, that is, it includes the idea that someone might be non-free to perform some actions rather than unfree, and in that sense everyone regards republican freedom as different from negative freedom. Trivalence makes (...) republican freedom moralized according to negative libertarians. Beyond that, negative libertarians argue that all the supposed advantages of republican freedom are compatible with those of pure negative-freedom measures. That is, losses and gains of republican freedom are captured in measures of pure negative freedom, and any protection for republican freedom also protects negative freedom, ensuring each is equally resilient. Since republican freedom has no advantages over negative freedom, but has other problems (is moralized and is trivalent), negative freedom is superior. I examine this debate in this article through the ‘coalition problem’ for republican freedom. The coalition problem is that since there is always a coalition of others who could dominate any agent in any sphere, all agents are subject to domination, and hence no one can ever have republican freedom. Pettit’s simple solution to this reductio ad absurdum distinguishes potential from actual coalitions. Individuals are only dominated by actual coalitions, and not by potential ones. The simple solution highlights moralization problems as it demonstrates that domination cannot be purely institutionally defined, but requires consideration of dispositions and expectations about others’ behaviour. I argue that the differences between the ‘free man’ and ‘unfree person’ paradigmatic to republican arguments is best captured not by the difference between domination and interference, but, rather, from familiar distinctions between different types of rights and freedoms. Resilience is a practical matter that might track some of these familiar distinctions. (shrink)

ARISTOTLE’S chapter-long digression in the Peri Hermenias to remark a restriction of the law of the excluded middle has touched off reams of commentary, logical, metaphysical and theological. For the theologian, God’s omniscience and human freedom were each at stake; for the metaphysician, the status of time; and logicians professed to find here an application for their remote exercises in trivalent logics. But whatever be the concern of the commentator, a glance at any one of them is likely to discourage (...) a student from pursuing a question which seems destined only to further entangle itself. (shrink)

RésuméLa probabilité présente, en Physique classique, des caractères paradoxaux; relative à un état de connaissance, et n'apportant de renseignements que sur un ensemble nombreux d'essais, elle semble pourtant se définir comme un caractère objectif de l'événement isolé; elle suppose qu'on doive répondre » peut‐ětre « à certaines questions, alors que la Logique classique n'admet d'autre réponse que » oui « ou » non «. Utilisée par la micro‐physique, elle brise l'unité des fondements physico‐mathématiques qui se manifestait dans la Logique bivalente, (...) conçue comme théorie de l'objet, de l'événement, de la théorie quelconques.L'analyse de ces caractères paradoxaux, ainsi que des notions d'objet et d'événement, montre que la Physico‐mathématique retrouve son unité de fondements si l'on substitue à la Logique bivalente, la Logique trivalente de la Composition , ou la Logique des Attitudes qui la généralise . Cette revision des notions fondamentales à laquelle la science en progrès s'oblige elle‐měme est une expérience de l'idonéisme en action.ZusammenfassungDie Wahrscheinlichkeit zeigt in der klassischen Physik paradoxe Züge; obwohl sie vom Stande des Wissens abhängt und nur über eine grosse Anzahl von Versuchen Auskünfte gibt, scheint sie als eine objektive Eigenschaft des Einzelereignisses definiert zu sein; sie nimmt an, dass gewisse Fragen mit » vielleicht « beantwortet werden müssen, obgleich die klassische Logik keine andere Antwort als » ja « oder » nein « zulässt. Durch die Mikrophysik nutzbar gemacht, durchbricht sie die Einheit der physikalisch‐mathematischen Fundamente, die sich kundtat in der bivalenten Logik, verstanden als Theorie des beliebigen Objektes, des beliebigen Ereignisses, der beliebigen Theorie. Die Analyse dieser paradoxen Züge, sowie der Begriffe des Objektes und des Ereignisses zeigt, dass Mathematik und Physik ihre einheitlichen Grundlagen wiederfinden, wenn man die bivalente Logik durch die trivalente der Komposition ersetzt oder durch die Logik der Einstellungen , welche sie verallgemeinert . Diese Revision der grundlegenden Begriffe, zu der sich die fortschrittliche Wissenschaft selbst verpflichtet, ist ein Beispiel des in die Tat umgesetzten Idoneismus.Probability shows paradoxical characteristics in classical physics; although dependent upon a state of knowledge, and yielding information only as the total result of numerous trials, it seems nevertheless to be definable as an objective characteristic of the isolated event; it assumes that one must reply » perhaps « to certain questions, whilst classical logic would allow no answer but » yes « or » no «. When used by micro‐physics, it breaks the unity of the physico‐mathematical foundations which became apparent in bivalent logic, in its conception as a theory of any object, any event or any theory.Analysis of these paradoxical characteristics and of the concepts » object « and » event « shows physics and mathematics regain their fundamental unity if bivalent logic is replaced by the trivalent logic of synthesis or the logic of attitudes, which generalises it . This revision of fundamental concepts which progressive science brings upon itself, is an example of the doctrine of suitability in action. (shrink)

Njik ka Amaka: Further Discussions on the Philosophy of Integrative Humanism (A Contribution to African and Intercultural Philosophies) presents philosophy from the view point of African thought system and logic. It presents African philosophy not as a reactionary to another brand of philosophy as is popularly the case among writers of African philosophy but as an unspoken, unstated rival of the positions of other philosophical traditions, with great emphasis on the importance of intercultural philosophizing. Somehow, the work manages to be (...) eclectic in welding orientations in the spirit of intercultural philosophy and it makes no pretense that it was written primarily for the contemporary lover of wisdom. The focal point which remains unwavering throughout was the further development of integrative humanism as a theory in African philosophy. The language style of this work was carefully knitted in simplistic and in some places bombastic forms. Simplistic to appeal to all class of readers and bombastic to demonstrate the emerging orientation that one way to hit back at the domineering colonial languages would be to speak them in deconstructive manner. The Englishness or the Frenchness of those colonial languages are in this book broken; traditional meanings of words are in some places altered to reflect the African senses; some other words are reshaped in various forms to reflect what the African has in mind and what he means; idioms are reinterpreted to reflect the unabashed and highest degree of indomitable Africanness; this is because a language that would bear forth African truths if it yet remains foreign can only emerge from the ruins of the one that bore Western truths. The benchmark of the theory of integrative humanism as a vanguard for intercultural philosophy is a passage from bivalent to trivalent mode of logical interpretation wherein the intermediate position is hoisted as the necessary missing link of reality. (shrink)

The valencies of the lanthanides vary more than was once thought. In addition to valencies associated with a half-full shell, there are valencies associated with a quarter- and three-quarter-full shell. This can be explained on the basis of Slater’s theory of many-electron atoms. The same theory explains the variation in complexing constants in the trivalent state. Valency in metallic and organometallic compounds is also discussed.

This paper gives a novel analysis of the logical structure underlying three classes of vague adjectival predicates (relative adjectives, i.e., tall; total adjectives, i.e., straight; and partial adjectives, i.e., wet) and the realisation of this structure in arguments formed with comparative constructions (i.e., John is taller than Mary). I analyse three classes of valid arguments that can be formed with different types of gradable predicates in comparative constructions: scalarity arguments (i.e., Mary is taller than John and John is tall Mary (...) is tall), maximality arguments (i.e., Stick A is straighter than Stick B Stick B is not straight), and evaluativity arguments (i.e., This towel is wetter than that towel this towel is wet) I show how a previously proposed multi-valued logical system (called Delineation TCS; see Burnett, 2012b; Burnett, 2013) based on the trivalent framework of (Cobreros et al., 2012) can be used to model the reasoning patterns described in the paper. In this system, we derive the scalarity, maximality and evaluativity properties of different classes of adjectives from statements about the properties of cognitive indifference relations associated with these predicates. I therefore conclude that systems such as DelTCS give us valuable insight into the relationship between cognitive judgements of indifference and reasoning patterns with scalar terms. (shrink)

The article discusses Georg Simmel’s theorizing on the social in the light of his treatment of the ‘dyad’ and the ‘triad’, constellations of two and three elements. What makes the dyad and the triad particularly interesting is the fact that they express the difference between the primary intersubjectivity immanent to the individuals and the objectified social forms in numerical terms, as quantitatively determined. In the article, it is argued that in its basic, methodologically simplest form, the social amounts for Simmel (...) to dyadic interaction between I and you, that can be conceptualized as ‘beingwith’. Nevertheless, a third element is always included also in the dyad, be it only as an excluded third. Therefore, it is claimed that in order to fully understand the dynamics of social relationships, one must look at the interplay of two socio-logics, bivalent and trivalent. The ‘third’ not only interrupts the supposedly immediate relation between the two elements of the dyad, but it is also capable of transforming it into a completely new figure: a social whole, a ‘we’, which obtains a supra-individual life independent of the individuals. (shrink)

Although it was traditionally thought that self-reference is a crucial ingredient of semantic paradoxes, Yablo (1993, 2004) showed that this was not so by displaying an infinite series of sentences none of which is self-referential but which, taken together, are paradoxical. Yablo's paradox consists of a countable series of linearly ordered sentences s(0), s(1), s(2),... , where each s(i) says: For each k > i, s(k) is false (or equivalently: For no k > i is s(k) true). We generalize Yablo's (...) results along two dimensions. First, we study the behavior of generalized Yablo-series in which each sentence s(i) has the form: For Q k > i, s(k) is true, where Q is a generalized quantifier (e.g., no, every, infinitely many, etc). We show that under broad conditions all the sentences in the series must have the same truth value, and we derive a characterization of those values of Q for which the series is paradoxical. Second, we show that in the Strong Kleene trivalent logic Yablo's results are a special case of a more general fact: under certain conditions, any semantic phenomenon that involves self-reference can be emulated without self-reference. Various translation procedures that eliminate self-reference from a non-quantificational language are defined and characterized. An Appendix sketches an extension to quantificational languages, as well as a new argument that Yablo's paradox and the translations we offer do not involve self-reference. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy and Literature 25.2 (2001) 335-346 [Access article in PDF] Aesthetic Incunabula Ellen Dissanayake Incunabula n. pl. (f. L swaddling clothes, cradle): Early stages of development of a thing.Over the past thirty years, developmental psychologists have discovered remarkable cognitive abilities in young infants. Before these investigations, common pediatric wisdom accepted that apart from a few innate "reflexes"--for crying, suckling, clinging, startling--babies were pretty much tabulae rasae for their elders (...) to inscribe as they (and their cultures) decreed. Today, however, it is well established that newborns come into the world with decided preferences and motivations, so that one can speak intelligently of "neonatal" (or even "fetal") psychology. 1The demonstrable existence of inherent psychological capacities in newborns and infants has important implications for twenty-first-century literary theory, which--also over the past thirty years--has operated under the theoretical assumption that all knowledge and experience are mediated by culture and that there is therefore no "natural" psychological or emotional experience. Yet if, as infant psychology confirms, babies in every culture show the same or similar cognitive abilities and preferences and the same or similar motivations and emotional responses, it is necessary to modify the dogmatic assertion that all experience, including literary response, is socially constructed. Rather, adult psychology and experience grow from and build upon inborn motives and preferences. Literary theorists can no longer ignore contemporary knowledge in psychobiology, ethology, and evolutionary psychology that directly pertains to conceptualizing their subject.The innate capacities of newborns predispose them specifically for social and emotional interaction with others, not simply for soliciting [End Page 335] their physical care. For example, newborns prefer human faces to any other sight (high-contrast colors, cartoon characters, cute stuffed animals) and human voices to any other sound (soft music, tinkling bells, The Chipmunks). 2 They can imitate face, mouth, and hand movements and respond appropriately to another person's emotional expressions of sadness, fear, and surprise. 3 At birth, infants can estimate and anticipate intervals of time and temporal sequences and remember these temporal patterns, categorizing them both in time and space and in terms of affect and arousal. 4 By six weeks of age, these perceptual and cognitive abilities of normal infants permit them to engage with adult partners in complex communicative interchanges--the playful behavior that is commonly or colloquially called "babytalk."Babytalk is not the trival or inane pastime that it might superficially seem but, rather, a cradle in which nascent psychosocial capacities can emerge and be developed. Interestingly for literary theory, babytalk achieves these effects through fundamentally aesthetic means. Adult-infant engagements are noteworthy for their use of stylization (formalization or simplification), repetition, exaggeration, and elaboration in visual, vocal, and gestural modalities. I call these "aesthetic incunabula"--that is, sources (or early developmental stages) of the operations used by artists in all media to attract attention and to provoke and manipulate emotional response. The existence of sensitivities to such features in the first months of life suggests that humans are born with natural (innate, universal) predispositions for aesthetic engagement from which cultures and individuals can go on to create their myriad elaborated forms of artistic expression.Although babytalk varies in minor details among individuals and cultures, it can be described in all instances as a multimedia performance. That is, it contains not only "talk" (or the characteristic high-pitched, undulant, breathy, patterned, repetitive vocalizations with pauses called "motherese," "parentese," or "infant-directed speech"). Along with the utterances are peculiarly stereotyped, repetitive, and exaggerated facial expressions (such as widened eyes, uplifted eyebrows, sustained open mouth, and smiles), head movements (e.g., distinct bobs backwards, slow rhythmic nods, movement toward and away from the baby), and gestures (such as touching, stroking, and rhythmic pats) to which infants respond with their own vocal sounds and face and body movements and, in many societies, sustained mutual gaze.Babytalk is not only a performance, but a duet that incorporates both synchrony of behavior and alternation (or turn-taking)--activities that [End Page 336] are made possible by the infant's remarkable inborn sensitivity to temporal sequence and pattern. Microanalyses of videotaped interactions (at 24... (shrink)

The paper’s novelty is in combining two comparatively new fields of research: non-transitive logic and the proof method of correspondence analysis. To be more detailed, in this paper the latter is adapted to Weir’s non-transitive trivalent logic \({\mathbf{NC}}_{\mathbf{3}}\). As a result, for each binary extension of \({\mathbf{NC}}_{\mathbf{3}}\), we present a sound and complete Lemmon-style natural deduction system. Last, but not least, we stress the fact that Avron and his co-authors’ general method of obtaining _n_-sequent proof systems for any _n_-valent logic (...) with deterministic or non-deterministic matrices is not applicable to \({\mathbf{NC}}_{\mathbf{3}}\) and its binary extensions. (shrink)

In the paper, Jan Łukasiewicz’s program of the logicization of philosophy is presented and discussed. Łukasiewicz, known mostly for his invention of trivalent logic as well as his achievements in propositional calculus and metalogic, had always been concerned with the methodological condition of philosophy. He finally found “the measure of exactness” in mathematical logic. According to him, only the use of logical tools may provide philosophical investigations with an appropriate level of exactness. He expressed his views most firmly and directly (...) in the paper, “A call for the method of Philosophy”. Łukasiewicz proposed giving philosophical theories the form of axiomatic systems by indicating the primitive terms of their language, selecting suitable axioms, and explicitly determining the applied rules of inference. All the theses of these systems should be consequences of the accepted axioms and confronted with the data of experience and the results of science. Łukasiewicz’s program is presented together with its inspirations and prospects. (shrink)

This paper presents the results of a qualitative investigation into the ethical and aesthetic values held by late- and mid-career landscape architects in the UK. It identifies the dominant discourses within three value areas, the aesthetic, the social and the environmental. Within the web of value discourses, some are clearly conflicting, while others are compatible or mutually supporting. The most prevalent values are those associated with 'technocentric accommodation'. A 'trivalent' approach to design is advocated which combines values from the three (...) main areas. (shrink)

Kripke’s theory of truth succeeded in providing a trivalent semantics for a language that contains its own truth predicate and means of self-reference; but it did so by radically restricting the expressive power of the logic. In Kripke’s analysis, the Liar (e.g. This very sentence is not true) receives the indeterminate truth value; but the logic cannot express the fact that the Liar is something other than true: in order to do so, a weak negation not* would be needed, but (...) it would also make the logic inconsistent (because the ‘Super Liar’ This very sentence is not* true could not be assigned any truth value). Taking a hint from the quantificational form of the problematic sentences (… is something other than true), we define a hierarchy of negations which each quantifies over a domain of truth values, assimilated to ordinals. The resulting logic has as many negations and truth values as there are ordinals. Unlike Kripke’s logic, it enjoys a form of expressive completeness. And although the logic is not monotonic, we show that under broad conditions we can construct a variety of fixed points; one of them emulates Kripke’s ‘least fixed point’, while another one assigns a different truth value to each Super Liar. (shrink)

This paper is a response to Enguehard (Natural Language Semantics 29(4):527–578, 2021 ), who observes that presuppositions project in the same way from coordinations of declaratives and coordinations of polar questions, but existing mechanisms of projection from declaratives (e.g. Schlenker in Theoretical Linguistics 34(3):157–212, 2008, Semantics and Pragmatics 2:1–78, 2009 ) fail to scale to questions. His solution involves specifying a trivalent inquisitive semantics for (coordinations of) questions that bakes the various asymmetries of presupposition projection into the lexical entry of (...) conjunction/disjunction. However, we argue that such a move faces both theoretical and empirical issues. Instead, we show that the data can be handled without moving to such an asymmetric inquisitive denotation, by adapting the novel pragmatic theory of _Limited Symmetry_ (Kalomoiros in Proceedings of the 52nd annual meeting of the North East linguistic society, GLSA, Amherst, 2022 ) to an inquisitive framework in a way that leaves the underlying semantics for conjunction symmetric and bivalent, while deriving the projection data. (shrink)