The paper presents generalizations of results on so-called Horn logic, well-known in universal algebra, to the setting of fuzzy logic. The theories we consider consist of formulas which are implications between identities (equations) with premises weighted by truth degrees. We adopt Pavelka style: theories are fuzzy sets of formulas and we consider degrees of provability of formulas from theories. Our basic structure of truth degrees is a complete residuated lattice. We derive a Pavelka-style completeness theorem (degree of (...) provability equals degree of truth) from which we get some particular cases by imposing restrictions on the formulas under consideration. As a particular case, we obtain completeness of fuzzy equational logic. (shrink)

The paper deals with fuzzy Horn logic which is a fragment of predicate fuzzy logic with evaluated syntax. Formulas of FHL are of the form of simple implications between identities. We show that one can have Pavelka-style completeness of FHL w.r.t. semantics over the unit interval [0, 1] with left-continuous t-norm and a residuated implication, provided that only certain fuzzy sets of formulas are considered. The model classes of fuzzy structures of FHL are characterized by closure properties. (...) We also give comments on related topics proposed by N. Weaver. (shrink)

In the modern age, the political secret has acquired a bad reputation. With modern democracy’s ideal of transparency, political secrecy is identified with political crime or corruption. The article argues that this repression of secrecy in modern democracies falls short of a substantial understanding of the structure and workings of political secrecy. By outlining a genealogy of political secrecy, it elucidates the logic as well as the blind spots of a current culture of secrecy. It focuses on two fundamental (...) logics of secrecy, deduced from the Latin terms ‘ arcanum’ and ‘ secretum’. Whereas the logic of arcanum regards secrecy as a legitimate dimension of government, a modern logic of secretum is marked by an inextricable dialectics between the withdrawal and communication of knowledge, between secrecy and publicity. Here, the secret is not so much a piece of withheld knowledge as a ‘secrecy effect’ that binds the realm of secrecy to the public sphere by a dialectics of permanent suspicion and scandal. Instead of falling into the trap of this ‘secrecy effect’ it is worth taking a closer look at the tradition of thought on the arcana imperii, from Tacitus to early modern doctrines of raison d’état to Carl Schmitt. What this tradition deals with is the functionality of secrecy and its complicated relation to the law. The arcana tradition elaborates the crucial point of secrecy: its potential, but also its profound ambivalence. Secrecy opens up a discretionary space of action exempt from the rule of law, and, according to Carl Schmitt, ignores the law so as to allow it to become effective. Secrecy serves to protect and stabilize the state, but at the same time it opens a space of exception from the rule of law that breeds violence, corruption and oppression. Instead of seeing secrecy as the opposite of a political culture of transparency, it is more productive to regard secrecy as transparency's complement – a counterpart, however, that is marked by the profound paradox of being both a consolidation of and a threat to democracy. (shrink)

The paper studies closure properties of classes of fuzzy structures defined by fuzzy implicational theories, i.e. theories whose formulas are implications between fuzzy identities. We present generalizations of results from the bivalent case. Namely, we characterize model classes of general implicational theories, finitary implicational theories, and Horn theories by means of closedness under suitable algebraic constructions.

Strong reasoning skills are an important aspect to cultivate in life, as they directly impact decision making on a daily basis. By examining the different ways the world views logic and order, new methods and techniques can be employed to help expand on this skill further in the future. -/- Philosophical Perceptions on Logic and Order is a pivotal scholarly resource that discusses the evolution of logical reasoning and future applications for these types of processes. Highlighting relevant topics (...) including logic patterns, deductive logic, and inductive logic, this publication is an ideal reference source for academicians, students, and researchers that would like to expand their understanding of how society currently employs the use of logical reasoning techniques. (shrink)

1. Implicature: some basic oppositions IMPLICATURE is a component of speaker meaning that constitutes an aspect of what is meant in a speaker’s utterance without being part of what is said. What a speaker intends to communicate is characteristically far richer than what she directly expresses; linguistic meaning radically underdetermines the message conveyed and understood. Speaker S tacitly exploits pragmatic principles to bridge this gap and counts on hearer H to invoke the same principles for the purposes of utterance interpretation. (...) The contrast between the said and the meant, and derivatively between the said and the implicated (the meant-but-unsaid), dates back to the fourth century rhetoricians Servius and Donatus, who characterized litotes—the figure of pragmatic understatement—as a figure in which we say less but mean more (“minus dicimus et plus significamus”; see Hoffmann 1987 and Horn 1991a for discussion). In the classical Gricean model, the bridge from what is said (the literal content of the uttered sentence, computed directly from its grammatical structure with the reference of indexicals resolved) to what is communicated is constructed through implicature. As an aspect of speaker meaning, implicatures are by definition distinct from the non-logical inferences that the hearer draws; it is a category mistake to attribute implicatures either to hearers or to sentences (e.g. P and Q) and subsentential expressions (e.g. some). But we can systematically (at least for generalized implicatures; see below) correlate the speaker’s intention to implicate q (in uttering p in context C), the expression p that carries the implicature in C, and the inference of q induced by the speaker’s utterance of p in C. (shrink)

In this paper we mainly study preservation theorems for two fragments of the infinitary languagesLκκ, withκregular, without the equality symbol: the universal Horn fragment and the universal strict Horn fragment. In particular, whenκisω, we obtain the corresponding theorems for the first-order case.The universal Horn fragment of first-order logic (with equality) has been extensively studied; for references see [10], [7] and [8]. But the universal Horn fragment without equality, used frequently in logic programming, has received (...) much less attention from the model theoretic point of view. At least to our knowledge, the problem of obtaining preservation results for it has not been studied before by model theorists. In spite of this, in the field of abstract algebraic logic we find a theorem which, properly translated, is a preservation result for the strict universal Horn fragment of infinitary languages without equality which, apart from function symbols, have only a unary relation symbol. This theorem is due to J. Czelakowski; see [5], Theorem 6.1, and [6], Theorem 5.1. A. Torrens [12] also has an unpublished result dealing with matrices of sequent calculi which, properly translated, is a preservation result for the strict universal Horn fragment of a first-order language. And in [2] of W. J. Blok and D. Pigozzi we find Corollary 6.3 which properly translated corresponds to our Corollary 19, but for the case of a first-order language that apart from its function symbols has only oneκ-ary relation symbol, and for strict universal Horn sentences. The study of these results is the basis for the present work. In the last part of the paper, Section 4, we will make these connections clear and obtain some of these results from our theorems. In this way we hope to make clear two things: (1) The field of abstract algebraic logic can be seen, in part, as a disguised study of universal Horn logic without equality and so has an added interest. (2) A general study of universal Horn logic without equality from a model theoretic point of view can be of help in the field of abstract algebraic logic. (shrink)

Litotes, “a figure of speech in which an affirmative is expressed by the negative of the contrary” has had some tough reviews. For Pope and Swift, litotes—stock examples include “no mean feat”, “no small problem”, and “not bad at all”—is “the peculiar talent of Ladies, Whisperers, and Backbiters”; for Orwell, it is a means to affect “an appearance of profundity” that we can deport from English “by memorizing this sentence: A not unblack dog was chasing a not unsmall rabbit across (...) a not ungreen field.” But such ridicule is not without equivocation, given that litotes, or “logical” double negation, may or may not be semantically redundant. When the negation of a logical contrary yields an unexcluded middle, it contributes to expressive power: someone who is not unhappy may not be happy either, and an occurrence may not be infrequent without being frequent. But if something is not possible, what can it be but possible? Why does Crashaw’s “not impossible she” survive rhetorically while Orwell’s “not unsmall rabbit” is doomed? How is Robbie being “not not friends” with Mary on 7th Heaven distinct from being friends with her, if not not-p reduces to p? The key is recognizing in litotes a corollary of MaxContrary, the tendency for contradictory sentential negation ¬p to strengthen pragmatically to a contrary ©p, as when the formal contradictory Fr. “Il ne faut pas partir” is reinterpreted as expressing a contrary. Just as the Law of Excluded Middle can apply where it “shouldn’t”, resulting in pragmatically presupposed disjunctions between semantic contraries, so that “p v ©p” amounts to an instance of “p v ¬p”, the Law of Double Negation can fail to apply where it “should”. When not not-p conveys ¬©p, the negation of a virtual contrary, the middle between p and not-p is no longer excluded, rendering the Fregean dictum that “Wrapping up a thought in double negation does not alter its truth value” not unproblematic. (shrink)

We describe which subdirectly irreducible flat algebras arise in the variety generated by an arbitrary class of flat algebras with absorbing bottom element. This is used to give an elementary translation of the universal Horn logic of algebras, and more generally still, partial structures into the equational logic of conventional algebras. A number of examples and corollaries follow. For example, the problem of deciding which finite algebras of some fixed type have a finite basis for their quasi-identities (...) is shown to be equivalent to the finite identity basis problem for the finite members of a finitely based variety with definable principal congruences. (shrink)

We introduce SO-HORNr which is a revised version of SO-HORN and show that SO-HORNr captures equation image on ordered finite structures. We also introduce second-order extended Horn logic SO-EHORN and a superclass SO-EHORNr of it. We show that both of them capture equation image on ordered finite structures by proving that SO-EHORN and SO-EHORNr have the same expressive power when only consider ordered structures.

In this paper we deal with infinitary universal Horn logic both with and without equality. First, we obtain a relative Lyndon-style interpolation theorem. Using this result, we prove a non-standard preservation theorem which contains, as a particular case, a Lyndon-style theorem on surjective homomorphisms in its Makkai-style formulation. Another consequence of the preservation theorem is a theorem on bimorphisms, which, in particular, provides a tool for immediate obtaining characterizations of infinitary universal Horn classes without equality from those (...) with equality. From the theorem on surjective homomorphisms we also derive a non-standard Beth-style preservation theorem that yields a non-standard Beth-style definability theorem, according to which implicit definability of a relation symbol in an infinitary universal Horn theory implies its explicit definability by a conjunction of atomic formulas. We also apply our theorem on surjective homomorphisms, theorem on bimorphisms and definability theorem to algebraic logic for general propositional logic. (shrink)

According to a wide-spread opinion, the history of moral philosophy is divided into two major parts. Whereas ancient ethics is concerned with agent-relative prudential volitions, modern ethics is based upon agent-neutral moral obligations. The article questions this opposition by indicating that the mainstream of ancient ethics should rather be characterized by its typical theory of action. This theory doesn’t rest on prudential rationality but on the idea that a rational agent has to follow a number of non-arbitrary principles which conduct (...) his or her deliberating, choosing and acting. The model in question starts with an intellectualist concept of the will , and develops an account of human flourishing which is the basis of subtle further distinctions between relevant and incidental goods. Finally, it opens the possibility to bridge the gap between the issue of human flourishing and the Kantian concept of morality. (shrink)

What is a number? Answering this will answer questions about its philosophical foundations - rational numbers, the complex numbers, imaginary numbers. If we are to write or talk about something, it is helpful to know whether it exists, how it exists, and why it exists, just from a common-sense point of view [Quine, 1948, p. 6]. Generally, there does not seem to be any disagreement among mathematicians, scientists, and logicians about numbers existing in some way, but currently, in the mainstream (...) arena only definitions, descriptions of properties, and effects are presented as evidence. Enough historical description of numbers in history provides an empirical basis of number, although a case can be made that numbers do not exist by themselves empirically. Correspondingly, numbers exist as abstractions. All the while, though, these "descriptions" beg the question of what numbers are ontologically. Advocates for numbers being the ultimate reality have the problem of wrestling with the nature of reality. I start on the road to discovering the ontology of number by looking at where people have talked about numbers as already existing: history. Of course, we need to know not only what ontology is but the problems of identifying one, leading to the selection between metaphysics and provisional approaches. While we seem to be dimensionally limited, at least we can identify a more suitable bootstrapping ontology than mere definitions, leading us to the unity of opposites. The rest of the paper details how this is done and modifies Peano's Postulates. (shrink)

The existing literature on the rational underdetermination problem often construes it as one resulting from the ubiquity of objective values. It is therefore sometimes argued that subjectivists need not be troubled by the underdetermination problem. But on closer examination, it turns out, they should. Or so I will argue. The task of the first half of this paper is explaining why. The task of the second half is finding a subjectivist solution the rational underdetermination problem. The basic problem, I argue, (...) is as follows. Idealizing subjectivism generates too many ideal selves to deliver determinate or commensurable options regarding what non-ideal deliberating agents ought to do. My solution: these idealized options should be assessed from the only perspective we can, in fact, occupy, namely, that of our non-ideal, actual selves. Deciding what to do therefore becomes, in part, an exercise in deciding who to be. But one might now worry this just moves the arbitrariness bump in the rug. Privileging the perspective of our actual self seems contrary to the rationale for idealizing in the first place. I consider two solutions to the problem, one democratic, the other modelled on trusteeship. In the end, I argue, our actual self has complete freedom to choose the ideal self it grants rational authority. In the final part of the paper, I present my positive proposal as a solution to the underdetermination problem confronting the idealizing subjectivist and then argue that, so understood, this account vindicates a tidied-up version of how some reflective people already do deliberate in their everyday lives. This, in turn, suggests that a decision-procedure closely connected to the account is both possible and attractive. (shrink)

This article engages with the work of Scotland-based artist Kate Davis (b.1977). The discussion begins to articulate a framework for understanding Davis's work within a feminist logic of re-visioning and re-citing, strategies that are explicated and suggested as paradigmatic to feminist art production since 1970. Fundamentally, the article explores Davis's complex strategies for adopting and adapting motifs from within the archives of art history, arguing that her work constitutes a mode of visual research and historiography.

This article brings Leibniz' metaphysics down to earth. The long sought genesis of thinking is based on the organic nature. Leibniz' metaphysics is comprehensible as universal, because it links logic with the ontological fundamentals of the real. The paradox of individual substance is solvable. Herefore fact and concept of thinking are to be stretched towards acting, self-evolution as well as becoming self-conscious. So individual substance becomes the principle of organic nature, of becoming self-conscious, of historicity as well as the (...) principle of person — by the very structure. The opposites of belief and knowledge, of nature and history can be derived from the same principle. All human beings, all living beings are inseparably linked by their common structure. They differ though in the various informations and contents which flow in this structure. (shrink)

In the epistemology of his middle period, Plato repeatedly describes the alleged ‘intellection of true reality’ in terms of sight, vision, illumination, or touch. Does this show more than Plato’s preference for optic and haptic metaphors? Should we assume that this goes back to a specific reason to be found in his underlying epistemological position? On the traditional reading, Plato actually wants to defend a sort of intuitionism. According to this still wide-spread reading, he claims that there is a very (...) particular way of grasping Forms: Philosophers manage to come into a sort of direct contact with intelligible objects. Intellection is thus conceived as a quasi-visual presence of intelligible Forms. In our contribution, we want to challenge this interpretation by raising several objections against it. Surprisingly, there exists a close parallel in Aristotelian scholarship: Although Aristotle’s epistemology is built on considerably different foundations, there is an interpretative tradition according to which one has to assume a strict discontinuity between the methodical procedure and the intellection of the highest epistemic objects. As for Plato, we want to refuse an interpretation of Aristotle based on a concept of intuition characterized by the following features: Intuition is a specific way of gaining knowledge. It is independent of empirical conditions and leads to apriori knowledge. It consists of some experience of evidence or is accompanied by it. Knowledge gained by this kind of intuition does not need any further argument or justification. (shrink)

This paper presents a logical Style painting classifier based on evaluated Horn clauses, qualitative colour descriptors and Explanations. Three versions of $\ell $-SHE are defined, using rational Pavelka logic, and expansions of Gödel logic and product logic with rational constants: RPL, $G$ and $\sqcap $, respectively. We introduce a fuzzy representation of the more representative colour traits for the Baroque, the Impressionism and the Post-Impressionism art styles. The $\ell $-SHE algorithm has been implemented in Swi-Prolog and (...) tested on 90 paintings of the QArt-Dataset and on 247 paintings of the Paintings-91-PIB dataset. The percentages of accuracy obtained in the QArt-Dataset for each $\ell $-SHE version are 73.3%, 65.6% $) and 68.9% $). Regarding the Paintings-91-PIB dataset, the percentages of accuracy obtained for each $\ell $-SHE version are 60.2%, 48.2% $) and 57.0% $). Our logic definition for the Baroque style has obtained the highest accuracy in both datasets, for all the $\ell $-SHE versions. An important feature of the classifier is that it provides reasons regarding why a painting belongs to a certain style. The classifier also provides reasons about why outliers of one art style may belong to another art style, giving a second classification option depending on its membership degrees to these styles. (shrink)