Proclus and the Neoplatonic Syllogistic

An investigation of Proclus' logic of the syllogistic and of negations in the Elements of Theology, On the Parmenides, and Platonic Theology. It is shown that Proclus employs interpretations over a linear semantic structure with operators for scalar negations (hypernegation/alpha-intensivum and privative negation). A natural deduction system for scalar negations and the classical syllogistic (as reconstructed by Corcoran and Smiley) is shown to be sound and complete for the non-Boolean linear structures. It is explained how Proclus' syllogistic presupposes converting the tree of genera and species from Plato's diairesis into the Neoplatonic linear hierarchy of Being by use of scalar hyper and privative negations.     

Between Square and Hexagon in Oresme's Livre du Ciel et du Monde

In logic, Aristotelian diagrams are almost always assumed to be closed under negation, and are thus highly symmetric in nature. In linguistics, by contrast, these diagrams are used to study lexicalization, which is notoriously not closed under negation, thus yielding more asymmetric diagrams. This paper studies the interplay between logical symmetry and linguistic asymmetry in Aristotelian diagrams. I discuss two major symmetric Aristotelian diagrams, viz. the square and the hexagon of opposition, and show how linguistic considerations yield various asymmetric versions of these diagrams. I then discuss a pentagon of opposition, which occupies an uneasy position between the square and the hexagon. Although this pentagon belongs neither to the symmetric realm of logic nor to the asymmetric realm of linguistics, it occurs several times in the literature. The oldest known occurrence can be found in the cosmological work of the 14th-century author Nicole Oresme.     

IV. Strawson on the traditional logic

In his Introduction to Logical Theory, Strawson argues that Aristotelian logic can be given a successful interpretation into ordinary English, but not into the symbolism of Principia Mathematica, on the grounds that Aristotelian logic and ordinary English share something absent in PM, namely, the doctrine of presupposition. It is argued that Strawson is mistaken. PM does justice to the logical rules of Aristotelian logic and also has a fully articulated doctrine of presupposition.     

Existence,Negation, and Abstraction in the Neoplatonic Hierarchy 1

The paper is a study of the logic of existence, negation, and order in the Neoplatonic tradition. The central idea is that Neoplatonists assume a logic in which the existence predicate is a comparative adjective and in which monadic predicates function as scalar adjectives that nest the background order. Various scalar predicate negations are then identifiable with various Neoplatonic negations, including a privative negation appropriate for the lower orders of reality and a hyper-negation appropriate for the higher. Reversion to the One can then be explained as the logical inference of hyper-negations from mundane knowledge. Part I develops the relevant linguistic and logical theory, and Part II defends Wolfson and the scalar interpretation against the more traditional Aristotelian understanding of Whittaker and others of reversion as intensional abstraction     

Internal Negation and the Principles of Non-Contradiction and of Excluded Middle in Aristotle

It has long been recognized that negation in Aristotle’s term logic differs syntactically from negation in classical logic: modern external negation attaches to propositions fully formed, whereas Aristotelian internal negation forms propositions from sentential constituents. Still, modern external negation is used to render Aristotelian internal negation, as may be seen in formalizations of Aristotle’s semantic principles of non-contradiction and of excluded middle. These principles govern the distribution of truth values among pairs of contradictory propositions, and Aristotelian contradictories always consist of an affirmation and a denial. So how should we formalize a false denial? In the literature, we find that a false denial is formalized by means of two negation signs attached to a one-place predicate. However, it can be shown that this rendering leads to an incorrect picture of Aristotle’s principles. In this paper, I propose a solution to this technical problem by devising a formal notation especially for Aristotelian propositions in which internal negation is differentiated from external negation. I will also analyze both principles, each of which has two logically equivalent forms, a positive and a negative one. The fact that Aristotle’s principles are distinct and complementary is reflected in my new formalizations.     

Affirmation and Denial in Aristotle’s De interpretatione

Topoi - Modern logicians have complained that Aristotelian logic lacks a distinction between predication (including negation) and assertion, and that predication, according to the Aristotelians,...     

Nelson's Negation on the Base of Weaker Versions of Intuitionistic Negation

Constructive logic with Nelson negation is an extension of the intuitionistic logic with a special type of negation expressing some features of constructive falsity and refutation by counterexample. In this paper we generalize this logic weakening maximally the underlying intuitionistic negation. The resulting system, called subminimal logic with Nelson negation, is studied by means of a kind of algebras called generalized N-lattices. We show that generalized N-lattices admit representation formalizing the intuitive idea of refutation by means of counterexamples giving in this way a counterexample semantics of the logic in question and some of its natural extensions. Among the extensions which are near to the intuitionistic logic are the minimal logic with Nelson negation which is an extension of the Johansson's minimal logic with Nelson negation and its in a sense dual version — the co-minimal logic with Nelson negation. Among the extensions near to the classical logic are the well known 3-valued logic of Lukasiewicz, two 12-valued logics and one 48-valued logic. Standard questions for all these logics — decidability, Kripke-style semantics, complete axiomatizability, conservativeness are studied. At the end of the paper extensions based on a new connective of self-dual conjunction and an analog of the Lukasiewicz middle value ½ have also been considered.     

Aristotelian diagrams for semantic and syntactic consequence

Several authors have recently studied Aristotelian diagrams for various metatheoretical notions from logic, such as tautology, satisfiability, and the Aristotelian relations themselves. However, all these metalogical Aristotelian diagrams focus on the semantic (model-theoretical) perspective on logical consequence, thus ignoring the complementary, and equally important, syntactic (proof-theoretical) perspective. In this paper, I propose an explanation for this discrepancy, by arguing that the metalogical square of opposition for semantic consequence exhibits a natural analogy to the well-known square of opposition for the categorical statements from syllogistics, but that this analogy breaks down once we move from semantic to syntactic consequence. I then show that despite this difficulty, one can indeed construct metalogical Aristotelian diagrams from a syntactic perspective, which have their own, equally elegant characterization in terms of the categorical statements. Finally, I construct several metalogical Aristotelian diagrams that incorporate both semantic and syntactic consequence (and their interaction), and study how they are influenced by the underlying logical system’s soundness and/or completeness. All of this provides further support for the methodological/heuristic perspective on Aristotelian diagrams, which holds that the main use of these diagrams lies in facilitating analogies and comparisons between prima facie unrelated domains of investigation.

     

Illusory inferences with quantifiers

The psychological study of reasoning with quantifiers has predominantly focused on inference patterns studied by Aristotle about two millennia ago. Modern logic has shown a wealth of inference patterns involving quantifiers that are far beyond the expressive power of Aristotelian syllogisms, and whose psychology should be explored. We bring to light a novel class of fallacious inference patterns, some of which are so attractive that they are tantamount to cognitive illusions. In tandem with recent insights from linguistics that quantifiers like “some” are treated as wh-questions, these illusory inferences are predicted by the erotetic theory of reasoning, which postulates that a process akin to question asking and answering is behind human inference making.     

Basic Quasi-Boolean Expansions of Relevance Logics

The basic quasi-Boolean negation (QB-negation) expansions of relevance logics included in Anderson and Belnap’s relevance logic R are defined. We consider two types of QB-negation: H-negation and D-negation. The former one is of paraintuitionistic or superintuitionistic character, the latter one, of dual intuitionistic nature in some sense. Logics endowed with H-negation are paracomplete; logics with D-negation are paraconsistent. All logics defined in the paper are given a Routley-Meyer ternary relational semantics.

     

Fallacies of Accident

In this paper I will attempt a unified analysis of the various examples of the fallacy of accident given by Aristotle in the Sophistical Refutations. In many cases the examples underdetermine the fallacy and it is not trivial to identify the fallacy committed. To make this identification we have to find some error common to all the examples and to show that this error would still be committed even if those other fallacies that the examples exemplify were not. Aristotle says that there is only one solution “against the argument” as opposed to “against the man”, and it is this solution the paper attempts to find. It is a characteristic mark of my analysis that some arguments that we might normally be inclined to say are fallacious turn out to be valid and that some arguments that we would normally be inclined to say are valid turn out to be fallacious. This is (in part) because what we call validity in modern logic is not the same as the apodicticity that Aristotelian syllogisms require in order to be used in science. The fallacies of accident, uniquely among the fallacies, are failures of apodicticity rather than failures of, in particular, semantic entailment. This makes sense in a tensed and token-based logic such as Aristotle’s. I conclude that the closest analogue to the fallacy of accident that we can point to is a fallacy in modal logic, viz., the fallacy of necessity.     

Constructive Logic with Strong Negation is a Substructural Logic. II

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL _ew of the substructural logic FL _ew . The main result of Part I of this series [41] shows that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL _ew (namely, a certain variety of FL _ew -algebras) are term equivalent. In this paper, the term equivalence result of Part I [41] is lifted to the setting of deductive systems to establish the definitional equivalence of the logics N and NFL _ew . It follows from the definitional equivalence of these systems that constructive logic with strong negation is a substructural logic. Presented by Heinrich Wansing     

Elementary categorial logic,predicates of variable degree,and theory of quantity

Developing some suggestions of Ramsey (1925), elementary logic is formulated with respect to an arbitrary categorial system rather than the categorial system of Logical Atomism which is retained in standard elementary logic. Among the many types of non-standard categorial systems allowed by this formalism, it is argued that elementary logic with predicates of variable degree occupies a distinguished position, both for formal reasons and because of its potential value for application of formal logic to natural language and natural science. This is illustrated by use of such a logic to construct a theory of quantity which is argued to be scientifically superior to existing theories of quantity based on standard categorial systems, since it yields realvalued scales without the need for unrealistic existence assumptions. This provides empirical evidence for the hypothesis that the categorial structure of the physical world itself is non-standard in this sense.I would like to thank my collegue Mark Brown and an anonymous referee for helpful comments on an earlier draft of this paper.     

A Debate About Anderson's Logic

This article is about the history of logic in Australia. Douglas Gasking (1911–1994) undertook to translate the logical terminology of John Anderson (1893–1962) into that of Ludwig Wittgenstein's (1921) Tractatus. At the time Gilbert Ryle (1900–1976), and more recently David Armstrong, recommended the result to students; but it is reasonable to have misgivings about Gasking as a guide to either Anderson or Wittgenstein. The historical interest of the debate Gasking initiated is that it yielded surprisingly little information about Anderson's traditional (syllogistic or Aristotelian) logic and its relation to classical (first-order predicate or Russellian) logic, the ostensible topic; but the materials now exist to interpret Anderson's logic in classical logic, possibly as an algebra of classes. This would be of little interest to contemporary logicians, but it might shed some light on Anderson's philosophy.     

Gentzen-Type Methods for Bilattice Negation

A general Gentzen-style framework for handling both bilattice (or strong) negation and usual negation is introduced based on the characterization of negation by a modal-like operator. This framework is regarded as an extension, generalization or re- finement of not only bilattice logics and logics with strong negation, but also traditional logics including classical logic LK, classical modal logic S4 and classical linear logic CL. Cut-elimination theorems are proved for a variety of proposed sequent calculi including CLS (a conservative extension of CL) and CLS_cw (a conservative extension of some bilattice logics, LK and S4). Completeness theorems are given for these calculi with respect to phase semantics, for SLK (a conservative extension and fragment of LK and CLS_cw, respectively) with respect to a classical-like semantics, and for SS4 (a conservative extension and fragment of S4 and CLScw, respectively) with respect to a Kripke-type semantics. The proposed framework allows for an embedding of the proposed calculi into LK, S4 and CL.     

The paradox of negation in Nāgārjuna's philosophy

This essay discusses the paradox of the Nāgārjunian negation as presented in his Vigrahavyāvartani. In Part One it is argued that as the Naiyāyika remarks, Nāgārjuna's speech act ‘No proposition has its own intrinsic thesis’ seemingly contradicts his famous claim that he has no negation whatsoever. In Parts Two and Three I consider the traditional as well as modem responses to this paradox and offer my own. I argue that Nāgārjuna's speech act does not generate a paradox for two reasons: (a) the equivalence thesis of the kind‐?P = ?P is obviously false; and (b) since Nāgārjuna's speech act is situated in the dialogical/conversational universe of discourse as opposed to the argumentative/systematic universe of discourse, the teaching of the non‐intrinsic thesis of all statements that it purports, holds for all statements in its class, including itself. Lastly, it is argued that even though the Nāgārjunian speech act is not a negation situated in the argumentative universe of discourse, it serves both philosophical and soteriological purposes.     

Hegel's notion of Aufheben

The paper is an attempt to make sense of Hegel's notion of aufheben. The double meaning of aufheben and its alleged ‘rise above the mere “either‐or”; of understanding’ have been taken, by some, to constitute a criticism of the logic of either‐or. It is argued, on the contrary, that Hegel's notion of aufheben, explicated in its primary and philosophical context, turns out to be a substantiation of that logic. The intelligibility of the formula of either‐or depends, for example, on the categories of Being and Not‐Being. But if these categories are regarded as particular finite determinations themselves subject to the formula of either‐or, then the formula, far from being intelligible, ‘falls apart’. Hegel is arguing, in other words, that if we are to substantiate the logic of either‐or, we must, at the same time, ‘rise above’ that logic. The role of aufheben is then considered in the special sciences. Here it is argued that we must distinguish between empirical transitions, governed by the finite determinations of things, and logical or dialectical transitions, governed by considerations of the intelligibility of the notions involved. Applying the notion of aufheben to the former transitions suggests wrongly that empirical transitions have an objective or philosophic necessity. Finally, the place of ‘immanent transformation’ in the context of aufheben is examined. It is concluded that if there is to be a transformation, then a distinction must be drawn between thought and its content, but then the transformation cannot be regarded as immanent.     

Negation: A theory of its meaning,representation, and use

This article presents a model-based theory of what negation means, how it is mentally represented, and how it is understood. The theory postulates that negation takes a single argument that refers to a set of possibilities and returns the complement of that set. Individuals therefore tend to assign a small scope to negation in order to minimize the number of models of possibilities that they have to consider. Individuals untrained in logic do not know the possibilities corresponding to the negation of compound assertions formed with if, or, and and, and have to infer the possibilities one by one. It follows that negations are easier to understand, and to formulate, when individuals already have in mind the possibilities to be negated. The paper shows that the evidence, including the results of recent studies, corroborates the theory.     

Constructive Logic, Truth and Warranted Assertability

Shapiro and Taschek have argued that simply using intuitionistic logic and its Heyting semantics, one can show that there are no gaps in warranted assertability. That is, given that a discourse is faithfully modelled using Heyting's semantics for the logical constants, then if a statement S is not warrantedly assertable, its negation ^∼ S is. Tennant has argued for this conclusion on similar grounds. I show that these arguments fail, albeit in illuminating ways. An appeal to constructive logic does not commit one to this strong epistemological thesis, but appeals to semantics of intuitionistic logic none the less do give us certain conclusions about the connections between warranted assertability and truth.     

Fuzzy intuitionistic quantum logics

Fuzzy intuitionistic quantum logics (called also Brouwer-Zadeh logics) represent to non standard version of quantum logic where the connective not is split into two different negation: a fuzzy-like negation that gives rise to a paraconsistent behavior and an intuitionistic-like negation. A completeness theorem for a particular form of Brouwer-Zadeh logic (BZL ³) is proved. A phisical interpretation of these logics can be constructed in the framework of the unsharp approach to quantum theory.