Semantic Triads (original) (raw)
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Journal of Symbolic Logic, Vol. 37, 159-169, 1972
Introduction. In what follows there is presented a unified semantic treatment of certain "paradox-free" systems of entailment, including Church's weak theory of implication (Church [7D and logics akin to the systems E and R of Anderson and Belnap (Anderson [3], Belnap [6D.1 We shall refer to these systems generally as relevant logics.
A Note on Goddard and Routley's Significance Logics
Australasian Journal of Logic, 2018
The present note revisits the joint work of Leonard Goddard and Richard Routley on significance logics (namely, logics able to handle nonsignificant sentences) with the aim of shedding new light on their understanding by studying them under the lens of recent semantic developments, such as the plurivalent semantics developed by Graham Priest. These semantics allow sentences to receive one, more than one, or no truth-value at all from a given carrier set. Since nonsignificant sentences are taken to be neither true nor false, i.e. truth-value gaps, in this essay we show that with the aid of plurivalent semantics it is possible to straightforwardly instantiate Goddard and Routley's understanding of how the connectives should work within significance logics.
Foreword: Three-valued logics and their applications
Journal of Applied Non-Classical Logics, 2014
Three-valued logics belong to a family of nonclassical logics that started to flourish in the 1920s and 1930s, following the work of (Lukasiewicz, 1920), and earlier insights coming from Frege and Peirce (see (Frege, 1879), (Frege, 1892), (Fisch and Turquette, 1966)). All of them were moved by the idea that not all sentences need be True or False, but that some sentences can be indeterminate in truth value. In his pioneering paper, Lukasiewicz writes: "Three-valued logic is a system of non-Aristotelian logic, since it assumes that in addition to true and false propositions there also are propositions that are neither true nor false, and hence, that there exists a third logical value."
MEANING: A LOGICAL DERIVATION, FORMULATION I
ABSTRACT Expressions, words and symbols without reference to something else which could be called their meanings are semantically helpless. But not all expressions and words refer; some even come with ambiguities and equivocations like Golden Mountain, Chimera etc., however, any symbol which does not refer could not properly be called a symbol. So because every symbol necessarily refers to something definite, it is not the case that ambiguities and equivocations would sneak into symbolic expressions. Hence, logic becomes that science which prefers symbolic or artificial or formal language to natural language. Therefore, since “meaning” or semantics is a central focus of logic together with syntax, we attempted in this work to obtain a logical derivation of it in the symbolic language of logic.
Four-Valued” Semantics for the Relevant Logic R
Journal of Philosophical Logic, 2004
This paper sets out two semantics for the relevant logic R based on Dunn's four-valued semantics for first-degree entailments. Unlike Routley's semantics for weak relevant logics, they do not use two ternary accessibility relations. Unlike Restall's semantics, they capture all of R. But there is a catch. Both of the present semantics are neighbourhood semantics, that is, they include sets of propositions in the specification of their frames.
Shedding new light in the world of logical systems
Lecture Notes in Computer Science, 1997
The notion of an Institution 5] is here taken as the precise formulation for the notion of a logical system. By using elementary tools from the core of category theory, we are able to reveal the underlying mathematical structures lying \behind" the logical formulation of the satisfaction condition, and hence to acquire a both suitable and deeper understanding of the institution concept. This allows us to systematically approach the problem of describing and analyzing relations between logical systems. Theorem 2.10 redesigns the notion of an institution to a purely categorical level, so that the satisfaction condition becomes a functorial (and natural) transformation from speci cations to (subcategories of) models and vice versa. This systematic procedure is also applied to discuss and give a natural description for the notions of institution morphism and institution map. The last technical discussion is a careful and detailed analysis of two examples, which tries to outline how the new categorical insights could help in guiding the development of a unifying theory for relations between logical systems.
Negation, Opposition, and Possibility in Logical Concept Analysis
Lecture Notes in Computer Science, 2006
... in Logical Concept Analysis Sébastien Ferré Irisa/Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France ferre@irisa.fr Abstract. ... a syntax or language, ie, a set L of formulas, a set of operations like conjunction (⊓, binary), disjunction (⊔, binary ...
A semantic analysis of logics that cope with partial terms
2012
Specifications of programs frequently involve operators and functions that are not defined over all of their (syntactic) domains. Proofs about specifications–and those to discharge proof obligations that arise in justifying steps of design–must be based on formal rules. Since classical logic deals only with defined values, some extra thought is required. There are several ways of handling terms that can fail to denote a value—this paper provides a semantically based comparison of three of the best known approaches.
Positive and Negative Properties. A Logical Interpretation
Published as: KaczmarekJ., [2003], Positive and Negative Properties. A Logical Interpretation, [in:] Bulletin of the Section of Logic, vol. 32, No 4, pp. 179 - 189 In the paper we construct a simple sentential logic (L BK) based on the ontology presented in [3]. L BK has internal and external strata, which yield a double characterization of the connectives. This leads to the correspondence of L BK to Bochvar's and Kleene's logics. The connectives of the internal level correspond to the connectives of Bochvar's internal logic and of Kleene's weak logic, while the connectives of the external level correspond to those of Bochvar's external logic and of Kleene's strong logic. The ontological interpretation shows that the former represent the so-called connectives " de re " , the latter the connectives " de dicto ". 1. Inspirations We present a particular problem concerning properties of individuals. Given some representation of statements about individuals by the language of many – valued logic we show simple relations that connect objects and their properties. The problem is a piece of more general topic of formal representation of three categories of objects: universals (that are treated as incomplete objects), individuals (complete objects) and concepts (that are treated by philosophers as incomplete objects; we do not investigate them here). For our analysis we distinguished two types of objects: complete and incomplete. Complete objects can be identified with individuals (like Socrates, Tarski) and incomplete objects with universals (man, horse). The division and terminology are attributed to Meinong [6]. According to him we can also distinguish properties such as redness and compliment properties such as non-redness. So, if redness belongs to an object, then redness is its positive property. Hence, non-redness is its negative one. In accordance with classic ontological approach (Aristotle, medieval philosophers, Wolff) the individuals we talk about are described by these properties. Philosophers point out that one can distinguish three kinds of properties in objects: essential, attributive and accidental properties. Rational is an essential property of the man Socrates, being able to smile is his attribute and being dark-haired is his accidental property. Taking into account the division of properties into positive and negative we propose to say that rational is a positive and essential property of an individual. Non-rational would be an essential and negative property of that individual. Next, if all animals are non-rational then non-rational is positive and (perhaps) essential property of the individual horse. Similarly, in the case of Socrates being a lowyer is his negative and accidental property. For Meinong and Łukasiewicz any property P is a positive or negative property of an object a. In this paper we propose to limit this thesis by showing it appears evident that the properties odd or even can not be referred to a man. It means that the sentences: (1) Socrates is even, (2) Socrates is not even (is odd)