page 135 LOGIC IN WHITEHEAD`S UNIVERSAL ALGEBRA
... There are various ways of ‘being together’, that is, various ways of building classes, and this makes ‘togetherness’ an ambiguous notion. The extensionalist Whitehead notes that the particular mode of ‘togetherness’ is already an intension that infects the composite entity. Therefore, in mathematics ...
... There are various ways of ‘being together’, that is, various ways of building classes, and this makes ‘togetherness’ an ambiguous notion. The extensionalist Whitehead notes that the particular mode of ‘togetherness’ is already an intension that infects the composite entity. Therefore, in mathematics ...
Linear Contextual Modal Type Theory
... can be modeled as resources, in programming language theory it is state, and in security simply messages that are being created and consumed. Traditionally one recovers intuitionistic logic from linear logic by singling out those resources that can be constructed from no other resources. They can be ...
... can be modeled as resources, in programming language theory it is state, and in security simply messages that are being created and consumed. Traditionally one recovers intuitionistic logic from linear logic by singling out those resources that can be constructed from no other resources. They can be ...
Partial Grounded Fixpoints
... We start by giving the central definition of this text, namely the notion of A-groundedness. Definition 3.1. Let A be an approximator of O. A point (x, y) ∈ Lc is A-grounded if for every v ∈ L with A(x ∧ v, y ∧ v)2 ≤ v, also y ≤ v. The definition of this concept is a direct generalisation of grounde ...
... We start by giving the central definition of this text, namely the notion of A-groundedness. Definition 3.1. Let A be an approximator of O. A point (x, y) ∈ Lc is A-grounded if for every v ∈ L with A(x ∧ v, y ∧ v)2 ≤ v, also y ≤ v. The definition of this concept is a direct generalisation of grounde ...
1992-Ideal Introspective Belief
... derivable from the premises alone. For example, consider the premise set {lLp > q,p V q}. We would like since there is no reasonable way of to conclude ‘Lp, coming to believe p. But an inference rule that would allow us to conclude 1Lp would have to take into account all possible derivations, includ ...
... derivable from the premises alone. For example, consider the premise set {lLp > q,p V q}. We would like since there is no reasonable way of to conclude ‘Lp, coming to believe p. But an inference rule that would allow us to conclude 1Lp would have to take into account all possible derivations, includ ...
Chapter 11: Other Logical Tools Syllogisms and Quantification
... previously, propositional logic ignores the different meanings these statements would have in different cultural contexts. Recall also the paradox of material implication discussed in Chapter 8. Our background knowledge about how the world works conflicts with some results in propositional logic. Th ...
... previously, propositional logic ignores the different meanings these statements would have in different cultural contexts. Recall also the paradox of material implication discussed in Chapter 8. Our background knowledge about how the world works conflicts with some results in propositional logic. Th ...
Formal deduction in propositional logic
... Intuitive meaning of rules • The elimination (introduction) of a connective means that one occurrence of this connective is eliminated (introduced) in the conclusion of the scheme of formal deducibility generated by the rule. • Remark: In (∨−) it is the ∨ between A and B in A ∨ B that is eliminated ...
... Intuitive meaning of rules • The elimination (introduction) of a connective means that one occurrence of this connective is eliminated (introduced) in the conclusion of the scheme of formal deducibility generated by the rule. • Remark: In (∨−) it is the ∨ between A and B in A ∨ B that is eliminated ...
Taming method in modal logic and mosaic method in temporal logic
... PALH ’s universe is any binary relation satisfying the conditions in H, and it’s the relativized version of PAL. PAL∅ is the completely relativized version of PAL. In PALSQ transitivity of the universe ensures that composition is associative and that causes Hilbert incompleteness and undecidability ...
... PALH ’s universe is any binary relation satisfying the conditions in H, and it’s the relativized version of PAL. PAL∅ is the completely relativized version of PAL. In PALSQ transitivity of the universe ensures that composition is associative and that causes Hilbert incompleteness and undecidability ...
page 139 MINIMIZING AMBIGUITY AND
... make use of at least two different predicates. Same thing for Einstein as a grown up, and Einstein at the age of three, and for expressions as “it rains”. Nevertheless, this view does not give an answer to the question “Which (occurrences of) non-logical terms should be precise enough?”. Obviously w ...
... make use of at least two different predicates. Same thing for Einstein as a grown up, and Einstein at the age of three, and for expressions as “it rains”. Nevertheless, this view does not give an answer to the question “Which (occurrences of) non-logical terms should be precise enough?”. Obviously w ...
Cylindric Modal Logic - Homepages of UvA/FNWI staff
... two α-tuples iff x and y differ at most in their i-th coordinate. The crucial observation, and in fact the basic observation underlying our whole enterprise, is that this truth definition is in fact of a modal nature: we may see Dij and ≡i as unary resp. binary accessibility relations on the α-dimen ...
... two α-tuples iff x and y differ at most in their i-th coordinate. The crucial observation, and in fact the basic observation underlying our whole enterprise, is that this truth definition is in fact of a modal nature: we may see Dij and ≡i as unary resp. binary accessibility relations on the α-dimen ...
Strong Logics of First and Second Order
... This will be the main purpose of Sections 1 and 2. In Section 1 we shall investigate the many facets of the absoluteness of first-order logic. In Section 2 we shall start by investigating two traditional strong logics (ω-logic and β-logic) that share many of these features of absoluteness, only now ...
... This will be the main purpose of Sections 1 and 2. In Section 1 we shall investigate the many facets of the absoluteness of first-order logic. In Section 2 we shall start by investigating two traditional strong logics (ω-logic and β-logic) that share many of these features of absoluteness, only now ...
Introduction to Discrete Structures Introduction
... • Definition: The ordered n-tuple (a1,a2,…,an) is the ordered collection with the element ai being the i-th element for i=1,2,…,n • Two ordered n-tuples (a1,a2,…,an) and (b1,b2,…,bn) are equal if and only if for every i=1,2,…,n we have ai=bi (a1,a2,…,an) • A 2-tuple (n=2) is called an ordered pair C ...
... • Definition: The ordered n-tuple (a1,a2,…,an) is the ordered collection with the element ai being the i-th element for i=1,2,…,n • Two ordered n-tuples (a1,a2,…,an) and (b1,b2,…,bn) are equal if and only if for every i=1,2,…,n we have ai=bi (a1,a2,…,an) • A 2-tuple (n=2) is called an ordered pair C ...
BASIC COUNTING - Mathematical sciences
... then q” or “p implies q”. In English this phrase carries many meanings. Sometimes it means that p causes q as in “if you eat too much you will get fat.” Sometimes it means that p guarantees q and vice versa as in “if you write a book report, I will give you five points extra credit” (tacitly assurin ...
... then q” or “p implies q”. In English this phrase carries many meanings. Sometimes it means that p causes q as in “if you eat too much you will get fat.” Sometimes it means that p guarantees q and vice versa as in “if you write a book report, I will give you five points extra credit” (tacitly assurin ...
A pragmatic dialogic interpretation of bi
... identify, among the mathematical models of bi-intuitionism, those which may be regarded as its intended interpretations. The quest for an intended interpretation of a formal system often arises when several mathematical structures have been proposed to characterise an informal, perhaps vague notion ...
... identify, among the mathematical models of bi-intuitionism, those which may be regarded as its intended interpretations. The quest for an intended interpretation of a formal system often arises when several mathematical structures have been proposed to characterise an informal, perhaps vague notion ...
Knowledge Representation and Reasoning
... (and with ‘reasonable’ use of other resources such as memory). Certain classes of logical problem are not only intractable but also undecidable. This means that there is no program that, given any instance of the problem, will in finite time either: a) find a solution; or b) terminate having determi ...
... (and with ‘reasonable’ use of other resources such as memory). Certain classes of logical problem are not only intractable but also undecidable. This means that there is no program that, given any instance of the problem, will in finite time either: a) find a solution; or b) terminate having determi ...
INTERPLAYS OF KNOWLEDGE AND NON
... the interdefinability of ∆ and ∇, it follows ∇p → ∇ K p. Given that p is, by assumption, contingent, we have that ∇ K p. This result by G. H. Von Wright on the metaphysical status of knowledge containing connections of knowledge and (non-)contingency, developed in the fusion, is relevant for the met ...
... the interdefinability of ∆ and ∇, it follows ∇p → ∇ K p. Given that p is, by assumption, contingent, we have that ∇ K p. This result by G. H. Von Wright on the metaphysical status of knowledge containing connections of knowledge and (non-)contingency, developed in the fusion, is relevant for the met ...
what are we to accept, and what are we to reject
... for work on non-classical logic), he takes care to present it in a language that the outsider can understand. While an Australian like me—who breathes in nonclassical approaches from my early logical education—learns how to work with paraconsistent logics in a culture in which they’re a regular topi ...
... for work on non-classical logic), he takes care to present it in a language that the outsider can understand. While an Australian like me—who breathes in nonclassical approaches from my early logical education—learns how to work with paraconsistent logics in a culture in which they’re a regular topi ...
article in press - School of Computer Science
... intuitionistic account of the notions studied in modal logic; and suitability of intuitionistic modal logic for modelling certain computational phenomena. There exists an extensive literature on intuitionistic modal logics, for example [5–7,10,12,13,15,20,22,23,26,27,32– 35]. A comprehensive survey ...
... intuitionistic account of the notions studied in modal logic; and suitability of intuitionistic modal logic for modelling certain computational phenomena. There exists an extensive literature on intuitionistic modal logics, for example [5–7,10,12,13,15,20,22,23,26,27,32– 35]. A comprehensive survey ...
this PDF file
... situations (cf. Remark 4 below). Remark 1: 1. The axiom m.p is sometimes labelled “pseudo-modus/ponens” in order to distinguish it from the rule modus ponens. 2. The variable-sharing property (vsp) is the following: A logic S has the vsp if in any theorem of S of the form A → B, A and B share at lea ...
... situations (cf. Remark 4 below). Remark 1: 1. The axiom m.p is sometimes labelled “pseudo-modus/ponens” in order to distinguish it from the rule modus ponens. 2. The variable-sharing property (vsp) is the following: A logic S has the vsp if in any theorem of S of the form A → B, A and B share at lea ...
The logic and mathematics of occasion sentences
... occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of occasion sentences and a mathematical (Boolean) foundation for such a logic, thus ...
... occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of occasion sentences and a mathematical (Boolean) foundation for such a logic, thus ...
full text (.pdf)
... logic, foreshadowing Kripke’s [1963; 1965] formulation of similar state-based semantics for these logics (see [Artemov 2001]). Kripke models also form the basis of the standard semantics of DL (see [Harel et al. 2000]), although as mentioned, DL does not realize the intuitionistic nature of partial ...
... logic, foreshadowing Kripke’s [1963; 1965] formulation of similar state-based semantics for these logics (see [Artemov 2001]). Kripke models also form the basis of the standard semantics of DL (see [Harel et al. 2000]), although as mentioned, DL does not realize the intuitionistic nature of partial ...
Internal Inconsistency and the Reform of Naïve Set Comprehension
... numbers. The next highest ordinal is itself an ordinal (because of its own description) and (by definition based on the concept of the ordering of the series of ordinals) larger not only than all ordinals including its immediate predecessor (the set of all ordinals) but also than itself. In this cas ...
... numbers. The next highest ordinal is itself an ordinal (because of its own description) and (by definition based on the concept of the ordering of the series of ordinals) larger not only than all ordinals including its immediate predecessor (the set of all ordinals) but also than itself. In this cas ...
THE ABUNDANCE OF THE FUTURE A Paraconsistent Approach to
... to its contradictory flavor. Indeed, this position is naturally embodied by a paraconsistent semantics (Section 4). Can this possibly make sense or it is just a “weird” conceptual alternative among all the possible temporal logics? The question becomes more pressing since this alternative seems to h ...
... to its contradictory flavor. Indeed, this position is naturally embodied by a paraconsistent semantics (Section 4). Can this possibly make sense or it is just a “weird” conceptual alternative among all the possible temporal logics? The question becomes more pressing since this alternative seems to h ...
arXiv:1410.5037v2 [cs.LO] 18 Jun 2016
... language. Below we shall consider logics FO(A), where the above syntax is extended by clauses of the type AQ (y 1 , ..., yk ). Here AQ is (a symbol corresponding to) a generalized atom in A and each yi is a tuple of variables. Before considering such novel atoms, let us define lax team semantics for ...
... language. Below we shall consider logics FO(A), where the above syntax is extended by clauses of the type AQ (y 1 , ..., yk ). Here AQ is (a symbol corresponding to) a generalized atom in A and each yi is a tuple of variables. Before considering such novel atoms, let us define lax team semantics for ...
Willard Van Orman Quine
Willard Van Orman Quine (/kwaɪn/; June 25, 1908 – December 25, 2000) (known to intimates as ""Van"") was an American philosopher and logician in the analytic tradition, recognized as ""one of the most influential philosophers of the twentieth century."" From 1930 until his death 70 years later, Quine was continually affiliated with Harvard University in one way or another, first as a student, then as a professor of philosophy and a teacher of logic and set theory, and finally as a professor emeritus who published or revised several books in retirement. He filled the Edgar Pierce Chair of Philosophy at Harvard from 1956 to 1978. A recent poll conducted among analytic philosophers named Quine as the fifth most important philosopher of the past two centuries. He won the first Schock Prize in Logic and Philosophy in 1993 for ""his systematical and penetrating discussions of how learning of language and communication are based on socially available evidence and of the consequences of this for theories on knowledge and linguistic meaning."" In 1996 he was awarded the Kyoto Prize in Arts and Philosophy for his ""outstanding contributions to the progress of philosophy in the 20th century by proposing numerous theories based on keen insights in logic, epistemology, philosophy of science and philosophy of language.""Quine falls squarely into the analytic philosophy tradition while also being the main proponent of the view that philosophy is not conceptual analysis but the abstract branch of the empirical sciences. His major writings include ""Two Dogmas of Empiricism"" (1951), which attacked the distinction between analytic and synthetic propositions and advocated a form of semantic holism, and Word and Object (1960), which further developed these positions and introduced Quine's famous indeterminacy of translation thesis, advocating a behaviorist theory of meaning. He also developed an influential naturalized epistemology that tried to provide ""an improved scientific explanation of how we have developed elaborate scientific theories on the basis of meager sensory input."" He is also important in philosophy of science for his ""systematic attempt to understand science from within the resources of science itself"" and for his conception of philosophy as continuous with science. This led to his famous quip that ""philosophy of science is philosophy enough."" In philosophy of mathematics, he and his Harvard colleague Hilary Putnam developed the ""Quine–Putnam indispensability thesis,"" an argument for the reality of mathematical entities.