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On modal logics of group belief
On modal logics of group belief

page 135 LOGIC IN WHITEHEAD`S UNIVERSAL ALGEBRA
page 135 LOGIC IN WHITEHEAD`S UNIVERSAL ALGEBRA

Mathematical Logic
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... know which numbers a, b we must take. Hence we have an example of an existence proof which does not provide an instance. An essential point for Mathematical Logic is to fix a formal language to be used. We take implication → and the universal quantifier ∀ as basic. Then the logic rules correspond to ...
Reasoning about Complex Actions with Incomplete Knowledge: A
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... represented by modalities, and we extend it by allowing sensing actions as well as complex actions definitions. Our starting point is the modal logic programming language for reasoning about actions presented in [6]. Such language mainly focuses on ramification problem but does not provide a formaliz ...
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Introduction to first order logic for knowledge representation

... are used to indicate the basic (atomic) components of the (part of the) world the logic is supposed to describe. The alphabet is composed of two subsets: the logical symbols and the non logical symbols. Examples of such atomic objects are, individuals, functions, operators, truth-values, proposition ...
minimum models: reasoning and automation
minimum models: reasoning and automation

Intuitionistic Logic - Institute for Logic, Language and Computation
Intuitionistic Logic - Institute for Logic, Language and Computation

... respect, see e.g. [4]. In fact, there is less of this kind of work going on now even than before. On the other hand, one might say that intuitionism describes a particular portion of mathematics, the constructive part, and that it has been described very adequately by now what the meaning of that co ...
Sets, Logic, Computation
Sets, Logic, Computation

... between and constructions using these. It will be good to have shorthand symbols for these, and think through the general properties of sets, relations, and functions in order to do that. If you are not used to thinking mathematically and to formulating mathematical proofs, then think of the first p ...
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brouwer`s intuitionism as a self-interpreted mathematical theory

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The Dedekind Reals in Abstract Stone Duality

... and finite intersections that makes it difficult to see the duality between open and closed phenomena. Intuitionistic foundations also obscure this symmetry by stating many results that are naturally about closed sets in a form that uses double negations. When we treat the lattice of opens of one sp ...
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... model of T . Does the converse hold? The question was posed in the 1920’s by David Hilbert (of the 23 problems fame). The answer is that remarkably, yes, it does! This result, known as the Completeness Theorem for first-order logic, was proved by Kurt Gödel in 1929. According to the Completeness Th ...
higher-order logic - University of Amsterdam
higher-order logic - University of Amsterdam

... In addition to its primitives all and some, a first-order predicate language with identity can also express such quantifiers as precisely one, all but two, at most three, etcetera, referring to specific finite quantities. What is lacking, however, is the general mathematical concept of finiteness. E ...
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... constant ⊥; 3) absurdity implies everything, ⊥ ⊃ p, or ¬p ⊃ (p ⊃ q). Note that item 3 implies, in fact, item 2. If any contradiction implies everything, all contradictions are equivalent, and so we may use a propositional constant do denote arbitrary contradictory or absurd statement. In minimal log ...
Gödel`s Theorems
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Deep Sequent Systems for Modal Logic
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Quadripartitaratio - Revistas Científicas de la Universidad de
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An Introduction to Proof Theory - UCSD Mathematics
An Introduction to Proof Theory - UCSD Mathematics

... (usually finite) language L — for the time being, we shall use the language ¬, ∧, ∨ and ⊃. A propositional formula A is said to be a tautology or to be (classically) valid if A is assigned the value T by every truth assignment. We write ² A to denote that A is a tautology. The formula A is satisfiab ...
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logic for the mathematical

... sometimes taken aback (or should I say “freaked out”, so as not to show my age too much) when confronted with a deduction theorem which appears to have a hypothesis missing. The final chapter relates the 20th century style of logic from earlier chapters to what Aristotle and many followers did, as w ...
Proof Technique
Proof Technique

Master Thesis - Yoichi Hirai
Master Thesis - Yoichi Hirai

Henkin`s Method and the Completeness Theorem
Henkin`s Method and the Completeness Theorem

... theory T (pick your favorite one!) is sound and complete in the sense that everything that we can derive from T is true in all models of T , and everything that is true in all models of T is in fact derivable from T . This is a very strong result indeed. One possible reading of it is that the first- ...
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Axiom

An axiom or postulate is a premise or starting point of reasoning. As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy.The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.' As used in modern logic, an axiom is simply a premise or starting point for reasoning. What it means for an axiom, or any mathematical statement, to be ""true"" is a central question in the philosophy of mathematics, with modern mathematicians holding a multitude of different opinions.In mathematics, the term axiom is used in two related but distinguishable senses: ""logical axioms"" and ""non-logical axioms"". Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, ""axiom,"" ""postulate"", and ""assumption"" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. As modern mathematics admits multiple, equally ""true"" systems of logic, precisely the same thing must be said for logical axioms - they both define and are specific to the particular system of logic that is being invoked. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Within the system they define, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow otherwise they would be classified as theorems. However, an axiom in one system may be a theorem in another, and vice versa.
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