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THE HISTORY OF LOGIC
THE HISTORY OF LOGIC

... Gregory of Rimini, and Albert of Saxony. Buridan also elaborates a full theory of consequences, a cross between entailments and inference rules. From explicit semantic principles, Buridan constructs a detailed and extensive investigation of syllogistic, and offers completeness proofs. Nor is Buridan ...
Exercise
Exercise

... P(x) it is not enough to show that P(a) is true for one or some a’s. 2. To show that a statement of the form x P(x) is FALSE, it is enough to show that P(a) is false for one a ...
Lecture 23 Notes
Lecture 23 Notes

CHAPTER 10 Gentzen Style Proof Systems for Classical Logic 1
CHAPTER 10 Gentzen Style Proof Systems for Classical Logic 1

... computers. Their emphasis is on logical axioms, keeping the rules of inference at a minimum. Gentzen systems reverse this situation by emphasizing the importance of inference rules, reducing the role of logical axioms to an absolute minimum. They may be less intuitive then the Hilbert-style systems, ...
Algebraic Laws for Nondeterminism and Concurrency
Algebraic Laws for Nondeterminism and Concurrency

... well developed in recent years [ 1, 111and applied successfullyto many nontrivial languages.Even languageswith parallel constructs have been treated in this way, using the power-domain constructions of [3], [7], and [lo]. Indeed for such languagesthere is no shortageof possibledenotational models. F ...
Document
Document

... If A and B are formulas, then the expressions ~A, (A∧B), (A∨B) , A →B and A↔B are statement formulas, where ~, ∧, ∨, → and ↔ are logical connectives If A is a formula and x is a variable, then ∀x A(x) and ∃x A(x) are formulas All formulas constructed using only above rules are considered formulas ...
Lesson 1 Contents - Headlee's Math Mansion
Lesson 1 Contents - Headlee's Math Mansion

Subintuitionistic Logics with Kripke Semantics
Subintuitionistic Logics with Kripke Semantics

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Artificial Intelligence

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First-Order Logic with Dependent Types

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Principle of Mathematical Induction

... 1. “The Towers of Hanoi” is a puzzle with 3 nails and 7 rings, all of different sizes. Initially all rings are on the same nail in decreasing order from the bottom to the top. The procedure is removing the top ring from any nail and placing it on another nail. It is not allowed to place a bigger rin ...
Constructive Mathematics in Theory and Programming Practice
Constructive Mathematics in Theory and Programming Practice

... later in this paper). When working in any axiomatic system, we must take care to use only intuitionistic logic, and therefore to ensure that we do not adopt a classical axiom that implies LEM or some other nonconstructive principle. For example, in IZF we cannot adopt the common classical form of th ...
Temporal Here and There - Computational Cognition Lab
Temporal Here and There - Computational Cognition Lab

ppt - Purdue College of Engineering
ppt - Purdue College of Engineering

... • A tautology is a formula that is true in every model. (also called a theorem) – for example, (A  A) is a tautology – What about (AB)(AB)? – Look at tautological equivalences on pg. 8 of text ...
PLATONISM IN MODERN MATHEMATICS A University Thesis
PLATONISM IN MODERN MATHEMATICS A University Thesis

... form. According to the original notion articulated by Plato, an idea (or form) is a changeless object of knowledge; form involves problems and relationships between questions of knowledge, science, happiness, and politics, and distinguishes between knowledge and opinion. From Plato’s original theory ...
IS IT EASY TO LEARN THE LOGIC
IS IT EASY TO LEARN THE LOGIC

... is richer in possible alternatives to obtain the desired result by applying logical rules. However, the logical operations are necessary both in the decision as in the proof, because they put in motion the mind to get the best possible level of abstraction of reasoning. Hence, we must insist on pres ...
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First-Order Default Logic 1 Introduction

Logic (Mathematics 1BA1) Reminder: Sets of numbers Proof by
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Factoring out the impossibility of logical aggregation
Factoring out the impossibility of logical aggregation

Hyperbolic 3
Hyperbolic 3

... The points of each h-line l are assigned to the entire set of real numbers, called coordinates, in such a manner that (1) Each point on l is assigned to a unique number (2) No two points have the same number (3) Any two points on l may be assigned the coordinates zero and a positive real number resp ...
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Slide 1
Slide 1

A short article for the Encyclopedia of Artificial Intelligence: Second
A short article for the Encyclopedia of Artificial Intelligence: Second

... then it is η-convertible to λx(M x), provided x is not free in M ). Many standard proof-theoretic results – such as cut-elimination (Girard, 1986), unification (Huet, 1975), resolution (Andrews, 1971), and Skolemization and Herbrand’s Theorem (Miller, 1987) – have been formulated for this fragment. ...
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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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