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BASIC COUNTING - Mathematical sciences
BASIC COUNTING - Mathematical sciences

... is proved or it is not. Thus to be a mathematician is to learn to prove. We will learn what constitutes a valid proof and why these proofs (and not others) establish truth. As a useful step along the way we will learn a taxonomy (naming scheme) of proofs. ...
The Complete Proof Theory of Hybrid Systems
The Complete Proof Theory of Hybrid Systems

... is provable2 [18, BFC p. 245] and we also call it B. Axiom V is for vacuous modalities and requires that no free variable of φ (written F V (φ)) is bound by α. The converse holds, but we do not need it. Rule G is Gödel’s necessitation rule for modal logic [18]. Note that, unlike rule G, axiom V cru ...
A Logical Expression of Reasoning
A Logical Expression of Reasoning

Supplemental Reading 1
Supplemental Reading 1

The Foundations
The Foundations

... Equivalences: pq  (p  q) Proving equivalences using: ...
P(x)
P(x)

... – A term (denoting a real-world object) is a constant symbol, a variable symbol, or a function e.g. left-leg-of ( ). For example, x and f(x1, ..., xn) are terms, where each xi is a term. – An atom (which has value true or false) is either an n-place predicate of n terms, or, if P and Q are atoms, th ...
P(x) - Carnegie Mellon School of Computer Science
P(x) - Carnegie Mellon School of Computer Science

The Expressive Power of Modal Dependence Logic
The Expressive Power of Modal Dependence Logic

... Väänänen [17] introduced modal dependence logic MDL. In the context of modal logic a team is just a set of states in a Kripke model. Modal dependence logic extends standard modal logic with team semantics by modal dependence atoms, =(p1 , . . . , pn , q). The intuitive meaning of the formula =(p1 , ...
PDF - University of Kent
PDF - University of Kent

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

... various sides of intuitionistic logic. In no way we strive for a complete overview in this short course. Even though we approach the subject for the most part only formally, it is good to have a general introduction to intuitionism. This we give in section 2 in which also natural deduction is introd ...
page 135 ADAPTIVE LOGICS FOR QUESTION EVOCATION
page 135 ADAPTIVE LOGICS FOR QUESTION EVOCATION

The Foundations
The Foundations

IM_FA16-03-PredicateLogic
IM_FA16-03-PredicateLogic

Logic Part II: Intuitionistic Logic and Natural Deduction
Logic Part II: Intuitionistic Logic and Natural Deduction

... in many elds of mathematics, there are contradictory propositions from which anything is derivable ...
Probability Captures the Logic of Scientific
Probability Captures the Logic of Scientific

cs-171-15-FOL-Inference
cs-171-15-FOL-Inference

... Problem: works if α is entailed, loops if α is not entailed.  The problem of semi-decidable: algorithms exist to prove entailment, but no algorithm exists to to prove non-entailment for every non-entailed sentence. ...
Formal Theories of Truth INTRODUCTION
Formal Theories of Truth INTRODUCTION

... A1 all axioms and rules of first-order predicate logic including the identity axioms. A2 aa b = ab, where a and b are arbitrary strings of symbols. A3 q(a) = a A4 sub(a, b, c) = d, where a and c are arbitrary strings of symbols, b is a single symbol (or, equivalently, a string of symbols of length 1 ...
The Foundations
The Foundations

... Propositional or Boolean operators operate on propositions or truth values instead of on numbers. Transparency No. 1-9 ...
The Project Gutenberg EBook of The Algebra of Logic, by Louis
The Project Gutenberg EBook of The Algebra of Logic, by Louis

... and Russell. Of course there is no hard and fast boundary-line between the domains of these two parties. Thus Peirce and Schröder early began to work at the foundations of arithmetic with the help of the calculus of relations; and thus they did not consider the logical calculus merely as an interest ...
Dedukti
Dedukti

... called the Logical framework [47], also known as the λΠ-calculus, and the λ-calculus with dependent types. Problems 3 and 4 have been solved in an extension of predicate logic, called Deduction modulo theory [40]. Combining the λΠ-calculus and Deduction modulo theory yields the λΠ-calculus modulo th ...
Completeness in modal logic - Lund University Publications
Completeness in modal logic - Lund University Publications

... necessity and ◊ for possibility. Φ is true at a point p iff Φ is true at every point p can see. ◊Φ is true at p iff Φ is true at some such point. The points may be called possible worlds or just worlds, since that is the most illustrative way to think of them in modal applications. We can now define ...
cl-ch9
cl-ch9

The Deduction Rule and Linear and Near
The Deduction Rule and Linear and Near

CHAPTER 1. SENTENTIAL LOGIC 1. Introduction In sentential logic
CHAPTER 1. SENTENTIAL LOGIC 1. Introduction In sentential logic

... Definition 3.1. Let A be a set of atomic sentences. A truth assignment for A is any function h : A → {T, F }. For any set A of atomic sentences we let A be the set of all sentences containing only atomic sentences from A. The following theorem states that every truth assignment for A extends uniquel ...
Consequence Operators for Defeasible - SeDiCI
Consequence Operators for Defeasible - SeDiCI

... The intuitive meaning of inclusion and idempotence should be clear without further comments. The cut rule states that expanding the information in ¡ by adding new propositions from C (¡ ) does not result in new conclusions being obtained. Cautious monotonicity constitutes somehow the `inverse' of cu ...
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Natural deduction

In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the ""natural"" way of reasoning. This contrasts with the axiomatic systems which instead use axioms as much as possible to express the logical laws of deductive reasoning.
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