Systems of modal logic - Department of Computing
... system Σ simply when A ∈ Σ. Which closure conditions? See below. Systems of modal logic can also be defined (syntactically) in other ways, usually by reference to some kind of proof system. For example: • Hilbert systems: given a set of formulas called axioms and a set of rules of proof, a formula A ...
... system Σ simply when A ∈ Σ. Which closure conditions? See below. Systems of modal logic can also be defined (syntactically) in other ways, usually by reference to some kind of proof system. For example: • Hilbert systems: given a set of formulas called axioms and a set of rules of proof, a formula A ...
Abella: A System for Reasoning about Relational Specifications
... effort [5, 22, 34, 35, 43, 63, 64, 69] to design an increasingly more flexible and powerful logic that would provide such a treatment of binding. The Abella prover has allowed this work on proof theory to be validated and exploited within an interactive theorem proving setting. ...
... effort [5, 22, 34, 35, 43, 63, 64, 69] to design an increasingly more flexible and powerful logic that would provide such a treatment of binding. The Abella prover has allowed this work on proof theory to be validated and exploited within an interactive theorem proving setting. ...
The Farey Sequence and Its Niche(s)
... I will now begin to to cover some of the basic properties of the Farey sequence. These properties will allow us to prove much more powerful things in the future. First things first let us start by finding an equation for the length of Fn . However, before define the equation we must do a quick revie ...
... I will now begin to to cover some of the basic properties of the Farey sequence. These properties will allow us to prove much more powerful things in the future. First things first let us start by finding an equation for the length of Fn . However, before define the equation we must do a quick revie ...
Integer Sequences Related to Compositions without 2`s
... i + j 6= 2. This correspondence is actually one-to-one since any occurrence of i + j among the compositions of n + j corresponds to an occurrence of i among the compositions of n obtained by replacing i + j by i, subject to the condition that i 6= 2. As a result of Theorem 2, we only need to genera ...
... i + j 6= 2. This correspondence is actually one-to-one since any occurrence of i + j among the compositions of n + j corresponds to an occurrence of i among the compositions of n obtained by replacing i + j by i, subject to the condition that i 6= 2. As a result of Theorem 2, we only need to genera ...
An explicit version of Birch`s Theorem
... diagonalisation method of Birch [1] first described in Wooley [15], where we restricted our investigations to systems of cubic and quintic forms. Although the size of our bounds may be aptly described as “not even astronomical” (an eloquent phrase of Birch), it seems that this paper contains the fir ...
... diagonalisation method of Birch [1] first described in Wooley [15], where we restricted our investigations to systems of cubic and quintic forms. Although the size of our bounds may be aptly described as “not even astronomical” (an eloquent phrase of Birch), it seems that this paper contains the fir ...
On Triangular and Trapezoidal Numbers
... This paper examined a special type of figurate number, the triangular number. In general, a triangular number is the number of dots that can be arranged to form an equilateral triangle. One purpose of this study is to provide a comprehensive explanation of this type of number. Though these numbers a ...
... This paper examined a special type of figurate number, the triangular number. In general, a triangular number is the number of dots that can be arranged to form an equilateral triangle. One purpose of this study is to provide a comprehensive explanation of this type of number. Though these numbers a ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.