A Horn Clause that Implies an Undecidable Set of Horn Clauses ⋆ 1
... the author of the theorem here, however following [22] we shall present a proof of it. Proof: Let H = (Q(t1 ) =⇒ Q(t2 )). Let Q(s1 ) be the (hypothetical) label of the premise of the derivation and Q(s2 ) be the one of the conclusion (s1 and s2 are ground terms). Following[22] we define H1 = H . Let ...
... the author of the theorem here, however following [22] we shall present a proof of it. Proof: Let H = (Q(t1 ) =⇒ Q(t2 )). Let Q(s1 ) be the (hypothetical) label of the premise of the derivation and Q(s2 ) be the one of the conclusion (s1 and s2 are ground terms). Following[22] we define H1 = H . Let ...
6.042J Chapter 1: Propositions
... one knows if it is true or false. But that doesn’t prevent you from answering the question! This proposition has the form P IMPLIES Q where the hypothesis, P , is “the Riemann Hypothesis is true” and the conclusion, Q, is “x 2 0 for every real number x”. Since the conclusion is definitely true, we ...
... one knows if it is true or false. But that doesn’t prevent you from answering the question! This proposition has the form P IMPLIES Q where the hypothesis, P , is “the Riemann Hypothesis is true” and the conclusion, Q, is “x 2 0 for every real number x”. Since the conclusion is definitely true, we ...
Section 2.2: Introduction to the Logic of Quantified Statements
... If P (x) is false for all x ∈ D, then the universal conditional statement is true. In such circumstances, we say it is true by default or vacuously true. ...
... If P (x) is false for all x ∈ D, then the universal conditional statement is true. In such circumstances, we say it is true by default or vacuously true. ...
Rules of inference
... “It is below freezing now (p). Therefore, it is either below freezing or raining now (q).” “It is below freezing (p). It is raining now (q). Therefore, it is below freezing and it is raining now. “if it rains today (p), then we will not have a barbecue today (q). if we do not have a barbecue t ...
... “It is below freezing now (p). Therefore, it is either below freezing or raining now (q).” “It is below freezing (p). It is raining now (q). Therefore, it is below freezing and it is raining now. “if it rains today (p), then we will not have a barbecue today (q). if we do not have a barbecue t ...
Distinguishing Cartesian powers of graphs
... = G, and V (G) = V (H) = {1, . . . , k}. Color the vertices (k − 1, k − 1), (k, k), the vertices (i, j) with 1 ≤ i < j ≤ k, and vertices (i, i − 2) for 3 ≤ i ≤ k black and the other ones white. Since G is prime, all automorphisms of G ¤ H are generated by automorphism of G or H or interchanges of th ...
... = G, and V (G) = V (H) = {1, . . . , k}. Color the vertices (k − 1, k − 1), (k, k), the vertices (i, j) with 1 ≤ i < j ≤ k, and vertices (i, i − 2) for 3 ≤ i ≤ k black and the other ones white. Since G is prime, all automorphisms of G ¤ H are generated by automorphism of G or H or interchanges of th ...
Strong Normalisation for a Gentzen-like Cut
... which there are two contexts, Γ and ∆, such that Γ . M . ∆ holds given the inference rules in Figure 1. We shall write T∧ for the set of well-typed terms. Whilst the structural rules are implicit in our sequent calculus, i.e., the calculus has fewer inference rules, there are a number of subtleties ...
... which there are two contexts, Γ and ∆, such that Γ . M . ∆ holds given the inference rules in Figure 1. We shall write T∧ for the set of well-typed terms. Whilst the structural rules are implicit in our sequent calculus, i.e., the calculus has fewer inference rules, there are a number of subtleties ...
Secondary English Language Arts
... 7.G.4.1 Compute the perimeter and area of common geometric shapes and use the results to find measures of less common objects. 7.G.4.2 Identify and describe the properties of two-dimensional figures: a. identify angles as vertical, adjacent, complementary, or supplementary and provide descriptions o ...
... 7.G.4.1 Compute the perimeter and area of common geometric shapes and use the results to find measures of less common objects. 7.G.4.2 Identify and describe the properties of two-dimensional figures: a. identify angles as vertical, adjacent, complementary, or supplementary and provide descriptions o ...
§1. Basic definitions Let IR be the set of all real numbers, while IR
... Definition 1.1. Let X be a set of elements of arbitrary nature. For any n ∈ IN, by xn we denote an element of X corresponding to n. This defines a set of numbered elements {x1 , x2 , . . . , xn , . . .} (or, for brevity, {xn }) called a sequence defined on X. If X = IR then {xn } is called a sequenc ...
... Definition 1.1. Let X be a set of elements of arbitrary nature. For any n ∈ IN, by xn we denote an element of X corresponding to n. This defines a set of numbered elements {x1 , x2 , . . . , xn , . . .} (or, for brevity, {xn }) called a sequence defined on X. If X = IR then {xn } is called a sequenc ...
Discrete Mathematics: Chapter 2, Predicate Logic
... valid , their validity is not due to combinatorial truth-functional logical structure, which is all that SL takes into consideration. Their validity depends rather upon the deeper internal logical structure of the sentences involved. SL judges such arguments not to be valid based on its criteria of ...
... valid , their validity is not due to combinatorial truth-functional logical structure, which is all that SL takes into consideration. Their validity depends rather upon the deeper internal logical structure of the sentences involved. SL judges such arguments not to be valid based on its criteria of ...
Document
... (b) If you tried to prove by a similar argument that “ 4 is irrational”: that there was no rational number x such that x2 = 4, where would the first mistake in the proof be? For which positive integers n do the methods actually work? That is, for which n can you prove that there is no rational x suc ...
... (b) If you tried to prove by a similar argument that “ 4 is irrational”: that there was no rational number x such that x2 = 4, where would the first mistake in the proof be? For which positive integers n do the methods actually work? That is, for which n can you prove that there is no rational x suc ...
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.