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The classic proof that the square root of 2 is irrational is a refutation by contradiction. [11] Indeed, we set out to prove the negation ¬ ∃ a, b ∈ . a/b = √ 2 by assuming that there exist natural numbers a and b whose ratio is the square root of two, and derive a contradiction.
Gödel's second incompleteness theorem also implies that a system F 1 satisfying the technical conditions outlined above cannot prove the consistency of any system F 2 that proves the consistency of F 1. This is because such a system F 1 can prove that if F 2 proves the consistency of F 1, then F 1 is in fact consistent.
In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. [1] [2] It is one of the three laws of thought, along with the law of noncontradiction, and the law of identity; however, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens ...
Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions. In other words, a logical truth is a statement which is not only true, but one which is true under all interpretations of its logical components (other ...
Logical reasoning is a form of thinking that is concerned with arriving at a conclusion in a rigorous way. [1] This happens in the form of inferences by transforming the information present in a set of premises to reach a conclusion. [2] [3] It can be defined as "selecting and interpreting information from a given context, making connections ...
Law of noncontradiction. In logic, the law of non-contradiction ( LNC) (also known as the law of contradiction, principle of non-contradiction ( PNC ), or the principle of contradiction) states that contradictory propositions cannot both be true in the same sense at the same time, e. g. the two propositions " p is the case " and " p is not the ...
A truth table is a structured representation that presents all possible combinations of truth values for the input variables of a Boolean function and their corresponding output values. A function f from A to F is a special relation, a subset of A×F, which simply means that f can be listed as a list of input-output pairs.
First-order logic —also called predicate logic, predicate calculus, quantificational logic —is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than ...