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Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statements that hold true when one statement logically follows from one or more statements.
In logical argument and mathematical proof, the therefore sign, ∴, is generally used before a logical consequence, such as the conclusion of a syllogism. The symbol consists of three dots placed in an upright triangle and is read therefore. While it is not generally used in formal writing, it is used in mathematics and shorthand.
In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics.
Syntax is concerned with the rules used for constructing, or transforming the symbols and words of a language, as contrasted with the semantics of a language which is concerned with its meaning. The symbols , formulas , systems , theorems and proofs expressed in formal languages are syntactic entities whose properties may be studied without ...
The double turnstile is a binary relation. It has several different meanings in different contexts: To show semantic consequence, with a set of sentences on the left and a single sentence on the right, to denote that if every sentence on the left is true, the sentence on the right must be true, e.g. .
When referring to hypothetical future circumstance, there may be little difference in meaning between the first and second conditional (factual vs. counterfactual, realis vs. irrealis). The following two sentences have similar meaning, although the second (with the second conditional) implies less likelihood that the condition will be fulfilled:
Writing a closed letter 'O' means that you are a private person and an introvert. If the dot on your 'i' lands high above the letter, you are considered to be imaginative.
Logical consequence, also known as a consequence relation, or entailment Consequent , in logic, the second half of a hypothetical proposition or consequences Consequentialism , a theory in philosophy in which the morality of an act is determined by its effects