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Statement (logic) In logic and semantics, the term statement is variously understood to mean either: a meaningful declarative sentence that is true or false, or. a proposition. Which is the assertion that is made by (i.e., the meaning of) a true or false declarative sentence. In the latter case, a statement is distinct from a sentence in that a ...
Atomic sentence. In logic and analytic philosophy, an atomic sentence is a type of declarative sentence which is either true or false (may also be referred to as a proposition, statement or truthbearer) and which cannot be broken down into other simpler sentences. For example, "The dog ran" is an atomic sentence in natural language, whereas ...
In logic, the logical form of a statement is a precisely-specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system. In an ideal formal language, the meaning ...
Sentence (mathematical logic) In mathematical logic, a sentence (or closed formula) [1] of a predicate logic is a Boolean -valued well-formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that must be true or false. The restriction of having no free variables is needed to make sure that ...
Colorless green ideas sleep furiously was composed by Noam Chomsky in his 1957 book Syntactic Structures as an example of a sentence that is grammatically well-formed, but semantically nonsensical. The sentence was originally used in his 1955 thesis The Logical Structure of Linguistic Theory and in his 1956 paper "Three Models for the ...
Sentence (linguistics) In linguistics and grammar, a sentence is a linguistic expression, such as the English example " The quick brown fox jumps over the lazy dog." In traditional grammar, it is typically defined as a string of words that expresses a complete thought, or as a unit consisting of a subject and predicate.
A statement is logically true if, and only if its opposite is logically false. The opposite statements must contradict one another. In this way all logical connectives can be expressed in terms of preserving logical truth. The logical form of a sentence is determined by its semantic or syntactic structure and by the placement of logical constants.
Wherever logic is applied, especially in mathematical discussions, it has the same meaning as above: it is an abbreviation for if and only if, indicating that one statement is both necessary and sufficient for the other. This is an example of mathematical jargon (although, as noted above, if is more often used than iff in statements of definition).
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