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In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value. If the values of all variables in a propositional formula are given, it determines a unique truth value. A propositional formula may also be called a propositional expression, a sentence, or a sentential formula. A propositional formula is constructed from simple propositions, such as "five is greater than three" or propositional variables such as ''P'' and ''Q'', using connectives such as NOT, AND, OR, and IMPLIES; for example: :(''P'' AND NOT ''Q'') IMPLIES (''P'' OR ''Q''). In mathematics, a propositional formula is often more briefly referred to as a "proposition", but, more precisely, a propositional formula is not a proposition but a formal expression that ''denotes'' a proposition, a formal object under discussion, just like an expression such as "" is not a value, but denotes a value. In some contexts, maintaining the distinction may be of importance. ==Propositions== For the purposes of the propositional calculus, propositions (utterances, sentences, assertions) are considered to be either simple or compound.〔Hamilton 1978:1〕 Compound propositions are considered to be linked by sentential connectives, some of the most common of which are "AND", "OR", "IF ... THEN ...", "NEITHER ... NOR...", "... IS EQUIVALENT TO ..." . The linking semicolon ";", and connective "BUT" are considered to be expressions of "AND". A sequence of discrete sentences are considered to be linked by "AND"s, and formal analysis applies a recursive "parenthesis rule" with respect to sequences of simple propositions (see more below about well-formed formulas). : For example: The assertion: "This cow is blue. That horse is orange but this horse here is purple." is actually a compound proposition linked by "AND"s: ( ("This cow is blue" AND "that horse is orange") AND "this horse here is purple" ) . Simple propositions are declarative in nature, that is, they make assertions about the condition or nature of a ''particular'' object of sensation e.g. "This cow is blue", "There's a coyote!" ("That coyote IS ''there'', behind the rocks.").〔PM p. 91 eschews "the" because they require a clear-cut "object of sensation"; they stipulate the use of "this"〕 Thus the simple "primitive" assertions must be about specific objects or specific states of mind. Each must have at least a subject (an immediate object of thought or observation), a verb (in the active voice and present tense preferred), and perhaps an adjective or adverb. "Dog!" probably implies "I see a dog" but should be rejected as too ambiguous. : Example: "That purple dog is running", "This cow is blue", "Switch M31 is closed", "This cap is off", "Tomorrow is Friday". For the purposes of the propositional calculus a compound proposition can usually be reworded into a series of simple sentences, although the result will probably sound stilted. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Propositional formula」の詳細全文を読む スポンサード リンク
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