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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 "x + y" is not a value, but denotes a value. In some contexts ...
A well-formed formula is any atomic formula, or any formula that can be built up from atomic formulas by means of operator symbols according to the rules of the grammar. The language L {\displaystyle {\mathcal {L}}} , then, is defined either as being identical to its set of well-formed formulas, [ 48 ] or as containing that set (together with ...
An atomic sentence (or possibly the meaning of an atomic sentence) is called an elementary proposition by Ludwig Wittgenstein and an atomic proposition by Bertrand Russell: 4.2 The sense of a proposition is its agreement and disagreement with possibilities of existence and non-existence of states of affairs. 4.21 The simplest kind of ...
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 sentences can have concrete, fixed truth values: as the free variables of a (general) formula can range over several values, the truth value of such a formula may vary.
If p is an elementary proposition, ~p is an elementary proposition. Pp 1.71. If p and q are elementary propositions, p ∨ q is an elementary proposition. Pp 1.72. If φp and ψp are elementary propositional functions which take elementary propositions as arguments, φp ∨ ψp is an elementary proposition. Pp
In a given propositional logic, a formula can be defined as follows: Every propositional variable is a formula. Given a formula X, the negation ¬X is a formula. Given two formulas X and Y, and a binary connective b (such as the logical conjunction ∧), the expression (X b Y) is a formula. (Note the parentheses.)
Sometimes precedence between conjunction and disjunction is unspecified requiring to provide it explicitly in given formula with parentheses. The order of precedence determines which connective is the "main connective" when interpreting a non-atomic formula.
atomic proposition A proposition of the form R(x,y,...) where R is a relation. Barbara A mnemonic for a certain syllogism. class A subset of the members of some type codomain The codomain of a relation R is the class of y such that xRy for some x. compact A relation R is called compact if whenever xRz there is a y with xRy and yRz concordant