A "'tautology "'is a propositional formula that is assigned truth value " 1 " by every truth assignment of its propositional variables to an arbitrary Boolean algebra ( or, equivalently, every truth assignment to the two element Boolean algebra ).
32.
Both formalisations have variables, but where the one-rule axiomatisation has schematic variables that are outside the logic's language, the substitutional axiomatisation uses propositional variables that do the same work by expressing the idea of a variable ranging over formulae with a rule that uses substitution.
33.
A proof of a tautology in an appropriate deduction system may be much shorter than a complete truth table ( a formula with " n " propositional variables requires a truth table with 2 " n " lines, which quickly becomes infeasible as " n " increases ).
34.
Propositional logic is derived from first-order logic by omitting data terms and reasons only about abstract propositions, which may be simple propositional variables or atoms or compound propositions built with such logical connectives as " and ", " or ", and " not ".
35.
The metavariables themselves are outside the reach of instantiation, not being part of the language of propositional calculus but rather part of the same language for talking about it that this sentence is written in, where we need to be able to distinguish propositional variables and their instantiations as being distinct syntactic entities .)
36.
:: Also, note that if \ alpha is of the form \ beta \ diamond \ gamma, where \ diamond is a binary logical connective, the sets of propositional variables in \ beta and \ gamma are disjoint .-- 71.175.24.57 04 : 39, 23 March 2007 ( UTC)
37.
The essential idea of a truth assignment is that the propositional variables are mapped to elements of a fixed Boolean algebra, and then the "'truth value "'of a propositional formula using these letters is the element of the Boolean algebra that is obtained by computing the value of the Boolean term corresponding to the formula.
38.
A formula written in the common propositional language ( logical connectives and propositional variables ) does not specify any particular logical system for its interpretation, hence such statements as " is logically equivalent to " should not be used in logical articles unless a concrete logical system, or some set of logical systems where such equivalence holds, is unambiguously determined by the context.
39.
For classical and intuitionistic logic, the " = " symbol means that corresponding implications " & ?! & " and " & ?! & " for logical compounds can be both proved as theorems, and the " d " " symbol means that " & ?! & " for logical compounds is a consequence of corresponding " & ?! & " connectives for propositional variables.
40.
If " predicate variables " are only allowed to be bound to predicate letters of zero arity ( which have no arguments ), where such letters actually represent propositions, then such variables are actually "'propositional variables "', and any predicate logic which allows second-order quantifiers to be used to bind such propositional variables is a second-order predicate calculus, or second-order logic.
