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The predicate calculus goes a step further than the propositional calculus to an "analysis of the inner structure of propositions" [4] It breaks a simple sentence down into two parts (i) its subject (the object (singular or plural) of discourse) and (ii) a predicate (a verb or possibly verb-clause that asserts a quality or attribute of the object(s)).
is a sentence. This sentence means that for every y, there is an x such that =. This sentence is true for positive real numbers, false for real numbers, and true for complex numbers. However, the formula (=) is not a sentence because of the presence of the free variable y.
Here, the numbers may come as close as they like to 12, including 11.999 and so forth (with any finite number of 9s), but 12.0 is not included. In some European countries, the notation [ 5 , 12 [ {\displaystyle [5,12[} is also used for this, and wherever comma is used as decimal separator , semicolon might be used as a separator to avoid ...
The method of truth tables illustrated above is provably correct – the truth table for a tautology will end in a column with only T, while the truth table for a sentence that is not a tautology will contain a row whose final column is F, and the valuation corresponding to that row is a valuation that does not satisfy the sentence being tested.
A closed formula, also ground formula or sentence, is a formula in which there are no free occurrences of any variable. If A is a formula of a first-order language in which the variables v 1, …, v n have free occurrences, then A preceded by ∀v 1 ⋯ ∀v n is a universal closure of A.
[1] [13] Examples of such compound sentences might include: Wikipedia is a free online encyclopedia that anyone can edit, and millions already have. (conjunction) It is not true that all Wikipedia editors speak at least three languages. (negation) Either London is the capital of England, or London is the capital of the United Kingdom, or both ...
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A more complex example is the lexer hack in C, where the token class of a sequence of characters cannot be determined until the semantic analysis phase since typedef names and variable names are lexically identical but constitute different token classes. Thus in the hack, the lexer calls the semantic analyzer (say, symbol table) and checks if ...