quantification the application of one or more quantifiers (e.g., ‘for all x’, ‘for some y’) to an open formula. A quantification (or quantified) sentence results from first forming an open formula from a sentence by replacing expressions belonging to a certain class of expressions in the sentences by variables (whose substituends are the expressions of that class) and then prefixing the formula with quantifiers using those variables. For example, from ‘Bill hates Mary’ we form ‘x hates y’, to which we prefix the quantifiers ‘for all x’ and ‘for some y’, getting the quantification sentence ‘for all x, for some y, x hates y’ (‘Everyone hates someone’).
In referential quantification only terms of reference may be replaced by variables. The replaceable terms of reference are the substituends of the variables. The values of the variables are all those objects to which reference could be made by a term of reference of the type that the variables may replace. Thus the previous example ‘for all x, for some y, x hates y’ is a referential quantification. Terms standing for people (‘Bill’, ‘Mary’, e.g.) are the substituends of the variables ‘x’ and ‘y’. And people are the values of the variables.
In substitutional quantification any type of term may be replaced by variables. A variable replacing a term has as its substituends all terms of the type of the replaced term. For example, from ‘Bill married Mary’ we may form ‘Bill R Mary’, to which we prefix the quantifier ‘for some R’, getting the substitutional quantification ‘for some R, Bill R Mary’. This is not a referential quantification, since the substituends of ‘R’ are binary predicates (such as ‘marries’), which are not terms of reference.
Referential quantification is a species of objectual quantification. The truth conditions of quantification sentences objectually construed are understood in terms of the values of the variable bound by the quantifier. Thus, ‘for all v, fv’ is true provided ‘fv’ is true for all values of the variable ‘v’; ‘for some v, fv’ is true provided ‘fv’ is true for some value of the variable ‘v’. The truth or falsity of a substitutional quantification turns instead on the truth or falsity of the sentences that result from the quantified formula by replacing variables by their substituends. For example, ‘for some R, Bill R Mary’ is true provided some sentence of the form ‘Bill R Mary’ is true.
In classical logic the universal quantifier ‘for all’ is definable in terms of negation and the existential quantifier ‘for some’: ‘for all x’ is short for ‘not for some x not’. The existential quantifier is similarly definable in terms of negation and the universal quantifier. In intuitionistic logic, this does not hold. Both quantifiers are regarded as primitive.
See also FORMAL LOGIC, PHILOSOPHY OF LOGI. C.S.