Explanation
In classical logic there are theorems which clearly presuppose that there is something in the domain of discourse. Consider the following classically valid theorems.
- 1. ;
- 2. (where r does not occur free for x in Ax and A(r/x) is the result of substituting r for all free occurrences of x in Ax);
- 3. (where r does not occur free for x in Ax).
A valid scheme in the theory of equality which exhibits the same feature is
- 4. .
Informally, if F is '=y', G is 'is Pegasus', and we substitute 'Pegasus' for y, then (4) appears to allow us to infer from 'everything identical with Pegasus is Pegasus' that something is identical with Pegasus. The problem comes from substituting nondesignating constants for variables: in fact, we cannot do this in standard formulations of first-order logic, since there are no nondesignating constants. Classically, ∃x(x=y) is deducible from the open equality axiom y=y by particularization (i.e. (3) above).
In free logic, (1) is replaced with
- 1b., where E! is an existence predicate (in some but not all formulations of free logic, E!t can be defined as ∃y(y=t)).
Similar modifications are made to other theorems with existential import (e.g. the Rule of Particularization becomes (Ar → (E!r → ∃xAx)).
Axiomatizations of free-logic are given in Hintikka (1959), Lambert (1967), Hailperin (1957), and Mendelsohn (1989).
Read more about this topic: Free Logic
Famous quotes containing the word explanation:
“What causes adolescents to rebel is not the assertion of authority but the arbitrary use of power, with little explanation of the rules and no involvement in decision-making. . . . Involving the adolescent in decisions doesn’t mean that you are giving up your authority. It means acknowledging that the teenager is growing up and has the right to participate in decisions that affect his or her life.”
—Laurence Steinberg (20th century)
“My companion assumes to know my mood and habit of thought, and we go on from explanation to explanation, until all is said that words can, and we leave matters just as they were at first, because of that vicious assumption.”
—Ralph Waldo Emerson (1803–1882)
“To develop an empiricist account of science is to depict it as involving a search for truth only about the empirical world, about what is actual and observable.... It must involve throughout a resolute rejection of the demand for an explanation of the regularities in the observable course of nature, by means of truths concerning a reality beyond what is actual and observable, as a demand which plays no role in the scientific enterprise.”
—Bas Van Fraassen (b. 1941)