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:
“Herein is the explanation of the analogies, which exist in all the arts. They are the re-appearance of one mind, working in many materials to many temporary ends. Raphael paints wisdom, Handel sings it, Phidias carves it, Shakspeare writes it, Wren builds it, Columbus sails it, Luther preaches it, Washington arms it, Watt mechanizes it. Painting was called “silent poetry,” and poetry “speaking painting.” The laws of each art are convertible into the laws of every other.”
—Ralph Waldo Emerson (1803–1882)
“There is no explanation for evil. It must be looked upon as a necessary part of the order of the universe. To ignore it is childish, to bewail it senseless.”
—W. Somerset Maugham (1874–1965)
“The explanation of the propensity of the English people to portrait painting is to be found in their relish for a Fact. Let a man do the grandest things, fight the greatest battles, or be distinguished by the most brilliant personal heroism, yet the English people would prefer his portrait to a painting of the great deed. The likeness they can judge of; his existence is a Fact. But the truth of the picture of his deeds they cannot judge of, for they have no imagination.”
—Benjamin Haydon (1786–1846)