axiom system for first order logic
Let FO be a first order language over signature . Furthermore, let be a countably infinite set of variables and the set extending to include the constants in . Before describing the axiom system for FO, we need the following definition:
Definition. Let be a wff. A generalization of is a wff having the form , for any . Thus, is a generalization of itself.
The axiom system of FO consists of any wff that is a generalization of a wff belonging to any one of the following six schemas:
, where is not free in
, where , , and is free for in
In schema 6, the wff is obtained by replacing all free occurrences of in by .
Logical symbols , and are derived: ; ; and .
Remark. Again, in schema 6, the reason why we want free for in is to keep the “intended meaning” of intact. For example, suppose is . Then is . Obviously the two do not have the same “intended meaning”. In fact, is not valid in any model. Similarly, in the schema 5, we want not occur free in to maintain its “intended meaning”. For example, if is the formula , then is , which is not true in any model with at least two elements.
The concepts of deductions and theorems are exactly the same as those found in propositional logic. Here is an example: if , then . To see this, we induct on the length of the deduction of . If , then is an axiom, and therefore so is its generalization , so that . Suppose now that the length is . If is an axiom, we are back to the previous argument. Otherwise, is obtained from earlier formulas and via modus ponens. Since the deduction of and are at most , by induction, we have and . Since is an axiom, we have by modus ponens, and then by modus ponens again.
Remark. Another popular axiom system for first order logic has 1, 2, 3, 6 above as its axiom schemas, plus the following schema
, where is not free in
and two rules of inferences: modus ponens, and universal generalization:
from we may infer , where
Note the similarity between the rule of generalization and axiom schema 5 above. However, the important difference is that is not permitted to be free in in the axiom schema.
where , such that for each ,
is either an axiom or in
is obtained by an application of modus ponens
is obtained by an application of generalization: is for some .
|Title||axiom system for first order logic|
|Date of creation||2013-03-22 19:32:22|
|Last modified on||2013-03-22 19:32:22|
|Last modified by||CWoo (3771)|