Insertion rule (logic)
The rule of substitution or dissipation through substitution is a rule of inference many logical calculations to derive which allows a set (a universal statement) further and to find equivalent statements to a statement:
Propositional logic
Be a general statement that includes the partial expression . If every occurrence of in is replaced by another expression , the result is a general statement.
Example :
The general statement is given . If you replace through , then what can be transformed results as a new, generally valid statement.
Application :
This rule can be used to transform expressions into simpler, equivalent ones.
If it is any expression, a partial expression contained in it can be replaced ( substituted ) by a new variable . If the resulting expression is converted equivalent according to other rules and the substitution is finally reversed, the result is a statement that is equivalent to the original expression.
Example:
Now substitute with s and get
Resubstition results in , so ( falsum, false ).
Why is this procedure correct?
Obviously, is generally valid for all expressions with partial expression . After substituting with by , we get . Be equivalent to , is also generally valid, i.e. also after resubstitution .
annotation
The now and then so called "installation rule"
Premises :
- (Replace partial expression t with s)
is not correct in every situation. For example, the "premises" s = "Socrates is a person" and a = "If Socrates is an animal, all people are animals." but not the statement created by replacing the partial statement t = "Socrates is an animal" with s = "If Socrates is a person, all people are animals."
However, the rule applies (as a special case of the replacement rule)
Premises :
- (Replace partial expression t with s)
Predicate logic
If in a (in a model ) valid statement for an all-quantified variable a term is inserted equally for every occurrence of , a (more specific) valid statement results.
Example :
If true, including (replaced by ) .
See also
Further meaning: