Substitution (logic)

Substitution is referred to as the logic generally the substitution of an expression by another.

More here four different expressions must be distinguished:

  • The Substituendum (Latin: " to the Substitute " ): the expression to be replaced
  • The Substituens (lat " the replacing " ): the expression of the replaced
  • The substitute base: the expression, by replacing
  • The substitution result: the result of the substitution.

Example:

If we replace in the expression

(read: " if, then and") the expression by

(read: 'or'), we obtain:

It is Substituendum, Substituens, substitution and base substitution result.

A distinction between universal and simple substitution, is also in the quantifier, the term " free substitution " of meaning.

Universal and simple substitution

In the universal substitution of all occurrences Substituendums need to be replaced with the simple substitution does not need all occurrences to be replaced. The difference between the two substitutional species is therefore only relevant if there are at least two occurrences of the Substituendums in the substitution basis. In the universal substitution, the Substituendum is no longer found in the substitution result, in the simple substitution, it can still happen.

Example:

If we replace in the expression

The expression by

We obtain with universal substitution:

In case of simple substitution, we could also obtain:

Universal and simple substitution play a role in different laws:

Law of universal substitution

, A statement is a theorem and is the result of the substitution of the universal by so again a theorem. It is important here that is universally substituted; with just a simple substitution is not guaranteed that a theorem is. A further requirement is that it is the Substituendum a " set parameters " is, that is a non - complex formula, which occurs also in any axiom. For the Substituens there is no corresponding restriction.

Example:

In the theorem

We can replace the term by universal

And again obtained a theorem, namely:

In case of simple substitution, we could also obtain:

What is not a theorem. If we dropped the requirement that the Substituendum is a set of parameters, then we could the whole expression by a formula, such as, replace, and received:

Which of course also is not a theorem.

The property that gets the universal substitution theorem property is exploited in some calculi by this is formulated as a rule of inference. The rule of universal substitution states that one can replace each set of parameters universally by any statement in any formula that has been gained with a proof.

Law of substitution of equivalent statements

Two statements or equivalent, and is a result of the simple substitution of In, and are also equivalent.

Example:

Two equivalent statements are for example:

And

,

If we in the statement

By simply substituting, we can obtain:

And then, again equivalent.

The term " free substitution "

A term is the substitution of a variable in a formula free when there is not in the scope of a quantifier or.

The background of this definition is the following: One wants to say that a statement is an all- or - existence generalization another in the quantifier. For example,

Formal:

An existential generalization of

Formal:

It now appears as a generalization would obtain if one of generalizing to Terms ( in the example " Frank " or ) replaced by universal and a quantifier or set before the statement. , The occurrences This gives a generalization but only under the additional condition that the term is too generalized for free replacement by.

Example

Consider the statement

Formally

Note that this is not the substitution by free since it occurs in the scope of the existential quantifier. Therefore, the following statement, no All- generalization of:

Because this statement means:

And this goes completely past the meaning of the original statement.

You can always make a generalization with another variable but in such a case. For example, in the substitution with free, so you can make the following All- generalization:

And this statement then has the desired meaning, namely:

  • Logic
  • Philosophy of Language
753187
de