Logical equivalence

A logical equivalence occurs when two logical expressions have the same truth value.

The term equivalent is used ambiguously in the logic:

• On the one hand in terms of substantive equivalence ( biconditional )
• To another in the sense of formal equivalence (logical equivalence ).

Biconditional ( substantive equivalence ) and logical equivalence (formal equivalence ) are significantly different terms. The biconditional is a term of the object language, the logical equivalence is a term of the metalanguage. The terms, however, are related to each other: the logical equivalence is a universal biconditional.

In the following, it 's all about the logical equivalence, but not the biconditional.

• 2.3.1 substantive equivalence ( biconditional )
• 2.3.2 definition
• 2.3.3 mathematical equation

Terminology and synonymy

It would appear no fixed terminology has so far trained. The Logical equivalence is also (mostly ) written logical equivalence and formal equivalence or simply known as equivalence ( with the likelihood of confusion with the substantive equivalence ).

Term

Here it 's all about the logical equivalence in the sense of classical, two-valued logic.

Definition

The logical equivalence is defined in two equivalent definitional primitives. The definition of logical equivalence is done here prototypical for the propositional equivalence. There is also a more advanced predicate logic equivalence.

The logical equivalence as the value chart equality of statement forms

A logical equivalence occurs when two logical expressions have the same truth value, are equivalent have the same truth -value entries in a truth table, "if they involve the same truth- functions, that is, the same possible values ​​to include or exclude. " if the progression of values ​​( truth table ) of the two statements is the same.

More generally - that is, not limited to propositional logic - are two statements P and Q of the classical two-valued logic if and only equivalent if both statements assume under any possible interpretation of the same truth- value

The logical equivalence as a general biconditional

A logical equivalence exists when a biconditional true, universal, is a tautology.

Depending on the terminology or precision of terminology, it is about the logical equivalence of statement form or statement compounds of sets, subsets, statements (complex) statements or expressions.

The Metasprachlichkeit the logical equivalence

The concept of logical equivalence is meta-linguistic or meta-theoretical. With it, a (meta- ) statement about the relationship ( relation) of two expressions of the object language is taken.

Accruals

Substantive equivalence ( biconditional )

From the equivalence as a meta-theoretical concept as the biconditional operator ( connective, connective ) is necessary to distinguish the respective logical object language, which is also often referred to as equivalence. This homonymy is unhappy insofar as they enticed to confuse an object and a metalinguistic concept or to mix, and because it forces you to pay very close attention to what "equivalent " is meant in context with the word. Details: biconditional.

Definition

" All definitions are in the form of logically true equivalences. "

Mathematical equation

The logical equivalence describes the value chart equality of statements, similar to the equal sign in algebra. So two propositions A, B of the classical propositional logic are logically equivalent if and only if the value sequence ( truth table ) of the two statements is the same.

" The function of the equivalence in the logic corresponds to the function of the equations in mathematics. ".

Example of the relation of logical equivalence and mathematical identity:

For all

Writing and ways of speaking

For "A equivalent to B" Double Arrow is in mathematical notation often used by the left and right ( ⇔, Unicode character U 21 D4 in Unicode block arrows )

It is said

• In mathematics: A is equivalent to B
• A holds if and only if B
• A true if and only if B

• A is logically equivalent to B
• A is worth running the same with B
• A is logically equivalent to B

It also writes

• A iff. B ( iff )
• A iff. B ( engl. if and only if)
• A = B.

This writing and speech for the logical equivalence is to be distinguished from that for the biconditional. For the object language statement " A iff B" ( Bikonditional! ) statement in the logic ( among other things):

The logical equivalence as a relation and their properties

The " equivalence is a relation " and that " a relation between two statements that are not identical in content, but are always together either true or false. ".

The equivalence can be used as a " three -place relation between two things and a property " or as a binary relation that is already relativized to a property.

The equivalence relation has the properties of reflexivity, symmetry and transitivity.

Set

• In classical logic, the Metatheorem is that two sets X and Y are equivalent if and only if the biconditional formed from them X ↔ Y is a tautology.
• Is the biconditional not introduced by definition, but as a separate connective according to the above truth table, then applies the Metatheorem that the two sets of the form X ↔ Y and (X → Y ) & (Y → X) are equivalent.