Logical biconditional

As a biconditional, Bisubjunktion or substantive equivalence, sometimes (but ambiguous ), simply referred to only one equivalence

• A compound statement that is only true if its two sub-statements have the same truth value, ie either both true or both false;
• The truth function appropriately defined;
• The linguistic sign ( the connective ), with which these two sub-statements are put together.

Spelling and reading

As a sign of the biconditional connective than the double arrow triple slash or the double arrow is usually with two transverse lines used occasionally the tilde ~. ( Almost every of these marks is of different authors and in different contexts also used with a different meaning, most commonly the tilde for the sentence negation and the double arrow with two transverse lines for the meta-language equivalence. ) In the Polish notation the biconditional by the capital letter E expressed.

In the natural language, there are several ways to express a biconditional, for example, the formulations "A if and only if B ' (abbreviated as " A iff. B ")," A if and only if B "or " A is sufficiently and necessary for B "; also the phrase " A if and only if B " used in English can be found abbreviated as " A iff B" occasionally even in German texts. Each of these formulations is suitable to read the print.

Importance

For the bivalent, truth-functional classical logic the truth value of the course (the truth table ) and thus the importance of Bikonditionals defined as follows by the eq function ( "w " stands for " true", "f " stands for " false"):

In classical logic, the statements and (that is the conjunction of the conditional and the conditional ) are equivalent, ie they have the same truth- value chart. For this reason, the biconditional is often not introduced as an independent connective, but returned the following definition of conjunction and conditional:

It should be ": =" the metalinguistic character for " is defined as" and be set and metalinguistic variables, ie variables which may stand for arbitrary sentences of the logical object language. As a concrete example, the expression would be dissolved according to this definition.

The above equivalence and definability above show in particular that the biconditional expresses a necessary and sufficient condition: indicates that A B is a sufficient condition for B, and that a necessary condition for A; and indicates that, B is a sufficient condition for A and that A is a necessary condition for B.

Examples

• Is a biconditional, that is always true, that is a tautology.
• Is a biconditional, that is never true.
• Is a biconditional, that may be true or false, depending on how it is to the truth of the statements part A, B, C.
• ". The moon is just then a light source when Isaac Newton was a German " is a true biconditional, as well: " Mars is a planet if and only if the oceans contain salt. ". This example shows that the paradoxes of material implication analogous occur during biconditional: It may be true, without any substantive relationship between the two statements is.

Ambiguity for multiple arguments

If more than two arguments by connected, it is not clear how the formula is meant:

May be the abbreviation for,

Or that all together are either true or false together:

This is the same for two arguments. The two truth tables show only rows with two arguments the same bit pattern:

The left Venn diagram below, and the line (AB) in these matrices are available for the same operation.

Venn diagrams

Red areas represent the truth ( such as in and for ).