# Logical equivalence

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

The term equivalence is used ambiguously in logic:

• on the one hand in terms of material equivalence (biconditional)
• on the other hand in the sense of formal equivalence (logical equivalence).

Biconditional (material equivalence) and logical equivalence (formal equivalence) are essentially different terms. The biconditional is a concept of the object language , the logical equivalence is a concept of the metalanguage . However, the terms are related to one another: the logical equivalence is a universally valid biconditional.

In the following, it is only about the logical equivalence, not about the biconditional.

## Terminology and Synonymy

As far as can be seen, no fixed terminology has so far developed. The logical equivalence is also (mostly) written logical equivalence and also called formal equivalence or simply equivalence (with the risk of confusion with the material equivalence).

## term

This is only about the logical equivalence in the sense of classical, two-valued logic .

### definition

The logical equivalence is defined in two equivalent definitional basic forms. The definition of the logical equivalence takes place here as a prototype for the propositional equivalence. There is also a logical predicate equivalence based on this.

#### The logical equivalence as the equality of values ​​in the form of statements

A logical equivalence exists when two logical expressions have the same truth value, are equivalent, have the same truth value entries in a truth table, “if they contain the same truth functions, i.e. H. Include or exclude the same possible values. ", if the value curve ( truth table ) of the two statements is the same.

In more general terms - d. H. not restricted to propositional logic - two propositions P and Q are equivalent to classical, two-valued logic if and only if both propositions assume the same truth value under every possible interpretation .

#### Logical equivalence as a generally valid biconditional

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

Depending on the terminology or the precision of the terminology, it is a matter of the logical equivalence of forms of statements or combinations of statements , of sentences , of partial sentences , statements , (complex) statements or expressions .

### The metalinguistic of logical equivalence

The concept of logical equivalence is metalinguistic or metatheoretical . It is used to make a (meta) statement about the relationship between two expressions in the object language.

### Delimitations

#### Material equivalence (biconditional)

The biconditional as an operator ( junction , connective) of the respective logical object language must be distinguished from the equivalence as a metatheoretical concept , which is also often referred to as equivalence. This homonymy is unfortunate in that it tempts one to confuse or confuse an object and a metalinguistic concept, and because it forces one to pay close attention to what is meant in the respective context with the word "equivalence". Details: biconditional .

#### definition

"All definitions have the form of logically true equivalences."

#### Math equation

Logical equivalence describes the equality of values ​​in statements, analogous to the equal sign in algebra. Two statements A , B of classical propositional logic are logically equivalent if and only if the value curve ( truth table ) of the two statements is the same.

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

Example of the connection between logical equivalence and mathematical identity:

For all true${\ displaystyle a, b \ in \ mathbb {R}}$${\ displaystyle (x = (a + b) ^ {2} \ Leftrightarrow x = a ^ {2} + 2ab + b ^ {2}) \ Rightarrow (a + b) ^ {2} \ equiv a ^ {2 } + 2ab + b ^ {2}}$

### Spelling and speaking styles

A double arrow pointing left and right is often used for " A equivalent B " in mathematical notation (⇔, Unicode character U + 21D4 in the Unicode block arrows )

${\ displaystyle A \ Leftrightarrow B}$

They say

• in mathematics:
• A is equivalent to B.
• A is true if and only if B
• A is true if and only if B
• in logic:
• A is logically equivalent to B.
• A has the same values ​​as B
• A is logically equivalent to B.

One also writes

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

This way of writing and speaking for the logical equivalence is to be distinguished from that for the biconditional . For the object language statement " A if and only if B " (biconditional!) One writes in logic (among other things):

${\ displaystyle A \ equiv B}$ or ${\ displaystyle A \ leftrightarrow B}$

## The logical equivalence as a relation and its properties

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

The equivalence can be used as a "three-digit relation between two things and one property" or as a two-digit relation that is already relative to one property.

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

## sentence

• In classical logic, the metatheorem holds that two sentences X and Y are equivalent if and only if the biconditional X ↔ Y formed from them is a tautology .
• If the biconditional is not introduced by definition, but as an independent junction according to the above truth table, then the metatheorem applies that the two sentences of the form X ↔ Y and (X → Y) & (Y → X) are equivalent.

• Implication - the difference between object-language and metalinguistic use is particularly well worked out here
• Predicate logic

## Individual evidence

1. Cf. Regenbogen / Meyer, Dictionary of Philosophical Terms (2005) / Equivalence: close relationship
2. Zoglauer, Thomas, Introduction to Formal Logic for Philosophers (1999), p. 47
3. ^ Salmon, Logic (1983), p. 88
4. Spies, Introduction [2004], p. 32
5. Seiffert, Logic (1971), p. 186
6. See Hilbert / Ackermann, Grundzüge, 6th edition (1972), p. 11 f.
7. Salmon, Logic (1983), p. 96; Bußmann, Lexikon der Sprachwissenschaft, 3rd edition (2002) / equivalence; Lohnstein, Formal Semantik (1996), p. 41
8. Salmon, Logic (1983), p. 96; Hilbert / Ackermann, Grundzüge, 6th edition (1972), p. 11 f.
9. Hilbert / Ackermann, Grundzüge, 6th edition (1972), p. 11 f.
10. Spies, Introduction to Logic (2004), p. 22
11. ^ Seiffert, Logic (1973), p. 186
12. Bußmann, Lexikon der Sprachwissenschaft, 3rd edition (2002) / Equivalence
13. ^ Lohnstein, Formal Semantik (1996), p. 41
14. Zoglauer, Thomas, Introduction to Formal Logic for Philosophers (1999), p. 47
15. Zoglauer, Thomas, Introduction to formal logic for philosophers (1999), p. 43
16. Reichenbach, Grundzüge der Symbolischen Logic (1999), p. 36
17. Zoglauer, Thomas, Introduction to formal logic for philosophers (1999), p. 43
18. Rainbow / Meyer, Dictionary of Philosophical Terms (2005) / Equivalence
19. ^ Seiffert, Wissenschaftstheorie IV (1997), Equivalence
20. Rainbow / Meyer, Dictionary of Philosophical Terms (2005) / Equivalence
21. Rainbow / Meyer, Dictionary of Philosophical Terms (2005) / Equivalence
22. Hilbert / Ackermann, Grundzüge, 6th edition (1972), p. 12