implication

The term implication (from Latin implicare , to entangle , verb : to imply ; adjective : to implicit ) is not used uniformly in logic for a specific logical context, in particular, distinctions are made

• a material implication as one of several possible logical connections ( connectors ) between two propositional variables : (see also article " Junctor "). This material implication, also called subjunction or conditional , can be defined truth- functionally (see section below ). It can already be found in Philo of Megara (3rd century BC) and is usually paraphrased colloquially as: "if  a , then  b ".${\ displaystyle a \ rightarrow b}$
• a formal implication as a form of logical connection, which should correspond more to an intuitive perception that can result from the habits of everyday language . In the course of time, various interpretations emerged in order to formalize the phenomenon as clearly as possible . The above formula is viewed in a more differentiated way, for example as read: "For every individual x, the following applies: If x has the property A , then it also has the property B. " The analysis of a statement with decomposition into the predicator and its argument , especially for the formal implication, is found similarly in Plato and Aristotle .${\ displaystyle \ bigwedge _ {x} (A (x) \ rightarrow B (x))}$

The intuitionistic implication or subjunction within the dialogical logic as well as the strict implication of Ackermann as well as the strict implication can also be viewed as variants of a deduction-based formal implication . The implication was formalized as a hypothetical judgment by Bruno von Freytag-Löringhoff and Albert Menne .

These more specific interpretations can also be referred to as object language implications. A distinction must then be made between the metalinguistic implications; they allow one to talk about the logical structure of these languages. Accordingly, they can be ascribed an even closer connection to the concept of derivability and the concept of inference .

Walther Brüning characterizes the deductive implication as transgressive , that is, logically not exact, and introduces a strict derivation concept in his strict logic .

Difference between object-language and metalinguistic implication

The object-language implication (material implication, conditional, subjunction) is a statement that is made up of two shorter statements by means of the juncture "(already) if ..., then ...". For example, "if it rains, the road is wet" is a material implication; this implication says something about the logical connection of the sentences: namely that the truth of the first sub-clause ( antecedent , also antecedent ) is a sufficient condition for the truth of the second sub-clause ( consequence ).

The metalinguistic implication, on the other hand, is a statement about statements , a meta-statement . A metalinguistic implication would be the statement "From the sentence 'It's raining' follows the sentence 'The street is wet'" ". Nothing is said here about rain, moisture or their connection, but rather two sentences of the object language and their logical relationship. In doing so, reference can be made to their meaning (for example: whether what one sentence says is present, if what the other says is present), or not, two sentences can be linked to one another by their logical form alone (so can say, for example: "If , then "). ${\ displaystyle a \ land b}$${\ displaystyle a}$

Object language implications

The object-language implication, a statement that is composed of two shorter statements by means of the juncture "(already) if ..., then ...", is referred to as material implication , subjunction and conditional .

Truth functional implication

The subjunction is false only if A is true and B is false. This area is white in the Venn diagram .
It's classic${\ displaystyle A \ rightarrow B \ Leftrightarrow \ neg A \ lor B}$

${\ displaystyle \ Leftrightarrow}$ ${\ displaystyle \ lor}$

In classical logic , only truth-functional statement combinations are used, that is to say only those in which the truth value of the statement combination depends solely on the truth value of the partial statements.Within a conditional , the first statement is referred to, among other things, as antecedent, implicant or fore-term, the second statement, among other things, as suffix, suffix , consequence, implicate, and rarely also succession. ${\ displaystyle a \ rightarrow b}$${\ displaystyle a}$${\ displaystyle b}$

Since ancient times - for the first time by Philo of Megara  - the truth-functional implication or seq-function has been defined by the following truth table:

${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle a \ rightarrow b}$
f f w
f w w
w f f
w w w

This truth-functional object-language implication is called material implication , subjunction or (increasingly) conditional , among other things . It expresses the sufficient condition , that is, it does not claim any causal or other substantive connection between and . ${\ displaystyle A}$${\ displaystyle B}$

In ancient times it was discussed to what extent and under what conditions the natural language “if ..., then ...” expresses a sufficient condition and thus corresponds to the material implication; But above all, whether and how the other meanings of the natural language "if ..., then ...", for example the causal one ("A causes B"), can be analyzed. Attempts to analyze other meaning than the purely truth-functional (“material”) meaning of the natural language “if…, then…” lead to non-classical implications, for example the strict implication and the intuitionistic implication .

A simple arrow is used as a symbol for the junction in the formal language of logic , especially in the Anglo-Saxon area, based on the Peano-Russell spelling, the curve (“horseshoe”, “horseshoe” , “ arch mark” (Reichenbach)) is occasionally used also the arrow with two slashes . ${\ displaystyle \ rightarrow}$${\ displaystyle \ supset}$${\ displaystyle \ Rightarrow}$

In the Polish notation , the capital letter C is used for the material implication , so that the statement “If a, then b” would be written as Cab .

Gottlob Frege expresses the conditional “If A, then B” through in his conceptual writing , the first formalization of classical predicate logic .

 Spellings ${\ displaystyle a \ rightarrow b}$ ${\ displaystyle a \ supset b}$ ${\ displaystyle a \ Rightarrow b}$ ${\ displaystyle Cab}$

Natural language and material implication

In the case of material implication, one often says briefly: “If a, then b.” This usage is somewhat unfortunate because the phrase “if ..., then ...” in German has a wide field of meaning and mostly not for material, that is to say here truth-functional , but rather for contextual connections ( causality or chronological sequence). Such connections cannot be expressed with the material implication. A very precise distinction must therefore be made between the material implication and the natural language "if ..., then ...". Sometimes one tries to avoid misunderstandings, which can result from the many meanings of the German "if ..., then ..." , by using formulations like " already if a, then b ..." or "a is a sufficient condition for b".

The implication for (a) “It's raining” and (b) “The road is getting wet” would be the statement

 When it rains, the road gets wet.

Alternative formulations that better emphasize the material character would be

 Even when it rains, the road gets wet.

or

 That it rains is a sufficient condition for the road to get wet.

The material implication is false if and only if the antecedent is true and the consequent is false . In any other case the implication is true . The conditional “If it rains, the street gets wet” is only wrong if it rains but the street does not get wet.

The stipulation that a material implication is only false if the antecedent (the if part) is true and the consequent is false leads to the following combinations of empirical statements being true :

 When London is in France, snow is white. false antecedent, true succession When London is in France, snow is black. wrong antecedent, wrong succession When London is in England, snow is white. true antecedent, true succession

These paradoxes of the material implication underline the extensional character ( see junction ) of the material implication: It does not claim any substantive connection between the antecedent (if part) and the succession (there is actually no connection between the geographic location of London and the color of snow) Rather, their truth value is traced back purely extensionally to the truth values ​​of their sub-clauses: "Even if the antecedent is true, the subsequent is also true."

Connection with the necessary condition

As already mentioned, the material implication expresses the sufficient condition . It is to be distinguished from the necessary condition , which says that one state of affairs is necessary but not sufficient for another state of affairs to occur.

example
“Only when a person is of legal age can they vote.” Legal age is a necessary condition for the right to vote, but it is not sufficient: You usually have to meet additional conditions, e. B. have the citizenship of the country.

The sufficient and the necessary condition are closely related. If a matter A is a sufficient condition for a matter B, then B is also a necessary condition for A. The example “Only when a person is of legal age can vote” is logically equivalent to “Even if a person is allowed to vote is they are of legal age. ”One can clarify this connection, which is often perceived as counterintuitive at first, by looking at the situation at a polling station. If you see a person voting there, then you can clearly conclude - even if he looks very young - that he must be of legal age; because only adults are allowed to vote.

Due to this contextual context , the material implication expresses the necessary as well as the sufficient condition:

${\ displaystyle A \ rightarrow B}$

is usually read as “A is a sufficient condition for B” or “If A, then B”; but since that is equivalent to “ B is a necessary condition for A ”, it can just as easily be read that way.

Properties and laws of logic

The material implication

${\ displaystyle a \ rightarrow b}$

is propositionally equivalent to the following statements, for example :

• ${\ displaystyle \ neg a \ lor b}$(read: "not a or b"). This equivalence can be used to define the material implication on the basis of disjunction and negation .
• ${\ displaystyle \ neg (a \ land \ neg b)}$(read: “it does not count: a and not b”). The material implication can also be defined on the basis of conjunction and negation.
• ${\ displaystyle \ neg b \ rightarrow \ neg a}$(read: "if not b, then not a"). So you can reverse the implication if you negate both antecedent and consequent at the same time. This logical law is also known as counterposition .

In addition, the statement a is equivalent to and the statement (read: “not a”) is equivalent to , where is any tautology and any contradiction . Furthermore, and are equivalent to . ${\ displaystyle \ top \ rightarrow a}$${\ displaystyle \ neg a}$${\ displaystyle a \ rightarrow \ bot}$${\ displaystyle \ top}$${\ displaystyle \ bot}$${\ displaystyle \ bot \ rightarrow a}$${\ displaystyle a \ rightarrow \ top}$${\ displaystyle \ top}$

Due to its extensional character, the material implication in predicate logic is well suited to formalizing statements of the type "All horses are mammals" as follows:

 Notation ${\ displaystyle \ forall x (P (x) \ rightarrow S (x))}$ Way of speaking "For all x the following applies: If x is a horse, x is a mammal"

Regarding the properties of the material implication, it should be noted: It is not associative , commutative , symmetrical , antisymmetrical or asymmetrical . But it is transitive , that is, the following applies:

out and follows${\ displaystyle a \ rightarrow b}$${\ displaystyle b \ rightarrow c}$${\ displaystyle a \ rightarrow c}$

It is also reflexive , so the following applies in general:

${\ displaystyle a \ rightarrow a}$

With the help of the implication and the negation , all propositional joiners can be represented.

Non-Classical Implications

Intuitionist implication

In intuitionism the expression means that a proof of (about the existence of which nothing is stated) can be supplemented to a proof of . This relationship cannot be defined in terms of the truth values of antecedents and succession, so it is not extensional or truth-functional . Instead, intensive semantics are used, the best known and first formalized being the Kripke semantics developed by Saul Aaron Kripke for modal logic . ${\ displaystyle a \ rightarrow b}$${\ displaystyle a}$${\ displaystyle b}$

The equivalences listed above apply intuitionistically "only in one direction", i.e. H. in particular:

• follows from , but not the other way around.${\ displaystyle a \ rightarrow b}$${\ displaystyle \ neg (a \ land \ neg b)}$
• follows from , but not the other way around.${\ displaystyle a \ rightarrow b}$${\ displaystyle \ neg b \ rightarrow \ neg a}$
• follows from , but not the other way around.${\ displaystyle \ neg a \ lor b}$${\ displaystyle a \ rightarrow b}$

Unlike the material implication, the intuitionist implication cannot be defined in terms of negation and conjunction or disjunction.

It is still true, however, that a is equivalent to and to , and that and are equivalent to . Like the material implication, the intuitionistic is transitive and reflexive. ${\ displaystyle \ top \ rightarrow a}$${\ displaystyle \ neg a}$${\ displaystyle a \ rightarrow \ bot}$${\ displaystyle \ bot \ rightarrow a}$${\ displaystyle a \ rightarrow \ top}$${\ displaystyle \ top}$

Strict implication

The strict implication is the combination of the modal logical necessity operator with the material implication.

 Notation ${\ displaystyle \ Box (a \ rightarrow b)}$, ${\ displaystyle \ Box (a \ supset b)}$ Way of speaking If a, then necessarily b

The strict implication was developed by Diodoros Kronos and in scholasticism as an attempt to circumvent the paradoxes of material implication and was reorganized in 1918 by Clarence Irving Lewis . This is intended to approximate the natural language "if ..., then ...". The strict implication is not already true if the antecedent is false or the consequent is true. There are numerous variants of the strict implication, depending on which modal calculus is used. The strict implication, like the material and the intuitionistic, is transitive and reflexive.

The concept of strict implication is also subject to criticism because, while it avoids the paradox of material implication, it leads to the analogous difficulty that every logically impossible statement implies any statement and that every statement strictly implies every logically necessary statement. Lewis' own use of strict implication was also accused of confusing object and metalanguage.

Metalinguistic implication

The metalinguistic implication is a statement about statements . A statement A implies a statement B if and only if A applies, B also applies. Analogously, several statements A 1 to A n imply a statement B if and only if the statements A 1 to A n also apply together with B. For example, the statements "All pigs grunt" and "Babe is a pig" imply "Babe grunts".

The concept of inference and thus the metalinguistic implication is formally specified in different ways. On the one hand, a distinction is made between the semantic inference , written down as, and the syntactic inference , the derivability, written down as : ${\ displaystyle A_ {1}, \ ldots, A_ {n} \ models B}$${\ displaystyle A_ {1}, \ ldots, A_ {n} \ vdash B}$

Semantic inference concept

A conclusion is semantically valid, written: if the truth of the statements A 1 to A n guarantees the truth of the statement B. In an interpretation semantics , this is the case if, for every interpretation in which each of the statements A 1 to A n is true, the statement B is also true. ${\ displaystyle A_ {1}, \ ldots, A_ {n} \ models B}$

Syntactic concept of inference
A consequence is syntactically valid, written , if and only if the statement B in a given logical calculus can be derived from the statements A 1 to A n , that is, if from the statements A 1 to A n using the rules of inference and axioms of the respective calculus allows the statement B to be generated.${\ displaystyle A_ {1}, ... A_ {n} \ vdash B}$

On the other hand, there are fundamentally different versions of the concept of inference and thus of the metalinguistic implication, for example that of classical logic or that of logic. These different definitions of inference or metalinguistic implication lead to fundamentally different calculi and semantic models . If it is not clear from the context what type of metalinguistic implication or conclusion is meant, it is therefore necessary to provide this information. One can therefore find formulations such as “A implies (semantic, syntactic) classic B” or “C implies intuitionistically (semantic, syntactic) D”. In the formal notation, the type of inference is usually indicated by a subscript at the inference sign. For example, “K” could stand for classical, “I” for intuitionistic conclusion, that is (semantic, classical), (syntactic, classical), (semantic, intuitionistic) and (syntactic, intuitionistic). ${\ displaystyle A \ models _ {K} B}$${\ displaystyle A \ vdash _ {K} B}$${\ displaystyle A \ models _ {I} B}$${\ displaystyle A \ vdash _ {I} B}$

In the vast majority of logics, there is a close connection between object-language and metalinguistic implications, which is expressed in the deduction theorem . If “if a, then b” is provable, then b can be derived from a; and if, conversely, b can be derived from a, then “If a, then b” is provable. For “c is provable” one also writes . The deduction theorem can thus be written down as follows: ${\ displaystyle \ vdash c}$

${\ displaystyle \ vdash a \ rightarrow b}$ iff. ${\ displaystyle a \ vdash b}$

The deduction theorem applies to both classical, intuitionist and strict implication. However, this is not a matter of course, but requires a (in most cases non-trivial) proof.

Web links

Wiktionary: Implication  - explanations of meanings, word origins, synonyms, translations

Individual evidence

1. a b Outline of formal logic . Paderborn: Universitäts-Taschen-Bücher-Verlag: 1983. Translated from the French by Joseph Maria Bocheński . Translated and expanded by Albert Menne .