Ex falso quodlibet

Ex falso quodlibet , actually ex falso sequitur quodlibet ( lat. "Anything wrong follows anything"), abbreviated to "efq", more clearly ex contradictione sequitur quodlibet (lat., Anything follows from a contradiction), denotes in the narrower sense one of the two in laws applicable to many logical systems:

Any statement follows from a logically - not just factually - incorrect sentence.
Any statement follows from two contradicting sentences.

A sentence is logically wrong if it cannot become true due to its logical form. In most logical systems, contradictions (or sentences from which a contradiction can be derived) meet this condition, hence the designation “ex contradictione sequitur quodlibet”. The term “ex falso sequitur quodlibet” is only synonymous if the “falsum” cited therein is understood as logical and not just factual falsehood.

In a broader sense, “ex falso quodlibet” also denotes the counterfactual (factual contradicting) material implication , i.e. H. a statement of the form “(already) if P, then Q”, where P is any factually untrue sentence, for example the statement “The earth is flat”. A counterfactual material implication would then be, for example, the sentence “(already) if the earth is flat, all cats are dogs”. Since material implication and logical conclusion are completely different concepts, the first object-based, the second metalinguistic, this usage of the language is not correct in the strict sense.

Formal definition

Ex contradictione sequitur quodlibet

If A and B are statements, then "ex contradictione sequitur quodlibet" denotes the fact that any sentence B follows from the contradicting statement or from the two contradicting statements : ${\ displaystyle A \ land \ neg A}$ ${\ displaystyle A, \ neg A}$

${\ displaystyle A \ land \ neg A \ models B}$
${\ displaystyle A, \ neg A \ models B}$

or can be derived:

${\ displaystyle A \ land \ neg A \ vdash B}$
${\ displaystyle A, \ neg A \ vdash B}$

In classical logic (and generally in systems in which the conjunction can be derived from two statements and vice versa), the state of affairs expressed by formulation (1) coincides with the state of affairs expressed by formulation (2).

Likewise, the above facts coincide in classical logic (and in general in systems in which there is a separation rule and in which the deduction theorem applies) with the validity or deduction of the following theorem:

{\ displaystyle (A \ land \ neg A) \ rightarrow B}

So:

{\ displaystyle \ models (A \ land \ neg A) \ rightarrow B}

respectively

{\ displaystyle \ vdash (A \ land \ neg A) \ rightarrow B}

The sentence itself is therefore often referred to as “ex contradictione sequitur quodlibet”. ${\ displaystyle (A \ land \ neg A) \ rightarrow B}$

This proposition can be derived from many logical systems (cf. calculus ), even in intuitionist logic . Logics in which it cannot be derived are called paraconsistent logics .

In some logical calculi the “Ex contradictione sequitur quodlibet” is used as an axiom or as a rule of conclusion. This happens regularly in calculi for intuitionist logic , where it could not otherwise be derived by renouncing the validity of . ${\ displaystyle \ neg \ neg A \ rightarrow A}$

Ex falso sequitur quodlibet

As "ex falso sequitur quodlibet", the validity of the following argument is usually formally designated:

{\ displaystyle \ neg P \ models (P \ rightarrow Q)}

or its derivability:

{\ displaystyle \ neg P \ vdash (P \ rightarrow Q)}

That is, from the fact that a sentence is false it follows that this sentence is a sufficient condition for any sentence Q.

Here, too, the term “ex falso sequitur quodlibet” is often applied to a single sentence, namely to

{\ displaystyle \ neg P \ rightarrow (P \ rightarrow Q)}

This theorem is also often used as an axiom in calculi of classical logic.

context

In logical systems with separation rule, in which the deduction theorem also applies (ie in particular, but not only, in classical logic), the "ex contradictione" and the "ex falso" coincide insofar as all the sentences that can be derived from the one are, can also be derived from the other; and in the sense that one follows the other. As a result, in practice there is often no precise distinction between the two.

Explanation

The contradiction is a valid premise: "Lemons are yellow and lemons are not yellow". Any statements can be inferred from this, e.g. For example, that Santa Claus exists, as follows: "Lemons are yellow or Santa Claus exists." For this statement to be true (in the context of classical logic), part of the statement must be true. So the statement “Lemons are yellow or Santa Claus exists” is valid (because lemons are yellow, see premise). But if this sentence is valid and lemons are not yellow (which is also secured in the premise) then there is only the possibility that Santa Claus exists.

Assume that the statements of the following set of premises are true:

All Greeks are brave.
Socrates is a Greek.
Socrates is not brave.

On the one hand the sentence follows from this

Socrates is brave.

(from "All Greeks are brave" and "Socrates is a Greek" can be derived), on the other hand, trivially, directly from the set of premises, the negation of this sentence,

Socrates is not brave.

Two contradicting statements can be derived from the set of premises, i.e. H. the set is inconsistent, at least one of the three statements must be wrong. If one accepts all three statements as true, however, after “ex falso quodlibet”, any statement follows from this set regardless of its truth, for example the factually true sentence “When it rains the ground gets wet”, but also untrue statements like “grass is black” or “Socrates has four eyes”. Even contradictions like “grass is black and is not black” can be derived from it.

philosophy

justification

The "ex falso quodlibet" applies in the common logics, especially in the classical propositional and predicate logic . However, at first glance it doesn't seem very intuitive and therefore needs justification. This can look like this: The inference relation is supposed to contain truth , ie that the truth of the premises is to be transferred to the truth of the conclusion. That is, if the premises are true, then the conclusion must always be true if the inference is valid . However, if the premises contain a contradiction, they cannot be true under any circumstances. In this case, the conclusion is no longer important. Hence, any conclusion can be drawn.

Another justification is the following: It can be assumed that contradictions should be avoided. If, for example, a contradiction follows from a scientific theory , this would be a good reason to reject the theory. The “ex falso quodlibet” now gives us a reason for this requirement that contradictions are to be avoided: After the “ex falso quodlibet”, any statement follows from a contradicting theory. However, this makes the theory pointless. A theory from which everything follows cannot be used to make distinctions, cannot answer our questions, and cannot help us make decisions. The "ex falso quodlibet" thus means that a contradicting set of premises is worthless in practice.

criticism

Nevertheless, the “ex falso quodlibet” has also been criticized. So-called paraconsistent logics were created that do not use the “ex falso quodlibet”. These logics do not presuppose that “ex falso quodlibet” is wrong, which would be an additional axiom, they just don't use it. The following arguments speak for paraconsistency:

The argument from everyday inference

One line of argument is that the "ex falso quodlibet" is not used in our everyday inferential actions. We all (presumably) have contradicting views (see cognitive dissonance ). However, that doesn't mean we believe every statement. A defender of the “ex falso quodlibet” could object that we often believe in contradicting statements, but that we do so unconsciously. As soon as someone points this out to us (“What you are saying now contradicts what you said earlier.”), Then we will probably not say “Yes, that is a contradiction, but what the heck”, but will try to resolve the contradiction dissolve.

The paradox argument

The other line of argument relies on the existence of paradoxes . A paradox consists of two apparently contradicting statements, but both seem equally plausible . Usually one tries to resolve a paradox; H. either to show that one of the two statements is not plausible, or to show that the statements do not contradict each other. But there are some paradoxes for which no really good resolution is known, such as B. the liar paradox . From the point of view of paraconsistent logic, one can in such a case hold the contradicting statements to be true, since one need not accept the devastating consequences of each statement following. However, if one accepts the “ex falso quodlibet”, this way out remains blocked, the criticism is therefore that the “ex falso quodlibet” blocks the most natural strategy of dealing with paradoxes.

The “ex falso quodlibet”, on the other hand, can be defended by pointing out that it generally seems more worthwhile to look for “real” solutions to the paradoxes than to simply accept the contradiction. The history of logic and mathematics has shown that the resolution of paradoxes has often led to an advance in knowledge. Thus, through the resolution of Russell's paradox, axiomatic set theories such as those of the Zermelo-Fraenkel set theory emerged; thinking about the paradoxes of infinity provided the basis for calculus . If it were always possible to accept contradicting statements, then the need to resolve the paradoxes would also disappear, and this would hinder the gathering of new knowledge.

In addition, there are paradoxes such as Curry's Paradox , in which all statements follow even in a paraconsistent logic, in which the paraconsistent logic is in the same situation as the classical logic. This makes it appear that paraconsistent logic only offers an ad hoc solution and does not tackle the problem of paradoxes at the root.

Propositional logic

The conclusion ( implication ) from a false statement ( premise ) is always correct, regardless of what is inferred ( conclusion - hence the designation from false follows anything ). If, on the other hand, the premise is correct, the implication is only correct if the conclusion is also correct (see subjunction ):

premise	Conclusion	→ conditional
not correct	true	→ true
not correct	not correct	→ true
true	true	→ true
true	not correct	→ wrong