Paradoxes of material implication

The paradoxes of material implication are a group of formulae that are truths of classical logic but are intuitively problematic.

The root of the paradoxes lies in a mismatch between the interpretation of the validity of logical implication in natural language, and its formal interpretation in classical logic, dating back to George Boole's algebraic logic. In classical logic, implication describes conditional if-then statements using a truth-functional interpretation, i.e. "p implies q" is defined to be "it is not the case that p is true and q false". Also, "p implies q" is equivalent to "p is false or q is true". For example, "if it is raining, then I will bring an umbrella", is equivalent to "it is not raining, or I will bring an umbrella, or both". This truth-functional interpretation of implication is called material implication or material conditional.

The paradoxes are logical statements which are true but whose truth is intuitively surprising to people who are not familiar with them. If the terms 'p', 'q' and 'r' stand for arbitrary propositions then the main paradoxes are given formally as follows:
 * 1) These, which are all equivalent to $$(p \lor \neg p) \lor q $$:
 * 2) $$(\neg p \land p) \to q$$, p and its negation imply q. This is the paradox of entailment.
 * 3) $$\neg p \to (p \to q)$$ or $$p \to (q \lor p)$$, if p is false then it implies every q, in which cases the statement $$p \to q$$ is said to be vacuously true; or if p is true then it implies itself or every q since $$p \to p$$ is equivalent to $$p \lor \neg p$$. This is referred to as 'explosion'.
 * 4) $$p \to (q \to p)$$ [$$\neg p \lor (\neg q \lor p)$$], if p is true then it is implied by every q. In other words, Begging the question is based on a paradox. By analogy with $$\neg p \to (p \to q)$$ being referred to as 'explosion', this could be referred to as 'implosion'. In these cases, the statement $$q \to p$$ is said to be vacuously true.
 * 5) $$p \to q,\; p/\neg q\;\; \vdash\ p \to (q \lor r) [p \to (\neg q \to r)] $$ {$$((p \to q) \land p/\neg q) \to (p \to (q \lor r) [p \to (\neg q \to r)])$$}, p implies q therefore it may alternatively imply any r which is not equivalent to q. Since r may be p or the negation of p or q, it is possible to infer a paradox from a straightforward implication.
 * 6) $$p \to (q \lor \neg q)$$ [$$\neg p \lor (q \lor \neg q)$$], either q or its negation is true, so their disjunction is implied by every p; and its extension $$(p \to [q \lor r]) \lor (q \to r)$$ [$$(\neg p \lor [q \lor r]) \lor (\neg q \lor r)$$], if p, q and r are three arbitrary propositions, then either p implies either q or r or q implies r; or equivalently either q or its negation implies r. This is because if q is true then p implies it, and if it is false then q implies any other statement. Since r can be p, it follows that given two arbitrary propositions, one must imply the other, even if they are mutually contradictory, in other words the principle of totality for material implication is a paradox. For instance, "Nadia is in Barcelona implies Nadia is in Madrid, or Nadia is in Madrid implies Nadia is in Barcelona." This truism sounds like nonsense in ordinary discourse.
 * 7) $$\neg(p \to q) \to (p \land \neg q)$$ [$$(\neg p \lor q) \lor (p \land \neg q)$$], if p does not imply q then p is true and q is false. NB if p were false then it would imply q, so p is true. If q were also true then p would imply q, hence q is false. This paradox is particularly surprising because it tells us that if one proposition does not imply another then the first is true and the second false.

The paradoxes of material implication arise because of the truth-functional definition of material implication, which is said to be true merely because the antecedent is false or the consequent is true. By this criterion, "If the moon is made of green cheese, then the world is coming to an end," is true merely because the moon is not made of green cheese. By extension, any contradiction implies anything whatsoever, since a contradiction is never true. (All paraconsistent logics must, by definition, reject (1) as invalid.) Also, any tautology is implied by anything whatsoever, since a tautology is always true.

To sum up, although it is deceptively similar to what we mean by "logically follows" in ordinary usage, material implication does not capture the meaning of "if... then".

Paradox of entailment
As the best known of the paradoxes, and most formally simple, the paradox of entailment makes the best introduction.

In natural language, an instance of the paradox of entailment arises:
 * It is raining

And
 * It is not raining

Therefore
 * George Washington is made of rakes.

This arises from the principle of explosion, a law of classical logic stating that inconsistent premises always make an argument valid; that is, inconsistent premises imply any conclusion at all. This seems paradoxical because although the above is a logically valid argument, it is not sound (not all of its premises are true).

Construction
Validity is defined in classical logic as follows:
 * An argument (consisting of premises and a conclusion) is valid if and only if there is no possible situation in which all the premises are true and the conclusion is false. 

For example a valid argument might run:
 * If it is raining, water exists (1st premise)
 * It is raining (2nd premise)
 * Water exists (Conclusion)

In this example there is no possible situation in which the premises are true while the conclusion is false. Since there is no counterexample, the argument is valid.

But one could construct an argument in which the premises are inconsistent. This would satisfy the test for a valid argument since there would be no possible situation in which all the premises are true and therefore no possible situation in which all the premises are true and the conclusion is false.

For example an argument with inconsistent premises might run:
 * It is definitely raining (1st premise; true)
 * It is not raining (2nd premise; false)
 * George Washington is made of rakes (Conclusion)

As there is no possible situation where both premises could be true, then there is certainly no possible situation in which the premises could be true while the conclusion was false. So the argument is valid whatever the conclusion is; inconsistent premises imply all conclusions.

Explanation
The strangeness of the paradox of entailment comes from the fact that the definition of validity in classical logic does not always agree with the use of the term in ordinary language. In everyday use validity suggests that the premises are consistent. In classical logic, the additional notion of soundness is introduced. A sound argument is a valid argument with all true premises. Hence a valid argument with an inconsistent set of premises can never be sound. A suggested improvement to the notion of logical validity to eliminate this paradox is relevant logic.

Simplification
The classical paradox formulae are closely tied to the formula,


 * $$(p \land q) \to p$$

the principle of Simplification, which can be derived from the paradox formulae rather easily (e.g. from (1) by Importation). In addition, there are serious problems with trying to use material implication as representing the English "if ... then ...". For example, the following are valid inferences:


 * 1) $$(p \to q) \land (r \to s)\ \vdash\ (p \to s) \lor (r \to q)$$
 * 2) $$(p \land q) \to r\ \vdash\ (p \to r) \lor (q \to r)$$

but mapping these back to English sentences using "if" gives paradoxes. The first might be read "If John is in London then he is in England, and if he is in Paris then he is in France. Therefore, it is either true that (a) if John is in London then he is in France, or (b) that if he is in Paris then he is in England." Using material implication, if John really is in London, then (since he is not in Paris) (b) is true; whereas if he is in Paris, then (a) is true. Since he cannot be in both places, the conclusion that at least one of (a) or (b) is true is valid.

But this does not match how "if ... then ..." is used in natural language: the most likely scenario in which one would say "If John is in London then he is in England" is if one does not know where John is, but nonetheless knows that if he is in London, he is in England. Under this interpretation, both premises are true, but both clauses of the conclusion are false.

The second example can be read "If both switch A and switch B are closed, then the light is on. Therefore, it is either true that if switch A is closed, the light is on, or that if switch B is closed, the light is on." Here, the most likely natural-language interpretation of the "if ... then ..." statements would be "whenever switch A is closed, the light is on", and "whenever switch B is closed, the light is on". Again, under this interpretation both clauses of the conclusion may be false (for instance in a series circuit, with a light that only comes on when both switches are closed).