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Subsection 7.2.12 “Paradoxes” of Material Implication

Material implication, the definition of → that we are using, is similar to the English expressions “implies” and “if/then”. But it isn’t identical. The differences lead to various kinds of confusion, sometimes thought of as paradoxes.

Define: A: Austin is in Texas

M: The Moon is made of green cheese.

C: Dallas is the coolest city in Texas.

Now we can look at some of these “paradoxes”.

The “Paradox” of Entailment (or the Principle of Explosion)

This sentence is true:

If Austin is in Texas and Austin isn’t in Texas then the moon is made of green cheese.

(A  A)  M

This is so even though there is no connection whatever between (A  A) and M.

More generally, for any claims p and q:

(p ∧ ¬p) → q

False → p

This observation, which follows directly from the truth table definition of →, is also called the Principle of Explosion (because, if we accept even one contradiction, we must accept every claim as true). It’s also called, in the Latin of classical logic, ex falso quodlibet or ex contradictione sequitur quodlibet.

If p is True

This sentence is true:

If Austin is in Texas then “The moon is made of green cheese implies that Austin is in Texas.”

A  (M  A)

Again, this is so even though there is no connection between A and M.

More generally, there is a simple truth table proof that, for any p and q, if p is true, then any q implies it:

p → (q → p)

One Implication Must Be True

This sentence (under the interpretation given by the parentheses shown here) is true:

(Austin being in Texas implies that Dallas is the coolest city in Texas) or (Dallas being the coolest city in Texas implies that the moon is made of green cheese).
(A  C)  (C  M)

More generally (and provable by truth table) for any p, q, and r:

(p → q) ∨ (q → r)

An English argument for the truth of this claim is: q must be either true or false. If it’s true, then (p → q) is true (regardless of the truth of p). If it’s false, then (q → r) is true (regardless of the truth of r).

If p Doesn’t Imply q

This sentence is true:

If it’s not true that Austin being in Texas implies that Dallas is the coolest city in Texas, then Austin is in Texas and Dallas isn’t the coolest city in Texas.

(A  C)  (A  C)

More generally (and provable by truth table) for any p and q:

¬(p → q) → (p ∧ ¬q)

An English argument for the truth of this claim is: The only way that the claim (p → q) can be false is if p is true, yet q is false.

Exercises Exercises

Exercise Group.

1. On the same subject as the old adage that, if there’s smoke there’s fire, let’s assert one premise:

[1] It’s not true that fire implies smoke.

For each of the following claims, indicate whether (given the premise) it must be true, it must be false, or it could be either true or false (depending on the truth of other claims).

Hint: Write out the premise and each of these claims in clear Boolean logic.

Part 1.

There’s fire and not smoke.

  • Must be true.

  • Must be false.

  • Could be either true or false.

Answer.

Correct answer is A

Solution.

Explanation: The premise can be written as (Fire  Smoke). Using conditional disjunction, De Morgan and double negation, we get that it is equivalent to Fire  Smoke.

Part 2.

If there’s smoke, it’s sunny.

  • Must be true.

  • Must be false.

  • Could be either true or false.

Answer.

Correct answer is A.

Solution.

Explanation: The premise can be written as (Fire  Smoke). Using conditional disjunction, De Morgan and double negation, we get that it is equivalent to Fire  Smoke. This claim can be written as Smoke  Sunny, which is equivalent to Smoke  Sunny, which must be true because Smoke is. Another way to see it is that, if we assume Smoke, then we have a contradiction with Smoke. So, by explosion, anything is true.

Part 3.

There’s smoke or there isn’t fire.

  • Must be true.

  • Must be false.

  • Could be either true or false.

Answer.

Correct answer is B.

Solution.

Explanation: The premise can be written as (Fire  Smoke). Using conditional disjunction, De Morgan and double negation, we get that it is equivalent to Fire  Smoke. This claim can be written as Smoke  Fire. But neither of those is possible.

Part 4.

It’s sunny and there’s a fire.

  • Must be true.

  • Must be false.

  • Could be either true or false.

Answer.

Correct answer is C.

Solution.

Explanation: As we’ve seen, the premise is equivalent to Fire  Smoke. This claim can be written as Sunny  Fire. We know that Fire must be true. So this claim will be true if Sunny is also true, but false if Sunny is false.

Part 5.

If there’s no fire, then it’s Tuesday.

  • Must be true.

  • Must be false.

  • Could be either true or false.

Answer.

Correct answer is A.

Solution.

Explanation: As we’ve seen, the premise is equivalent to Fire  Smoke. This claim can be written as Fire  Tuesday, which can be rewritten as Fire  Tuesday, which must be true since Fire is. Another way to see it is that, if we assume Fire, then we have a contradiction with Fire. So, by explosion, anything is true.

Part 6.

If it’s Tuesday, there’s no fire.

  • Must be true.

  • Must be false.

  • Could be either true or false.

Answer.

Correct answer is is C.

Solution.

Explanation: As we’ve seen, the premise is equivalent to Fire  Smoke. This claim can be written as Tuesday  Fire, which can be rewritten as Tuesday  Fire. We know that Fire must be true, so Fire isn’t. So this claim will be true if Tuesday is true, but false if Tuesday is false.