Continuing the Debate on Paradoxes of Material Implication

This article continues an interesting debate about the paradoxes of the material implication. The debate started when Emanuel invented two examples that convincingly reveal a paradox. One of his examples is expressed by the following two sentences in natural language:

[1] If Jan is older than 20 years and younger than 30 years then Jan is between 20 and 30 years old.

[2] Whether Jan is between 20 and 30 years old either follows from the fact alone that Jan is older than 20 or from the fact alone that Jan is younger than 30.

Introducing three propositional symbols:

• P: Jan is older than 20 years;
• Q: Jan is younger than 30 years;
• R: Jan is between 20 and 30 years old;

Emanuel renders the argument [1]-[2] as follows in propositional logic:

• [1*] P ∧ Q → R
• [2*] P → R ∨ Q → R

Note that in all situations in which [1*] holds, [2*] holds as well. In modern logic, the logical inference operator ⊨ has been defined to capture this notion of valid reasoning. Indeed, it can be shown that the following holds:

[1*] P ∧ Q → R   ⊨   [2*] P → R ∨ Q → R (EQ1)

Therefore, Emanuel concludes that the reasoning [1]-[2] is logically valid.

My idea was to render the formalization differently. To avoid confusion around the adjunctive and connective reading of the implication, I invoked the logical inference operator rather than the material implication. The paradox is then formalized as follows:

P ∧ Q ⊨ R   ⇒   P ⊨ R or Q ⊨ R (EQ2)

Interestingly, though EQ1 holds, EQ2 does not hold. Hence, EQ2 being false shows us, as expected, that if P and Q entails R then neither P entails R nor Q entails R. To get convinced of the existence of a paradox, we need something along the lines of EQ1.

In his response, Emanuel dismissed EQ2 as a proper formalization by claiming that it is much too strong. But is he justified to make this claim? Certainly not. When proving, say B follows from A, an indirect proof does not suffice. Such a proof needlessly demystifies how exactly A entails B. Once the mystery is clarified, one can see how B follows from A and the paradox is resolved. This is the general idea which I will further explain in the sequel.

So how is one supposed to refute a proponent that starts his reasoning from the premise P ∧ Q -> R? The answer is straightforward. In the dialectical method it is very reasonable to expect the proponent to defend his or her premises. With P, Q and R the selected proposition symbols, the proponent lacks the logical context to be sufficiently justified. The opponent can prove ⊭ P ∧ Q -> R to oppose against the premise. Hence, it appears the proponent starts from a false premise. He or she must show how R follows exactly from P and Q; thereby demystifying the implication.

The correct premise is that Jan must be of a certain age; this clearly cannot be refuted. Now, it immediately follows that exactly one of the following must hold:

P': Jan is 20 years old or younger (i.e., P' ↔ ¬P)

Q': Jan is 30 years old or older (i.e., Q' ↔ ¬Q)

R: Jan is between 20 and 30 years old (R)

Note that it follows that ¬P' ∨ ¬Q', or equivalently P ∨ Q. Also note that P becomes R ∨ Q' and Q becomes R ∨ P'. Suppose R ∨ P' (i.e., Q) and R ∨ Q' (i.e., P). Because either P' or Q' (or both) must be false, it follows that R must hold. This is how the proponent ought to show the implication: (R ∨ P') ∧ (R ∨ Q') → R. Formally, we arrived at the following result:

G ⊨ P ∧ Q → R (EQ3)

where G is the background theory containing at least:

1. P ↔ R ∨ Q'
2. Q ↔ R ∨ P'
3. P ∨ Q

Finally, using EQ1 as an inference rule we obtain the more complete formalization of the paradox:

G ⊨ (P → R) ∨ (Q → R) (EQ4)

Thus far, we have only demystified the material implication in the premise. In order to demystify the two implications in the conclusion, the key is to realize that material implications in a propositional formula may well be conditional. An explicit example is A → (B → C). Here, B → C is a conditional implication meaning that C is only implied by B under the condition that A holds.

In a disjunctive conclusion, implicit conditions of a material implication may occur. Because either P or Q (or both) must hold, both P and Q can become implicit conditions of material implications. The natural deduction proof reveals this:

1. G ⊨ P ∨ Q (from the premise)
2. G ⊨ P ∧ Q → R (from EQ3)
3. G, P ⊨ P (hypothesis)
4. G, P, Q ⊨ Q (hypothesis)
5. G, P, Q ⊨ P ∧ Q (∧-intro, 3, 4)
6. G, P, Q ⊨ R (→-elim, 2, 5)
7. G, P ⊨ Q → R (→-intro, 4, 6)
8. G, Q, P ⊨ P (hypothesis)
9. G, Q, P ⊨ R (reiteration, 6)
10. G, Q ⊨ P → R (→-intro, 8, 9)
11. G, P ⊨ P → R ∨ Q → R (∨-intro, 7)
12. G, Q ⊨ P → R ∨ Q → R (∨-intro, 10)
13. G ⊨ P → R ∨ Q → R (∨-elim, 1, 11, 12)

Note how the natural deduction proof reveals the implicit conditions at steps 7 and 10, because P and Q occur as condition at the left-hand side of ⊨, respectively. How does R follow from P in the conclusion of EQ4? The answer is: it follows conditionally given that Q holds. And how does R follow from Q? Again: it follows conditionally given that P holds.

So yes, R can be implied from P or Q, but otherwise as explicitly stated in natural language (see [2]), R is not implied from P alone or from Q alone. The word alone is just one bridge too far and makes the meaning of [2] very strong. So strong that indeed my formalization (see EQ2) is needed to detect the conditional implications. Hence, EQ2 is not too strong at all. It is EQ1, Emanuel's rendering, that is too weak: corresponding only with the weaker conclusion:

[2'] Whether Jan is between 20 and 30 years old is (conditionally) implied by the fact that Jan is older than 20 or the fact that Jan is younger than 30.