Tuesday, August 7, 2012

A Defense of the Material Conditional

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This is part 3 of my series on logic and critical thinking.
  1. Introductory Logic, Part 1—Introducing both logic in general (such as the difference between a deductive and inductive argument) and propositional logic in particular
  2. Introductory Logic, Part 2—More propositional logic
  3. Introductory Logic, Part 3—A defense of the material conditional
You don’t necessarily need to read parts 1 and 2 to understand this article. If you’re already familiar with propositional logic, or you think you might understand the crash-course in some rules of inference (e.g. modus ponens) that I’ll provide here, feel free to read on.

The Problem

One sort of statement propositional logic deals with is the conditional, i.e. an “If P, then Q” statement. In symbolic logic, “If P is true, then Q is true” is symbolized as P → Q (though sometimes also symbolized as P ⊃ Q), where P is the antecedent and Q is the consequent. Propositional logic uses what’s called a material conditional, which means “If P, then Q” is equivalent to “It is not the case that P is true and Q is false.” Thus, whether the material conditional P → Q is true is determined entirely by the truth of P and Q as follows, where the truth table below exhausts all possible true/false combinations of P and Q:

pq  
TTT  
TFF  
FTT  
FTF  


The truth table above has the following counterintuitive implications:
  1. Whenever the consequent (q) is true, the statement “If p is true, then q is true” is true.
  2. Whenever the antecedent (p) is false, the statement “If p is true, then q is true” is true.
These counterintuitive properties result in the so-called paradoxes of material implication, e.g. “If there is a married bachelor, then the earth is round” is true because the antecedent (“there is a married bachelor”) is false, and “If grass is air, then two plus two equal four” is true because the consequent (“two plus two equal four”) is true. Such implications seem strange. One way to resolve the paradox is to just note that the material conditional doesn’t match the conditionals (“if-then” statements) of ordinary language. In ordinary language, there’s the requirement that the consequent in some sense relevantly follows from the antecedent. For example, “If I am a man, then I am human” is a case where the consequent (“I am human”) more relevantly follows from the antecedent (“I am a man”) compared to “If I grass is air, then I am human.” Thus, ordinary language conditionals don’t have counterintuitive properties #1 and #2, even if material conditionals do. There’s one problem that strategy though. As counterintuitive as properties #1 and #2 are, it turns out that if we follow certain logical rules of inference, any “If p is true, then q is true” statement must have these strange properties, and I’ll prove that in this very article.

Recapping Some Rules of Inference

Some propositional logic rules I’ll use:

simplification
 
In English In Symbolic Logic
p and q

Therefore, p
p ∧ q

∴ p
p and q

Therefore, q
p ∧ q

∴ q
Disjunctive Syllogism
 
In English In Symbolic Logic
p or q
Not-p

Therefore, q
p ∨ q
¬p

∴ q
p or q
Not-q

Therefore, p
p ∨ q
¬q

∴ p

modus ponens
 
In English In Symbolic Logic
If p then q
p

Therefore, q
p → q
p

∴ q
conjunction
 
p
q

∴ p ∧ q
addition
 
p

∴ p ∨ q

conditional proof
 
a) p conditional proof assumption
b)
 ...
 q
c) p → q a-b, conditional proof

indirect proof
 
a) p indirect proof assumption
b)
 ...
 q ∧ ¬q (or ¬q ∧ q)
c) ¬p a-b, indirect proof
a) ¬p indirect proof assumption
b)
 ...
 q ∧ ¬q (or ¬q ∧ q)
c) p a-b, indirect proof


Indirect proofs and conditional proofs have a “What is true if such-and-such is true?” approach, and use the rules of logic to derive further statements. Given some proposition p, such proofs use the rules of logic (and possibly prior premises) to show what is true if p is true. For example:
  1. (H ∧ R) → ¬(H ∧ R)

  1. H ∧ R indirect proof assumption
    1. ¬(H ∧ R) 1, 2, modus ponens
    2. (H ∧ R) ∧ ¬(H ∧ R) 2, 3, conjunction
  1. ¬(H ∧ R) 2-4, indirect proof
Thus, if H ∧ R is true, then ¬(H ∧ R) is true, which leads to a self-contradiction, and therefore ¬(H ∧ R) is true.

In propositional logic, theorems are statements you can prove without premises. For example:
  1. A conditional proof assumption
    1. A ∨ B 1, addition
  1. A → (A ∨ B) 1-2, conditional proof
A slightly more complicated logic example:
  1. (C ∨ D) → Z

  1. B conditional proof assumption
    1. C conditional proof assumption
      1. C ∨ D 3, addition
      2. Z 1, 4, modus ponens
    1. C → Z 3-5, conditional proof
  1. B → (C → Z)2-6, conditional proof
One nice thing about these rules of logic is that they resemble how human beings naturally reason, e.g. we tend to think that a belief that leads to a self-contradiction is false. That seems especially noteworthy when we consider that “If P is true, then Q is true” must have those counterintuitive properties if we are to accept the above rules of inference as valid, as I’ll prove next.

Line by Line

One way to show that the two counterintuitive properties hold for any “If P, then Q” statement that follows the rules of inference is to go through each line of the truth table and show that each line of the truth table is needed if the rules of inference are to be accepted as valid.

The Second Line

pq
TFF


I’m starting a bit out of order, but what I’ll say here is important for the other stuff to build upon. A rule of inference is valid if and only if true input statements guarantee a true output statement. For example, consider the rule of modus ponens:
  1. P → Q
  2. P

  1. Q 1, 2, modus ponens
Modus ponens is a valid inferential form only if it’s impossible to have true premises (lines 1 and 2) and a false conclusion (line 3). That is, for modus ponens to be a valid rule of inference, true input statements (lines 1 and 2) must guarantee a true output statement (line 3). The only way this can happen is if the second line of the truth table is true. Proof: suppose it could be that P being true and Q being false makes line 1 true. Not only would line 1 be true in that case, but with P true and Q false, line 2 would also be true and line 3 would be false, which means we would have true input statements and a false output statement. So for modus ponens to be a valid rule of inference, the second line of the truth table must be true.

We can buttress the need for the second line of the truth table further by proving the following theorem:
(P ∧ ¬Q) → ¬(P → Q)
A proof of the theorem goes as follows:
  1. P ∧ ¬Q conditional proof assumption
    1. P → Q indirect proof assumption
      1. P 1, simplification
      2. Q 2, 3, modus ponens
      3. ¬Q 1, simplification
      4. Q ∧ ¬Q 4, 5, conjunction
    1. ¬(P → Q) 1-6, indirect proof
  1. (P ∧ ¬Q) → ¬(P → Q) 1-7, conditional proof
Because theorems are necessarily true, the above theorem means that it is impossible for the statement in line 8 to have a true antecedent (P ∧ ¬Q) with a false consequent ¬(P → Q), so what we have here is an example of something called strict implication. Given some “If P, then Q” statement, a material conditional merely says it is the case that “P is true and Q is false” is false, whereas a strict conditional (also referred to as strict implication or entailment) says it’s impossible that P is true and Q is false. To illustrate, “If the earth is round, then Abraham Lincoln existed” is true as a material conditional, but there is a possible world where the earth was round and Abraham Lincoln never existed, so this isn’t a true strict implication. In contrast, “If there are more than five particles, then there are more than three particles” is a legitimate example of entailment, because there is no possible world where “there are more than five particles” is true and “there are more than three particles” is false. According to the above theorem, “P is true and Q is false” entails (or strictly implies) that “If P, then Q” is false.

The First Line

pq
TTT


A theorem that illustrates the first line of the truth table:
(P ∧ Q) → (P → Q)
A proof of that theorem goes as follows:
  1. P ∧ Q conditional proof assumption
    1. P conditional proof assumption
      1. Q 1, simplification
    1. P → Q 2-3, conditional proof
  1. (P ∧ Q) → (P → Q) 1-4, conditional proof
Thus, the rules of inference entail that whenever the antecedent and consequent are both true, the conditional is true.

The Third Line

The third line of the truth table:

pq
FTT


A theorem that illustrates the third line of the truth table:
(¬P ∧ Q) → (P → Q)
A proof of that theorem goes as follows:
  1. ¬P ∧ Q conditional proof assumption
    1. P conditional proof assumption
      1. Q 1, simplification
    1. P → Q 2-3, conditional proof
  1. (¬P ∧ Q) → (P → Q) 1-4, conditional proof
Thus, the rules of inference entail that whenever the antecedent is false and the consequent is true, the conditional is true.

The Fourth Line

The fourth line of the truth table:

pq
FTF


A theorem that illustrates the fourth line of the truth table:
(¬P ∧ ¬Q) → (P → Q)
A proof of that theorem goes as follows:
  1. ¬P ∧ ¬Q conditional proof assumption
    1. P conditional proof assumption
      1. P ∨ Q 2, addition
      2. ¬P 1, simplification
      3. Q 3, 4, disjunctive syllogism
    1. P → Q 2-5, conditional proof
  1. (¬P ∧ ¬Q) → (P → Q) 1-6, conditional proof
Thus, the rules of inference entail that whenever the antecedent is false and the consequent is false, the conditional is true.

Conclusion

The material conditional has the following two properties:
  1. Whenever the consequent (q) is true, the statement “If p is true, then q is true” is true.
  2. Whenever the antecedent (p) is false, the statement “If p is true, then q is true” is true.
Odd as it may seem, for any “If P is true, then Q is true” statement to follow the set of inference rules I mentioned earlier (modus ponens, disjunctive syllogism etc.), it must have these two strange properties. For those rules of inference to be valid, it cannot be otherwise. We might think of an if-then statement in the ordinary sense as having more than just the “it is not the case that the antecedent is true and the consequent is false” requirement; e.g. one might want to add the requirement that the consequent in some sense relevantly follows from the antecedent. For example, “If I am a man, then I am human” is a case where the consequent (“I am human”) more relevantly follows from the antecedent (“I am a man”) compared to “If I grass is air, then I am human.” But there’s a catch: the “consequent must more relevantly follow from the antecedent” requirement for an if-then statement means such an if-then statement can’t follow the rules of inference. If you’re like me, that’s rather interesting and surprising; one would actually have to give up that set of logic rules to avoid counterintuitive properties #1 and #2, but those rules of logic seem very rational to use in our everyday lives.

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