Justification of the material conditional truth function in Introduction to Formal Logic

Question

Pages 150-151 of §18.3 of Introduction to Formal Logic by Peter Smith provide two justifications for the truth table of the material conditional.

In the first justification (paragraph (a) - (c) on pg. 150), Smith shows that the material conditional must have the truth table it has if it is to preserve the four basic properties of ordinary conditional: (1) modus ponens, (2) modus tollens, (3) the falsehood condition, (4) non-reversibility.

Question 1

I'm unclear about where these properties come from:

• Are they simply being inferred from the natural language usage of conditionals (Jackson uses a similar line of reasoning in which he uses the term "logical intuitions")?
• How do we know that these are the only four properties that a conditional-like truth-functional connective must satisfy?

The second justification (paragraph (d) on pg. 151) is as follows (emphasis added):

(d) For a fast-track argument, take the wff (P ∧ Q) → P. Assuming the arrow is conditional-like, this wff should always be true (since necessarily, if P and Q are both true, then P in particular is true). But the values of the antecedent/ consequent in this wff can be any of T/T, or F/T, or F/F (depending on the values of P and Q). And we've just said that our conditional needs to evaluate as T in each of those three cases. So that forces the same completion of the table for our conditional-like truth function.

Question 2

I don't know if I'm following this correctly, especially the last two lines.

Say P = "I have apples" and Q = "I have oranges", then:

• If I have apples, then I have apples, regardless of whether I have oranges.
  • So (P ∧ Q) → P and (P ∧ ¬Q) → P are both true.
• Similarly, if I don't have apples, then I don't have have apples, regardless of whether I have oranges.
  • So (¬P ∧ Q) → ¬P and (¬P ∧ ¬Q) → ¬P are both true.

Therefore, if → is conditional-like, then (P ∧ Q) → P must always be true (we can also verify this using logical equivalences: (P ∧ Q) → P ⇔ ¬(P ∧ Q) ∨ P ⇔ (¬P ∨ ¬Q) ∨ P ⇔ (¬P ∨ P) ∨ ¬Q ⇔ ⊤ ∨ ¬Q ⇔ ⊤)).

Given the definition of conjunction, the antecedent/consequent can only take the values F/F, F/T, or T/T, and, as I tried to show in the apples and oranges example above, (P ∧ Q) → P must be true in each of these cases. From this we can infer lines 1, 2, amd 4 of the material conditional's truth table:

A	C	A → C
F	F	T
F	T	T
T	F	?
T	T	T

But how does this force the same completion of line 3 as the first justification (which used the falsehood condition)? Is it F simply due to the fact that the antecedent/consequent can never be T/F (since the only possible combinations are F/F, F/T, and T/T)?

Answer 1

A fairly common approach to understanding logical connectives is to suppose that their meaning is completely determined by the role they play in logical inferences. This position is called inferentialism. Without going into detail, this does not entirely work, but it is reasonable to suppose that a connective must conform to at least a core of inferential relations for it to mean what it does. For example, if someone told you they accepted "A and B" as true, but didn't accept that A was true, you would think they didn't understand the meaning of 'and'.

Similarly, with conditionals quite generally, it is widely (though not universally) held that a conditional must obey modus ponens or else it is not a conditional. Another important characteristic that Peter Smith calls non-reversibility is commonly called the fallacy of affirming the consequent. A true "if A then B" and a true B does not guarantee a true A. Another characteristic is modus tollens, that a true "if A then B" taken with a false B yields a false A. There may be other characteristics that we would expect a conditional to have, but these are the most fundamental ones, and it turns out these are all we need for the present purpose.

Just like the classical connectives of conjunction and disjunction, the material conditional is a truth function in two variables, and it is bivalent, i.e. it is either true or false. This means it can be characterised by a truth table as follows:

A  B   A → B

T  T    a

T  F    b

F  T    c

F  F    d

The values for a, b, c and d can then be deduced from the inferential relationships mentioned above. The validity of modus ponens requires a T in position a and an F in position b, because in the event A is true, the conditional must have the same truth value as B. Affirming the consequent requires a T in position c, since if A → B were false whenever B is true and A is false, then there could be no instances of affirming the consequent, since there could be no cases of A → B being true, B true and A false. Finally, modus tollens requires a T in position d, since A → B must be true for there to be valid inferences where B is false and A is false.

(An alternative way to get a T in position d is simply to observe that we cannot have an F here since that would make the truth table for the material conditional identical with the truth value of B, which would make the conditional trivial. This was pointed out long ago by Peter Geach. So we could even dispense with modus tollens as a defining characteristic if we wished.)

Peter Smith offers a shortcut to putting a T in positions a, c and d by saying, as you show, that it is required if we understand that (A ∧ B) → B is always true. The F in position b is a separate requirement that follows from the fact a conditional cannot be true if its antecedent is true and its consequent is false.

An important corollary to all of this, is that it only justifies that out of all the bivalent and dyadic truth functions, the material conditional is the only one that serves as a conditional. It does not mean that all conditionals are material conditionals, which is very far from the truth. In practice, most conditionals that you meet in ordinary language are modal, either explicitly or implicitly, and as such are not truth functions at all.

Answer 2

Question 1: I'm unclear about where these properties come from.

The Truth Table for A => B

The truth table for logical implication can be seen as a convenient presentation of four related theorems in propositional logic:

Line 1 (TTT): A & B => [A => B]

Line 2 (TFF): A & ~B => ~[A => B]

Line 3 (FTT): ~A & B => [A => B]

Line 4 (FFT): ~A & ~B => [A => B]

Following are formal proofs deriving each line of the table using a form of natural deduction. They make use of the following properties of logical implication:

1. The introduction of '=>' by discharging a premise (the Conclusion Rule here, which can also introduce '~')
2. The elimination of '=>' by detachment or modus ponens (the Detach Rule here)

They make no use of any other properties of the '=>' operator.

Derivation of Line 1

Derivation of Line 2

Derivation of Line 3

Derivation of Line 4