What does the truth-value of a material implication represent?

Question

This question comes from my attempts to understand what the truth value for a material implication with a false antecedent represents. I have seen several justifications for this convention, usually through example, but each one seems to imply that the truth value of the material implication represents something different. I will provide three examples below to illustrate my reasoning.

(1) "'If I am healthy, I will come to class.' We can symbolize it, p -> q. The question is: when is this statement false? When will I have broken my promise?"

Two examples of this scenario are given when the antecedent is false.

Ex 1: I am not healthy, but I have come to class anyway. I did not violate my promise; the conditional is true.

Ex 2: I am not healthy, and I did not come to class. I did not violate my promise; the conditional is true.

Analysis: In this example, the truth-value of the material implication seems to represent whether or not an event falsifies the given implication, or "promise" in this case. If a scenario shows the promise was broken, the material implication gets labeled false. Otherwise, it is labeled true. Note that the truth value here does not represent whether or not p actually implies q in some sense, but only if the implication has been falsified or not.

(2) "If 3 is a perfect square, then 3 is not prime."

Analysis: This example was given as a way to make sense of having a true material implication when the antecedent is false. I should note that no explicit reasoning was given to show why this translates into a true material implication, so I will be assuming the only interpretation of it that I see as sensible for this purpose. In this example the truth-value of the implication seems to represent whether or not a sentential function is stating a true fact or false fact for all possible cases. I see it as analyzing whether or not the sentence "If x is a perfect square, then x is not prime." is true or false. This again means that the truth-value of the material implication does not represent whether p actually implies q, nor does it represent whether or not a given statement has been falsified as the last example did.

(3) The vacuous argument. "Another way to show that the paradoxes are acceptable to some of our intuitions about implication is to consider statements like, "If Congress passes serious campaign finance reform, then I'm a monkey's uncle!" Or, "...I'm a Dutchman!" Or, "...I'm the Pope!" These expressions are flamboyant ways of asserting the antecedent to be false. But note how we do it. We say that if this falsehood is allowed to stand, then anything follows —even the absurdity that I'm the Pope. And this sarcasm exactly follows the logic of material implication. False antecedents create true conditionals."

Analysis: The problem with this argument is that it appeals to convention rather than reason, and the underlying principle of the convention is not a “vacuous” argument as the comparison would like to imply. In everyday language we provide a ridiculous consequent as a way demonstrate how improbable the antecedent is, not because we think the antecedent is false and therefore anything follows. This reasoning does seem to imply that the truth-value of the material implication should represent whether or not q follows from p. This stands in contrast to the previous two examples.

Summary: The three examples provided above each seem to imply that the truth-value of a material implication represents something different. These differences can be seen below.

In (1) the truth-value of the material implication represents whether or not an implication has been falsified by a given scenario.

In (2) the truth-value of the material implication represents whether or not a sentential function is true for all cases.

In (3) the truth-value of the material implication represents whether or not q follows from p.

There are other arguments and examples that imply additional contrasting representations for the truth-value of a material implication, but I think these examples will suffice to show where my question is coming from. Any insight you can offer would be greatly appreciated!

Answer 1

I share your concern about the application of material implication. It is a useful connective in mathematics or other highly mathematical branches of science but it does little to capture what we mean by conditionals in natural languages. This is why there are so many so-called paradoxes of material implication, which are not really paradoxes at all, just examples of where material implication does not adequately account for ordinary usage.

To take your three examples...

1. I would call this a conditional promise rather than a conditional statement. As such, it doesn't have a truth value. In the event I am healthy, I fulfil my promise by coming to the class; in the event I am not, the conditions for the promise are not present - I didn't break the promise, but it seems a little odd to say I kept it.

2. This doesn't work as an example of material implication, because it fails to distinguish cases that are plausible from those that are not. "If 3 is a perfect square, then 3 is not prime" is plausible not because 3 is not a perfect square, but because all perfect squares are not prime. After all, "If 3 is a perfect square, then 3 is a transfinite number" is hardly plausible, but it has the same false antecedent.

3. Dutchman conditionals may well be treated as special cases of conditionals. They have a rhetorical purpose in which typically someone might assert some proposition P and someone else responds with, "if P then I'm a Dutchman" - meaning they consider P to be absurd and so they counter with an absurdity of their own.

There are many classes of case where material implication does not work at all as an account of English conditionals. Some of these are:

4. Bets on conditionals. If I offer to bet with you that "if X is nominated for the presidential election then X will win the election," I'm not betting that the material implication "X nominated -> X elected" comes true. If X is not nominated then there is no bet.

5. Conditional commands. If a doctor commands a nurse, "if the patient is still alive tomorrow morning, change the dressing," this is not a command to make it the case that "patient alive -> change dressing" - if it were, the nurse could comply by killing the patient.

6. Statements about probabilities. If you say that it is probable that "if A then B", you do not mean that it is probable that A -> B, since that would be trivially true in the event that A is highly improbable. What is meant by it is probable that "if A then B" is typically that the conditional probability P(B | A) is high. If I roll a fair, six-sided die, the probability of "if even then a 6" is P(6 | even) which is 1/3; whereas P( even -> 6 ) is 2/3.

7. Claims about causal relations. Material implication is just a truth function: it says nothing about how the content of the antecedent and consequent are related. When making causal claims we are always making a substantial claim about such a relationship. Typically we have a kind of internal model of how we think the world works and what things are true in it, and we use that to make projections about what would happen in hypothetical circumstances. "If I drink this poison, it will kill me," is not plausible because I have no intention of drinking it, but because my model of how my body works and what drinking poison would do to it strongly suggests to me that death will be the causal consequence.

8. Claims about conditionals where we know (or at least believe) the antecedent is false. Commonly called counterfactuals, these are not trivially true, even though the material implication is true.

Making sense of conditionals has spawned an enormous literature and there is still no generally accepted account of their meaning.

Answer 2

As you can see in the post What is the origin of the truth table in logic, the truth-functional definition of material implication (or conditional) was codified by ancient Greek Stoic logicians with the so-called philonian conditional.

In modern time, it was "re-discovered" by Gottlob Frege in his Begriffsschrift (1879).

It corresponds to point (1) of your analysis:

a conditional with true antecedent and false consequent is false; in all other cases it is true.

Your case (2) is correctly analyzed as:

∀x (Perf_Sq(x) → ¬ Prime(x));

it is a generalization of a conditional (in the past called "formal implication"; see the post: is there an "immaterial" implication ?).

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For (3), you have to consider that the relation of logical consequnce (or entailment) is what defines:

A good argument is one whose conclusions follow from its premises; its conclusions are consequences of its premises.

It is a relation that holds between a set Γ of sentences (the premises) and a sentence A (the conclusion) symbolized with:

Γ ⊨ A

and "discovered by Aristotle with his notion of deduction (sullogismos):

A deduction is speech (logos) in which, certain things having been supposed, something different from those supposed results of necessity because of their being so. (Prior Analytics I.2, 24b18–20)

Each of the “things supposed” is a premise of the argument, and what “results of necessity” is the conclusion.

The relation between logical consequence and conditional is:

A ⊨ B iff ⊨ A → B,

but we have to consider that the two are not the same: for example, the definition of logical consequence does not change also when the language has no conditional (→) connective.

Answer 3

Material Implication, Logical Equivalencies, Proof by Contraposition:

I demonstrate that the contraposition of a material implication (X), called the contrapositive of that implication (X*), is logically equivalent to that material implication (X): X ≡ X*.

Given the material conditional P -> Q, P is referred to as "antecedent" and Q is referred to as consequent in this form (forward implication from P to Q).

If the antecedent (P) is true, then the material implication (->) holds only if the consequent (Q) is also true. That is, a true antecedent/premise/condition (P) can only imply a true /consequent/conclusion/consequence (Q).

If the antecedent (P) is true and the consequent (Q) is false, then the implication does not hold (true). That is, a true antecedent cannot imply a false consequent: truth cannot imply falsity. This is the only option for which the material implication does not hold, i.e., the operator/connective (->) outputs false if and only if the antecedent (P) is true and the consequent (Q) is false.

If the antecedent (P) is false, then P materially implies Q, regardless of whether Q is true or false. From falsity anything follows. That is, a false antecedent (P) implies the consequent (Q) in the case where Q is false as well as in the case where Q is true. For more information, please look up the "principle of explosion" which states in Latin: "Ex falso sequitur quodlibet" = "From falsity follows anything".

Example 1 (with a false consequent): "If 2+2 = 5, then I am god" both P and Q are false, yet the implication holds (true).

Example 2 (with a true consequent): "If Julius Caesar invades North America, I speak some Latin" also holds true. (I do speak a little Latin).

Example 3: "If you write a great post, then I('ll) give you $10." This conditional constitutes a promise. Let us see for which truth value combinations of P and Q, the promise (implication) holds (true).

Let: P := You write a great post, and

Let: Q := I give you $10.

• Case 1. P is true and Q is true.
• Case 2. P is true, and Q is false.
• Case 3. P is false, and Q is true.
• Case 4. P is false, and Q is false.

Suppose case 1 is the case: "You write a great post, and I give you $10." Does the implication hold or have I broken my promise? I have fulfilled my promise in response to your great post. The implication holds (true).

Suppose case 2 is the case: "You write a great post, but I do not give you $10". Does the implication hold or have I broken my promise? In fact, I have broken my promise, because I have not fulfilled the consequent of the conditional, given a true antecedent. Therefore, for this option, the implication does not hold (true), i.e., the implication outputs a truth value of false.

Suppose case 3 is the case: "You do not write a great post, and I give you $10." Have I broken my promise? My promise was predicated on your writing a great post, and it says nothing about what should happen if the antecedent were false. My promise (implication) merely states what should happen if the antecedent were true. If you do not write a good post, but I nonetheless give you $10, then, strictly speaking, I have not violated my promise. Therefore, the implication holds (true), with a false antecedent (P) and a true consequent (Q).

Suppose case 4 is the case: "You do not write a great post, and I do not give you $10". Here, even though both antecedent (P) and consequent (Q) are both false, the implication nonetheless holds (true). A falsehood can imply a falsehood, because from falsity anything follows.

P -> Q means "P materially implies Q" which is stated as the following material conditional (if-then) statement: "If P, then Q". The material conditional P -> Q implies that P is a sufficient condition for Q.

Given the material conditional (if-then) statement (P -> Q), the operator/connective (->) is called material implication, which sets up a material conditional "If P (is the case), then Q (follows)", which can equivalently stated as "Q if P", which in its turn is equivalent to stating "P only if Q", which implies that P is a sufficient condition for Q, which is represented as follows: P => Q.

Moreover, the sufficiency of P for Q is logically equivalent to the necessity of Q for P: [P => Q] <= logically equivalent to => [Q <= P].

Consider the following four options:

1. P => Q: P is sufficient for Q.
2. Q <= P: Q is necessary for P.
3. P <= Q: P is necessary for Q.
4. Q => P: Q is sufficient for P.

Henceforth, let the symbol (≡) denote logical equivalence, and (≡|≡) denote logical non-equivalence.

• Options (1) and (2) are logically equivalent: (1) ≡ (2)
• Options (3) and (4) are logically equivalent. (3) ≡ (4)

Consider the original conditional (P -> Q) with the (forward) material implication (->), connecting P to Q, so that P is set up as a sufficient condition for Q.

Let us refer to P -> Q as the "original" material conditional:

• A1. Original material conditional: (P -> Q), with a "forward implication", i.e., from P to Q.
• A2. Converse of Original: (Q -> P), with a "reverse implication", i.e., from Q to P. reverse of "forward".
• A3. Inverse of Original: (~P -> ~Q), with a "forward implication" and with both antecedent (P) and consequent (Q) negated.
• A4. Contrapositive of Original: (~Q -> ~P), with a "reverse implication" and with both antecedent (P) and consequent (Q) negated; therefore, contrapositive = the inverse of the original in reverse; that is, contrapositive (of original) = converse of inverse (of original).

Note that:

• The contrapositive (A4) is logically equivalent to the original (A1): A1 ≡ A4.
• The converse (A2) is logically equivalent to the inverse (A3): A2 ≡ A3.

The inverse (A3) can be derived by the contraposition of the converse (A2): by reversing the direction of the implication in (A2) and negating the P and Q. Note, that just as the contraposition of the original yields the contrapositive (of the original), where the contrapositive is logically equivalent to the original, so too does the contraposition of the converse yield the inverse (of the original), which is logically equivalent to the converse. Therefore, the contraposition of a conditional (C) validly yields another conditional (C*) that is logically equivalent to the conditional (C): C ≡ C*.

Answer 4

The truth value of a material conditional P -> Q represents a claim that Q is not less true that P. Given premises of P and a true material conditional P -> Q, a conclusion of Q is deductively valid, that is, it will not introduce a new error.