# What is the relation between the material conditional in logic and conditionals that we use every day?

The material conditional has a truth-value of T in every case except where the antecedent proposition is true and the consequent is false. However, this means that many conditionals are true (if only vacuously so), that we would never use on a day to day basis. Is there some analysis demarcating ordinary language conditionals as a subset of material conditionals that can shed any light on this issue, or are we, as of now, limited to saying that vacuously true statements are true but useless to everyday life?

• Some contemporary multi-value logics evaluate a conditional with a false antecedent to "null." The choice of true is somewhat arbitrary but given the law of the excluded middle, you have to pick either true or false. Jul 9 '14 at 22:36
• It is strange though that in ordinary speech we would say that a conditional such as "If Budapest is a neighborhood in London then Budapest is in England" is true, while one such as "If grass is blue then I am a donkey's mother" is false. In both cases the antecedent and consequent are both false but the first conditional is intuitively true while the other is false. Is this simply a case where our intuition is flawed, or is there some deeper disconnect between ordinary language and logic? Jul 10 '14 at 4:05
• I think it's more of a confusion about what logic is and does. Logic is not best understood as a translation for normal speech but rather as a formal system for thinking through problems that is more precise than normal speech. But the precision comes at a price -- bivalency, etc., have to be introduced in ways that lose information or force us to draw conclusions that are not yet definite. Jul 10 '14 at 4:17
• That definitely makes sense, and definitely explains why it is difficult to reconcile ordinary language with first order logic. With that said, how should we analyze ordinary language conditionals and other ordinary language "connectives" if we cannot use formal logic as a tool? Jul 10 '14 at 4:22
• When material, you can translate the if statements differently, say as biconditionals, or you can translate them as conjunctions. Jul 10 '14 at 4:55

I think that this explanation of Stephen Cole Kleene, Mathematical Logic (1967 - Dover reprint) [pag.10 - footnote 12] is a good short elucidation of the "formalization" of conditional in a truth-functional setting :

The ordinary usage certainly requires that "If A, then B" to be true when A and B are both true, and to be false when A is true but B is false. So only our choice for T in the third and fourth lines [of the truth-table entered for A and B, i.e. the lines F-T and F-F] can be questioned. But if we changed T to F in both these lines, we would simply get a synonym for ["and"]; in the third line only, for [i.e. the bi-conditional]. If we changed T to F in the fourth line only, we would loose the useful property of our implication that "If A, then B" and "If not B, then not A" are true under the same circumstances [...].

The truth-functional definition of propositional connectives is a "model" that in some cases "fit" quite well with our usage in natural language (negation, disjunction, conjunction) and not so well in other cases (conditional).

When we assert a sentence A we are expressing the fact that we "judge it" to be true.

Thus, asserting the conditional A → B means to "judge" it true.

When mathematicians (like Frege) introduced the truth-functional conncetive , they have in mind one characteristic property of the connective, viz., the rule of modus ponens. With this rule, we assert A → B and A; in this case, the first assertion "exclude" the case when A is true and B false, while the second assertion "exclude" the two cases where A is false.

Thus, we have only one possibility left : B true, and this is what we expected.

In our "ordinary" use of the language we seldom assert a conditional "if ..., then ___" when we know the antecedent to be false; but the "modelling" of mathematical logic fit quite well with the use in ordinary mathematics.

The very important "context" in mathematics is the following :

Σ⊨φ;

in this case we say that Σ entails φ. The condition validating the relation of "entailment" is that : every interpretation that satisfy (all the sentences in) Σ will also satisfy φ; or, equivalently, there is no interpretation such that all of Σ are true and φ is false.

This "context" is commonly used when we assert that some thorem (φ) follows from a set Σ of sentences, e.g.the axioms of a theory.

When Σ={σ}, from σ⊨φ we have that : ⊨σ→φ.

This result establish a strict connextion between the conditional (→) and the relation of entailments (⊨). The two are different relations, but the above link between them is so useful that we "accept" the "not perfect" fit of the conditional with our natural language habits.

• First thank you for the response. I understand some of the benefits of the particular truth table that the material conditional has, but I am more interested in what the relationship is between the material conditional and ordinary language conditionals. I think that what you are trying to get at is that the material conditional, though bearing a resemblance to ordinary language conditionals, was created for very different purposes than those for which the ordinary language conditional was created. But then if logic can't provide insight into ordinary language conditionals what can? Jul 10 '14 at 4:11
• @leibnewtz - there are a lot of discussions in Analitycal Philosophy regarding natural language; my point of view is : natural language is not "logically regimented". When in natural language we use "or", it is exclusive or inclusive ? Both: it depends on the context; for mathematical usages we need "precision"; thus we have "separated" vel and aut. Jul 10 '14 at 6:59
• What does this mean in that quote " in the third line only, for 'not' "? Apr 28 '17 at 23:20
• @barlop - teh usual truth-table for : T-T is T; T-F is F; F-T is T; F-F is T. May 1 '17 at 14:10
• @MauroALLEGRANZA so the third line is `F-T is T` and if we change `is T` to `is F` in the third line only, we get "not". Not What? May 1 '17 at 19:34

From everyday usage, we have:

1. If we assume A is true and, without making any other assumptions, we can infer that B must also be true, then we can infer that A implies B is true. There may be intervening premises that were discharged and deactivated. (The Conclusion Rule)

2. If we assume A is true and, without making any other assumptions, we can obtain a contradiction, then A must be false. Here, too, there may be intervening premises that were discharged and deactivated. (The Indirect Conclusion Rule)

Using these rules from everyday usage, we can prove that for any logical, true-or-false propositions A and B we have: A is true implies that A is false implies B is true. ( A => [~A => B] )

Proof:

1. Let A and B be logical, true-or-false propositions. (Law of Excluded Middle applies)

2. Premise: Suppose A is true.

3. Premise: Suppose A is false. (Assumptions need not be consistent.)

4. Premise: Suppose B is false.

5. Joining (2) and (3), we obtain the contradiction A is true and A is false.

6. Applying the Indirect Conclusion Rule, (4) must be false, i.e. B must be true.

7. Applying the Conclusion Rule for (2) and (6), A implies B must be true. (Premise on line 4 has been discharged and deactivated.)

8. Applying the Conclusion Rule once more for (1) and (6), we obtain, as required, that A is true implies that A is false implies B is true. (Premise on line 3 has been discharged and deactivated.) ( A => [~A => B] )

From a falsehood, all things follow.

Further details at my blog posting Material Implication: If Pigs Could Fly.

Dan

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