Conditionals

As with any mathematical model, it is important to understand the assumptions that are made when using propositional logic. These assumptions can be illustrated by examining the relationship between some formal arguments (i.e., arguments expressed using formulas) and arguments with the same logical form expressed in a natural language, such as English.

First of all, there may be a formal argument that is invalid, but there is a way to translate the formulas involved in the argument that leads to an argument expressed in English that is very secure or even valid. Often this happens because there are aspects of the logical form of English sentences that cannot be captured with the very simple formulas used in propositional logic. For example, formulas of propositional logic do not have any way to represent the contribution of words such as "all" and "some" (see the chapter on First Order Logic).

Second, there may be a formal argument that is valid, but there is a way to translate the formulas involved in the argument that leads to an argument expressed in English that is not very secure. We have already seen examples of this in the previous section when discussing Byrne's Suppression Task in which certain premises seem to suppress the application of Modus Ponens. Other examples arise because of a mismatch between the way certain connectives are interpreted in propositional logic and the way that these connectives are used in English.

The first example concerns conjunction. As we have already noted, for all formulas $X$ and $Y$, the following are tautologically equivalent.

$X\wedge Y\qquad$ and $\qquad Y\wedge X$.

However, consider the following two sentences:

S1. Ann had a drink and went to the police station.
S2. Ann went to the police station and had a drink.

If $P$ is "Ann had a drink" and $Q$ is "Ann went to the police station", then the translation of the above two sentences is:

F1. $P\wedge Q$
F2. $Q\wedge P$

Although F1 and F2 are tautologically equivalent, their translations---sentences S1 and S2---do not express the same statement. The issue is that the connective "and" in sentence S1 suggests that the first event occurred before the second event. That is, S1 can be paraphrased as: "Ann had a drink and then went to the police station". Of course, this describes a very different situation than Ann first going to the police station then having a drink. Another example of a pair of sentences that illustrates this phenomenon is:

1. Ann ran the marathon and collapsed.
2. Ann collapsed and ran the marathon.

The most striking examples of mismatches between the interpretation of a connective in propositional logic and the use of that connective in English involve conditionals. Since conditionals play a prominent role in many different areas, such as in mathematical, practical and causal reasoning, it should not be surprising that the material conditional is not the best way to represent certain conditional statements. For instance, in propositional logic, each of the following conditional statements would be represented by the formula $P\rightarrow Q$ (allowing $P$ and $Q$ to represent different statements in each of the 5 cases).

1. If $x>5$, then $x+1>6$.
2. If that object is a square, then that object is a rectangle.
3. If you strike the match, then it will light.
4. If J.K. Rowling didn't write Harry Potter, then someone else did.
5. If J.K. Rowling didn't write Harry Potter, then someone else would have.

To illustrate the issues that can arise, consider the last two statements 4 and 5. Since J.K. Rowling did write Harry Potter, the antecedent of both conditional statements 4 and 5 is false (i.e., it is false that J.K. Rowling didn't write Harry Potter). According to the truth table for the conditional, this means that both conditional statements 4 and 5 are true. However, while statement 4 is clearly true (after all, someone must have written the Harry Potter books), it is not at all clear that statement 5 is true (indeed most people would say that 5 is certainly false).

Taking advantage of the mismatch between the way that the conditional is used in English and how the conditional is interpreted in propositional logic, we can find examples of formal arguments and arguments expressed in English with the following three properties:

1. the argument expressed in English has the same logical form as the formal argument;
2. the formal argument is valid; and
3. there is something clearly deficient about the argument expressed in English.

The remainder of this sections briefly presents the most prominent examples:

Valid argument: $X\models\neg X\rightarrow Y$
Deficient argument expressed in English: College Park is in Maryland. So, if College Park is not in Maryland, then Obama is a Republican.

Valid argument: $X\rightarrow Y\models\neg Y\rightarrow\neg X$
Deficient argument expressed in English: If Gödel had lived past 1978, he would not be alive today. So, if Gödel was alive today, then he would not have lived past 1978.

Valid argument: $X\rightarrow Y\models(Z\wedge X)\rightarrow Y$
Deficient argument expressed in English: If this match is struck, then it will light. So, if this match is soaked overnight in water and struck, then it will light.

Valid argument: $Y\rightarrow Z, X\rightarrow Y \models X \rightarrow Z$
Deficient argument expressed in English: If I quit my job, I won't be able to afford my apartment. But if I win 10 million dollars, I will quit my job. So, if I win 10 million dollars, then I won't be able to afford my apartment.

While it is interesting to think through the above examples and to find additional mismatches between valid/invalid inferences in propositional logic and certain common sense reasoning patterns, we should not conclude that there is something wrong with the definition of the truth table for the conditional. Indeed, one can show that, assuming that the interpretation of the connectives is truth-functional (i.e., the connectives are interpreted using truth tables), the definition of the truth table for the conditional is the only one that makes sense (a similar observation applies to all the connectives).