If/Then Truth Table

One of the difficulties new students of symbolic logic must overcome is understanding the defining truth table for the conditional, the “if/then” logical operator. The defining truth table tells us what the truth value of the proposition is, given the truth value of its component parts. For the conditional, it looks like this:

p    q     p ⊃ q
T    T         T
T    F         F
F    T         T
F    F         T

One way to defend this is to look at real-life conditional propositions with known truth values, for which we also know the truth value of the component parts. We will take our examples from the Bible.

The first row of the defining truth table states that a conditional with a true antecedent and a true consequent is true. In Genesis 44:26, Judah says about Benjamin, “If our youngest brother is with us, then we will go down.” The antecedent “Our youngest brother is with us” is true, and the consequent, “We will go down” was also true. We also know this is a true statement; Judah is speaking truthfully. There is the first row: If true then true is true.

The second row says a conditional with a true antecedent and a false consequent is false. In Judges 16:7 Samson says to Delilah, “If they bind me with seven fresh bowstrings, not yet dried, then I shall become weak, and be like any other man.” In verse 8 they bind him with seven fresh bowstrings (i.e. the antecedent is true) , but in verse 9 he breaks them easily (the consequent is false). So in verse 10, Delilah recognizes that Samson had lied to her; that is, she knows the conditional was false. Thus, if true then false is false.

The third row says a conditional with a false antecedent and a true consequent should be considered true. In Genesis 24:41, the servant quotes Abraham: “You will be clear from this oath when you arrive among my family; for if they will not give her to you, then you will be released from my oath.” The antecedent of the conditional (they will not give her to you) is false (they do give her), but the consequent (you will be released from my oath) is true (given the statement in verse 41 prior to the conditional). And we know the conditional was true; Abraham was speaking the truth. So if false then true is true. (See also Genesis 34:17).

Finally, the fourth row says conditionals with false antecedents and false consequents are also true. Here is one example: “If I find in Sodom fifty righteous within the city, then I will spare all the place for their sakes” (Genesis 18:26). The conditional is true, since it is God speaking. But note that the antecedent (God finds in Sodom fifty righteous people) is false, and the consequent (God spares the city) is also false. (See also Genesis 42:38). Thus, if false then false is true.

We see that it is not difficult to find examples of if/then statements in the Bible that support the traditional defining truth table for the conditional logical operator.


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