Daily Archives: January 13, 2018

Must we do every unit of Intermediate Logic?

A common question for new parents, teachers, or tutors going into Intermediate Logic:

“Intermediate Logic is a challenging course, especially trying to complete it all in one semester. Is each unit equally important, or can I skip something if I can’t fit it all in?”

The short answer is “You don’t have to do it all.” Unit 1 on Truth Tables is foundational to propositional logic, as is Unit 2 on Formal Proofs. Both of these are essential and must be completed by every student. Unit 3 teaches the Truth Tree method. A truth tree is another tool that does the same job as a truth table: determining consistency, equivalence, validity, etc. Some people like truth trees more than truth tables, since they are more visual. But Unit 3 could be considered an optional unit. Unit 4 covers Applying the Tools to Arguments. This is where the rubber meets the road for propositional logic, showing how to apply what has been learned up to this point to real-life reasoning. Consequently, Unit 4 should be completed by every student. Note that if you skip Unit 3, one question in Unit 4 will have to be skipped (namely, Exercise 28c #1). Unit 5 on Digital Logic – the logic of electronic devices – is entirely optional. Like Unit 4, this unit covers a real-life application of the tools of propositional logic, but one that is more scientific (though ubiquitous in this age of computers and smart phones). Though optional, many students find that they really enjoy digital logic.

It is common for teachers to skip either truth trees or digital logic. In fact, only the best classes successfully complete both Unit 3 and Unit 5. The Teacher Edition of the Intermediate Logic text includes two different schedules, one for completing every unit, and another for skipping Unit 5.

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.