How Truth-Functional Connectives Work (pages 154–155)
I would like you to have memorized the truth tables for conjunction, inclusive disjunction, negation, and conditional. See notes on Lecture 5 for these truth tables.
In this section, we’ll explore the concept of a truth-functional connective. In the next section, we’ll give you a taste of how to test for validity using all that we’ve learned in Lesson 4 as background. Lesson 6 gets into more detail and considers application to real-life examples, as, up until now, we’ve been dealing with simple sentences and silly examples. We’re using these simples and sillies because the real world is messy (and hence harder to symbolize and test for validity). So we’re starting slow and easy (i.e., simple and silly). The authors make two points about connectives (page 155): first, the connectives are used to construct new propositions from old ones; second, the truth value of the new proposition is determined by the truth value of the original propositions plus the rules of connection. That is why we call the connectives truth-functional connectives. To see how this works, page 155 takes you through a whole bunch of examples of how to determine the truth or falsity of new propositions that are built up from old propositions using the connectives we’ve already learned.
This section is easy once you realize that the authors are assigning the truth value for each variable (which represents any proposition), and then all you are asked to do is determine the truth value of the entire proposition. If you don’t understand by now that propositions are simply sentences with true or false values (something we discussed in Lesson 3), then you need to review! So, for example, if your eye scans page 155 and detects a list of a bunch of symbols “A & B True,” etc., and you
are confused about how the authors could ever determine that these propositions are true or false, then read the paragraph before, where they already assign true to propositions A and B and false to G and H.
Go through the example on page 155.