The type of deductive logic that we’ve been discussing so far in this Logic 101 series is called truth-functional, or symbolic logic. As I said in the post on the logical symbols, a few of these symbols are called logical connectives, and more precisely, truth-functional connectives.
But what is “truth-functional” logic, and what use is it for us? This type of logic is called truth-functional because it is concerned solely with the truth-value of propositions (expressed as variables like ‘P’ and ‘Q’) and their truth-functional connectives. Where this all comes together and makes sense is with truth tables, which I will discuss in this post.
A truth table is a just that: a table that shows how propositions joined by truth-functional connectives turn out to be true or false. I know this is a bit abstract right now, but I’ll be giving plenty of examples in this post to show exactly what this means.
Truth Tables for the Logical Connectives
To start at the most basic level, let’s take a look at how truth-functional logic defines each of the logical connectives based on its truth table. For each one of these, I will give the truth table, a brief explanation that describes what you’re looking at, and an example.
Explanation. Suppose we have a statement that is either true or false (call it ‘P’). If P itself is true, then ~P is false. The truth value of ~P is the opposite of the truth value of P itself. So, if P itself is false, then ~P turns out to be true.
Example. Suppose that I claim that I am currently at the library (P). The opposite of this claim would be that I’m not at the library (~P). If I turn out to be at the library when I say I am, then P is true and ~P is false. But if I’m not at the library when I say I am (maybe I’m lying or have amnesia), then ~P is true.
|P||Q||P & Q|
Explanation. The conjunction of two claims is true if and only if both claims are both true.
Example. Suppose I claim that I am currently at the library (P) and that I am studying (Q). I could say to you, “I am currently at the library and I am studying.” This is true if and only if it is the case that I am (1) actually at the library (rather than somewhere else) and (2) I am actually studying (rather than goofing off).
|P||Q||P v Q|
Explanation. The disjunction of two claims is true when either (1) both claims are true or (2) only one of the claims is true. In truth-functional logic, then, an “or” (disjunction) statement is an inclusive “or” statement. What this table is NOT saying is that either P is true or Q is true, but not both. That would be an exclusive “or” (disjunction) statement.
Example. Consider the claims from the previous example again. It is true that I am currently at the library (P) or I am studying (Q) if either both statements are true or only one of them is. I could be studying, but be in my car outside the library, and P v Q would still be true (because Q is). Or I could be in the library and browsing the web instead of studying, and P v Q would still be true (because P is). If I’m really on top of things, though, P v Q would be true because I’m actually at the library and I’m actually studying.
|P||Q||P → Q|
Explanation. A conditional (“if…then”) statement is defined as true in truth-functional logic in every case except where the antecedent is true and the consequent is false. This means that we can show that a conditional claim is false if and only if P is true and Q is false. This is incredibly important to know!
I have to take a bit of a detour here and make myself clear about something, just to make sure I’m not misunderstood. My entire purpose for discussing symbolic logic on this website is to give readers the tools they need to evaluate arguments. I do not think that symbolic logic is adequate, in itself, to capture all of the arguments we might give in plain language.
This is because, as I think anybody should be aware, the conditional is where truth-functional logic starts to get weird. Once you read this post, you have enough knowledge to understand the Appendix to Socratic Logic that Trent Dougherty wrote on this very idea. The fact is this: there are issues with the truth-functional interpretation of the conditional, because it turns out that any two true or false statements imply each other.
As I’ve already said, the main takeaway from learning about the truth-functional understanding of the conditional is that objecting to a conditional statement requires us to show that the antecedent can be true while the consequent is false. It should be obvious that I don’t think truth-functional logic is the only game in town because I will devote many more posts to explaining Aristotelian logic. Symbolic is only one tool that the careful thinker needs when evaluating logic.
Example. All we care about is the first two lines of this truth table for this example. Suppose I say “If I am currently in the library (P), then I am studying (Q).” This conditional statement is true if it’s true that I’m currently in the library and that I am studying. But if it’s the case that I am in the library but I am not studying, then the conditional statement is false. That’s because P is true but Q is false.
|P||Q||P ↔ Q|
Explanation. Remember that a biconditional statement is logically equivalent to this statement: (P→Q) & (Q→P). (I’ll prove this in the next post.) If it’s the case that a conditional statement is false when the antecedent is true and the consequent is false, then the second line can’t come out true. But since Q→P is part of the conjunction that makes up P↔Q, it can’t be the case that Q can be true as an antecedent and for P as the consequent to be false. So, the only case where P ↔ Q comes out true is if P and Q are either (1) both true or (2) both false.
Example. In a biconditional statement, both P and Q are necessary and sufficient for the other’s truth value. Suppose I say that “I am currently in the library (P) if and only if I am studying (Q).” This is a biconditional statement. If it’s the case that I am currently in the library, then it must also be the case that I am studying, since P is sufficient for Q. But if I’m not in the library, it must also be the case that I’m not studying, since P is necessary for Q. The same goes for whether or not I’m studying: depending on if that’s true, it must also be true that I am or am not in the library.
Tautologies and Contradictions
These five truth-functional connectives and their tables provide the entire basis for symbolic logic. But there’s a couple of additional concepts you should be aware of.
The first is what a tautology is. Simply put, a tautology is a statement that is always true in virtue of its propositions and truth-functional connectives. Here’s a truth table of a tautology:
|P||~P||P v ~P|
It should now make sense to you why P v ~P is always true, given what you now know about the disjunction and its truth table. Again, a disjunctive statement is true if either proposition on each side of it (its “disjuncts”) is true, or if both are. In each line of this particular truth table, it’s always the case that either P or ~P is true, so P v ~P is always true. It is a tautology.
A contradiction is the opposite of a tautology: it is a statement that is always false in virtue of its propositions and truth-functional connectives. Here’s an example:
|P||~P||P & ~P|
Given what you now know about the conjunction and its truth table, it should also make sense why P & ~P is always false. There simply isn’t a case where both P and ~P are both true. That’s why it’s a contradiction.
I recognize that this post has possibly been strange and foreign to many readers. It’s obviously foreign because people don’t normally think “how can I express that statement in a truth table?” At least, unless they are logicians, philosophers, or just plain annoying. And this can all seem strange because, like the proverbial fourth grader learning math, it seems disconnected from anything you’ll ever do in your life.
So why talk about this at all, especially on a website devoted to discussing theology? The truth is, with this post out of the way, we can now talk about a few ways that truth tables are actually useful when we go through the R.E.A.D. Method. That’s what I’ll discuss in the next post.