Truth Values & Truth Tables

Truth Preservation

The lessons on arguments & truth in logic introduced the difference between truth and validity in evaluating an argument. As you may have noticed, neither the language of logic nor the ways of forming and analyzing an argument guarantee the truth of the statements (the premises and conclusion) within that argument. The toolkit of logic doesn't generate truth. Instead, it allows statements to remain true throughout the process of developing an argument. We might call it truth preserving rather than truth producing.

We can analyze the function of every one of the operators you learned in the logic as a language series within the context of truth-preservation. This is normally done using truth tables.

So far, speaking generally, I have been careful to separate content from form. Now let's move towards integrating these two aspects of statements by considering truth values. We'll be using just two basic values, true and false. To start with a simple analogy, this is like a switch with the value ON or OFF. The switch sends a signal to a light, which may also be ON or OFF. The switch is a variable in a statement, like a proposition P. The light is the value of the entire statement.

If I give you two statements P and Q, can you tell me what they mean? No, because they're variables and you don't know what they represent. So then, can you say nothing about their content? Think for a moment. In the lesson on arguments, we looked at an argument with the form P⊃Q, P, ∴Q. It is well-formed, meaning if we fill it with true statements, the conclusion will be entailed. It is possible to say something about P then - it can be true or false. The same goes for Q.

By looking at the operators and the form of the argument, we can pass that truth down to the conclusion. While we don't know the specific value of P or Q, we can say things that hold for every possible truth value in combination, and declare the resulting truth value of the statement as a whole. If P is true, then the statement 'P' is true. If P is true and Q is true, the statement 'P & Q' is true. If P is false and Q is false, 'P & Q' is false. See, you can say something about the truth of these statements!


Quiz

We're going to dive into specifics, but let's test to see if you get the gist of logical values. Assume P and Q stand for two different random propositions or sentences. I'm going to give you a value for P and a value for Q, and you will tell me if the statement that follows is true.

  1. P = true, Q = false. P and Q = ________
  2. P = false, Q = false. P and Q = ________
  3. P = true. not-P = ________
  4. P = false, Q = false. P if and only if Q = ________
  5. P = false, Q = true. P or Q = ________
  6. P = true, Q = false. If P then Q = ________
  7. P = false, Q = true. If P then Q = ________

Truth of Propositions & Operators

Let's say we have a proposition P. In our light analogy, P is a switch, and P can be ON or OFF.

The negation NOT turns the value of P around. If P is True, then ~P is False, and vice versa. If we add another NOT, the negation is negated, so ~~P is true when P is true and false when P is false.

The conjunction AND joins two propositions, so you have to pay attention to both of their truth values. P may be true when Q is true, false when Q is true, and so on. As we've seen before, the statement P•Q will only be true if both P and Q are true. In all other cases, it is false. The light is lit only when both switches are on!

Inclusive OR is true when either P, Q or both are true. The light goes on when at least one switch is flipped.

On the other hand, XOR (exclusive OR) expects that either one or the other statement is true, but is false when both statements are true. As a reminder, this is the kind of 'or' you use when you ask, "Do you want to leave today or tomorrow?", where you exclude the possibility today and tomorrow. The light turns on when either one of the switches is flipped.

The logical implication is frequently used and, for some learners, a bit tricky. 'If P is true, then Q is true' (meaning 'P implies Q') holds if P and Q are both true. 'P implies Q' is false if P is true and Q is false - in that case, P wouldn't imply Q, but ~Q! If P is false and Q is false, then P could very well imply Q, so the implication holds true. The last case, where P is false and Q is true, is an interesting one. Something besides P might imply Q, so the implication still holds.

'P implies Q' holds true unless P is true and Q is false. This is like a light that goes on when switch P is on. If the switch is on, then we expect the light to be on. But the light might also turn on from another switch, so the switch P being off doesn't make the statement 'if P's flipped on, then the light is on' untrue. Only one thing can make the statement 'if P's on, the light is on" false - turning P on and finding the light off!

Finally, the biconditional means that the two propositions P and Q have the same value. The statement 'P if and only if Q' is true when both P and Q are true or both P and Q are false. In any other case, the biconditional is false. This makes the two propositions logically equivalent, so they must return the same value.

Conclusion & Application

I encourage you to go back through these operators, filling in P and Q with propositions you take to be true or false, and spend some time thinking about the logical truth values of the resulting statements. This is a good exercise that will help you familiarize yourself with truth tables and clear up some ambiguities in your logic.

Please visit the Youtube channel or the lessons on nativlang.com for more logic & language resources. As always, thanks for learning with me.