# What is meant by the 'truth' of a statement in a truth table? [duplicate]

The focus of propositional logic is said to be argument schemas that lead to valid conclusions, and not with the contents of the arguments themselves. This implies that an argument can consist of empirically false premises (i.e. factually false) but can still lead to a valid conclusion. For example consider the following valid argument:

All cups are green.
Socrates is a cup.
Therefore, Socrates is green.

Given then that we are not concerned with 'empirical truth' of statements in propositional logic, what do the 'truth values' assigned to statements in a truth table represent/indicate? What is meant be 'truth' in propositional logic?

Please could you ensure any answers are stated in simple terms as my understanding is minimal.

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• Your example is not propositional logic. It is an example of syllogistic logic, also known as term logic. Mathematical logic tends to view this as a restricted kind of predicate logic. Truth tables alone are not particularly useful or illuminating for analyzing that. – Henning Makholm Nov 4 '18 at 15:52
• @HenningMakholm Thanks and noted. That aside are you able to elucidate on what the 'truth' in truth tables refers, if not empirical truth? – seeker Nov 4 '18 at 15:55

Every row in a truth-table represents a type of 'world' or 'situation'. E.g. in our world, it is not true that all cups are green, but we can also think of worlds where all cups are green. Or, in terms of situations: there are situations where all cups are green (maybe someone notices that all the cups on the table are green and, restricing our 'domain' to just those cups, it would indeed be true that 'all cups are green'. But obviously, there are other scenarios or situations where not all cups are green.

This is exactly why this kind of statement is said to be a contingency: the truth-value of the statement is dependent on what the world/situation is like.

Also, in this sense, you are exactly right that logic does not care whether some statement is actually true (i.e. true in our world) or not: our world is just one of many possible worlds, and in logic we give no preference of one over the other. Logicians can contemplate worlds where pigs can fly just as easily as one where they don't.

A truth-table systematically exhausts all possible types of worlds relevant to the statements at hand. The truth-values in the 'reference colums' indicate what kind of world we are dealing with, e.g. if statements $$P$$ and $$Q$$ are set to True, then that tells us we are dealing with a kind of world where they are oth True. And, once we have set those values, we can work out the truth-conditions for the statement we are interested in ... and hopefully learn something from that (e.g. whether some argument is logically valid, or whether two statements are equivalent or ...)

• Is this a canonical viewpoint in formal logic? I haven't heard it before, but I love it. – alex811 Nov 4 '18 at 15:19
• How would you use a truth table to illuminate the syllogism in the question? – Henning Makholm Nov 4 '18 at 15:54
• @alex811: It looks quite mainstream to me. – Henning Makholm Nov 4 '18 at 15:55

First, as Henning Makholm mentions, the example you've given is not an example from propositional logic. Truth tables only apply to propositional logic and don't directly help for the example you've given.1 I'm going to restrict to talking about propositional logic.

A truth table is one particular but important example of a semantics for (classical) propositional logic. A semantics is a function that takes a formula of propositional logic, e.g. $$P\to Q$$, and maps it to some mathematical object satisfying some rules. For truth table semantics, this is a mapping into some fixed two element set, e.g. $$\{0,1\}$$. What the two element set is really doesn't matter. It could be $$\{\mathbb Q,\mathbb Z\}$$ or $$\{\mathsf{cat},\mathsf{dog}\}$$ or whatever. We arbitrarily call formulas that get mapped to one element, say $$\mathsf{dog}$$, "true" and formulas that get mapped to the other element, $$\mathsf{cat}$$ in this case, "false". That's what "truth" in a truth table means.

To define a semantics, we need to say where atomic propositions like $$P$$ and $$Q$$ get mapped. The mapping from atomic propositions to the two element set is typically called an assignment, and given the assignment the semantics is determined by the rules it must follow which I haven't stated. We usually consider a semantics relative to an assignment or a semantics as a function taking an assignment as well as a formula. We say that a formula is valid or a tautology if it is "true" (in the sense of the previous paragraph) for all assignments, or rather the semantics generated from those assignments. It's a contradiction if instead it is "false" for all assignments. If it is neither a tautology nor a contradiction, it is called contingent.

Absolutely nothing I've said above has any philosophical content, at least no more than defining polynomials and operations on polynomials. (Less arguably.) It is a completely separate question whether classical propositional logic and/or truth table semantics is a good model for reasoning. Coming to a conclusion on this will depend on what you mean by "reasoning" and likely "truth" which will probably be philosophical. Your philosophical views on truth don't change classical propositional logic; they just change whether classical propositional logic is a good model of them. If it is not, then you use a different model, in this case a different logic such as modal logic or constructive logic or paraconsistent logic or maybe some other logic that you build yourself.

1 Incidentally, this is an extremely common confusion. Someone will ask about truth tables and then immediately bring up examples from predicate logic. Usually this takes the form of a statement like: "How is '$$x$$ is even' either true or false? Doesn't it depend on $$x$$?" The answer is "$$x$$ is even" is a predicate, not a proposition, and truth tables don't apply. Logic is not about truth tables.

What do the 'truth values' assigned to statements in a truth table represent/indicate?

I find it helpful to think of each row of a truth table as corresponding to a theorem in propositional logic that can be proved using natural deduction.

Example

Consider this truth table:

Row 1: $$\space\space a \land b \land c \to (a\land (b \to c))$$

Row 2: $$\space\space a \land b \land \neg c \to \neg (a\land (b \to c))$$

Row 3: $$\space\space a \land \neg b \land c \to (a\land (b \to c))$$

$$\space\space\space\space \vdots$$

Row 8: $$\space\space \neg a \land \neg b \land \neg c \to \neg (a\land (b \to c))$$

The $$\neg$$ symbol corresponds to a falsehood (F) in the truth table.