I'll start this discussion with a digression. That's something I'm prone to do, so bear with me. Almost a month ago, a friend of a friend asked me if I could explain "what are truth tables?" This explanation is supposed to be suitable for someone with a minimal background in math or science, perhaps even suitable for a poet. Such an explanation will lack the rigor of a masterful exposition ( such as Rosser ) but I hope it is a little more accessible. I was dismayed to find that the math/logic section of my bookshelf consists of books that take the importance of truth tables for granted, so I came up with this page.
Truth tables are a way of determining, from syntax alone, the truth value of a sentence.
That's a starting point, but it's rather like getting back the answer "because Rayleigh scattering goes as the higher harmonics of the frequency" to the question "why is the sky blue?"
So I'll start to unpack what I've babbled above.
Logic is many things to many different people. There are some interesting conclusions drawn in Computability and Logic that I'm just beginning to appreciate eleven years after I first made my acquaintance with the book. But for many people, at least at first, logic is something like the search for "truth" with a capital "T". Many children are fascinated by the axiomatic approach to Euclid's geometry in school. . . or at least they would be if it were taught properly! You may have vague memories of an argument that goes:
Let "A" and "B" be statements. Suppose A implies B. Suppose you can prove A. Then you've proven B! Oh, I can see you're really underwhelmed. But this little bit of reasoning (called "modus ponens" for those of you who care) is always true. That is to say, whenever you know that A implies B, and you know A to be true, you know B to be true. This is a powerful thing, something not to be scoffed at. If you're ready for a little stretch, consider this: If A implies B, and you know that B is false, then you know that A is false! You know this because a true statement can never imply a false statement. And you know that from the truth table of "implies".
Everything I said above used language, and language is a slippery thing.
But let's use language to illustrate a little further with an example.
Let A be the statement " The feds are watching ."
Let B be the statement "Our freedoms are illusory."
A implies B reads " 'The feds are watching' implies 'Our freedoms are illusory'."
If you grant "A implies B", and you follow the link behind "A", then you must conclude "B". ;-)
What does all of this have to do with truth tables? One of the
biggest reasons for investigating truth tables is for finding statements
that are universally valid. That is to say, for finding statements
that, REGARDLESS OF THE TRUTH OF THE CONSTITUENTS OF THE STATEMENT, the
statement as a whole is always true. Our rule of modus ponens is
a consequence of the fact that the compound statement
((A implies B)& A) implies B
is universally valid, i.e., the statement is always true regardless of the truth values of the statements "A" and "B". This probably isn't obvious, and we'll write out the truth table (that is, the listing of all possible logical values of the constituents of the compound statement) later (but only if you "petition the Fred with email"). I've started with the rule of modus ponens to convince you that truth tables can lead to something useful (a method of argument, or a method of proof). Let's begin with the truth table of something a little simpler.
Another use of truth tables is in spotting compound statements that are always FALSE. Can you think of one? The simplist is the compound statement "P&~P" (where & means boolean "AND" and "~" means "NOT"). Let's write out this truth table. First we need the truth table for "&", which is, for any statements "A" and "B":
A B A&B
T T T
T F F
F T F
F F F
That is to say, in words, the statement "A&B" is true if and only if both "A" and "B" are true.
Let's write out the truth table for "P&~P", keeping in mind that if "P" is true, then "~P" is false (by definition!):
P ~P P&~P
T F F
F T F
There you have it, a compound formula involving only "P" and "~P" that is ALWAYS FALSE. If "P" is true, "P&~P" is false. If "P" is false, "P&~P" is false. So the truth table shows us that the statement "P&~P" is always false. That is to say, whatever the heck "P" stands for, the compound statement "P&~P" is always false!
Trust me when I say the truth table for modus ponens would take a little more room.
What do truth tables mean to a poet? I think it's truly awesome that complicated statements can be TRUE, provably TRUE, simply due to their form (their syntax). True in every world. Someone ought to be able to write a poem that catches that sense of awe. It's also pretty cool that some statements are always FALSE, provably FALSE.
An insightful poet might realize how many assumptions are smuggled into our discussion of truth tables. The most obvious: that a statement is "TRUE" or "FALSE". By restricting ourselves to statements that take on truth values, we eliminate almost everything that is interesting about our language. A man who only considered logic would live in a very sterile world indeed.
That's all I've got to say about truth tables for now.
If you have comments or suggestions, email me at
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