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Truth-functional logic was discovered by the ancient Stoics. They realized that
lurking just under the surface of language is a rich logical architecture.
It begins with declarative sentences. Only declarative sentences can be true
or false. "Shut the door!" can't be true or false— it's just a command. "What time is
it?" can't be true or false— it's just a question.
So declarative sentences are special. They put forward a claim and that claim
has a truth value. That's just a fancy way of saying that every claim is either
true or false. For instance, "The Manatee is the world's
largest (aquatic) herbivore," has a truth value of true but, "The manatee is the world's
largest Gangster," has a truth value of false. By the way get used to these T's
and F's... we're gonna use them a lot. The sentences I just gave are simple: they
cannot be broken down into a smaller unit of language and still have a truth
value. "The manatee is the world's largest..." can't be true or false because it
doesn't even make sense. Same goes for "... is the world's largest
herbivore." Not true, not false, because not really a claim. In contrast, here is a
compound sentence: it is made up of two simple sentences: "The manatee is the
world's largest (aquatic) herbivore," and "The manatee can weigh up to 1,300 pounds."
Each of these simple statements has a truth-value, and in fact they're both
true. One quick clarification: in normal
conversation we would probably just say, "The manatee is the world's largest (aquatic) herbivore
and can weigh up to 1,300 pounds." Now you might think this doesn't count as a
simple sentence because it's not a complete claim. But that's not quite
right. It is here, just broken up. "The manatee"... "can weigh up to 1,300 pounds." So
there really are two claims here in a single sentence. Now here is what
the Stoics discovered: the truth-value of the compound sentence is a function of
the truth-values of the simple sentences within it. Put plainly, if you know
whether the parts of the sentence are true you can figure out whether the
whole sentence is true. It's easy to get caught up on the word "function" but let's
not overcomplicate it. A function is just a rule of inputs and outputs. Like this:
you're familiar with the function of addition. Throw in a two, throw in another two—
those are the inputs—and the addition function will spit out a four. That's the
output. Multiplication is a function, too. If we toss in, oh, say 1000, and a 7 the
multiplication function will produce an output of... 7,000. That's right. That is a
function. A rule that generates outputs based on the inputs. Now here's the
genius of the Stoics. They realized that some words have logical clout. For
instance, the word "and". When two simple sentences are connected by the word "and",
then when the first simple sentence is true and the second simple sentence is
true, then the sentence as a whole is true. That's why, "The Manatee is the
world's largest (aquatic) herbivore," and "The Manatee can weigh up to 1,300 pounds," has
the truth-value of true. Because its component sentences are each true. It's a
function. But let's say we have an "and" compound sentence like this one where
one of the component sentences is true and one is false. In that case, the truth
function for "and" will spit out false because the truth function for "and"
requires that both inputs be true. If they're not, then the sentence as a whole
will be false. So even though it's true that manatees
are endangered, the sentence as a whole is false. We can summarize the truth
function for "and" in a truth-table. But first,
just a bit of vocabulary. An "and" sentence is called a "conjunction" and
each of its parts are called a "conjunct". Let's write an H for "herbivore". That
stands for "The manatee is the world's largest (aquatic) herbivore"— the left conjunct. And
let's write in W for "weigh", as in, "The manatee can weigh up to 1,300 pounds"—that's the
right conjunct. Now each of these conjuncts have a truth-value of true. So
according to the conjunction function, the sentence as a whole is true. But now
let E stand for, "Manatees are endangered" and P stands for, "Manatees are
polka-dotted." Well, one is true but the other is false. With these inputs the
conjunction function tells us that the sentence as a whole is false. Now we
reach a crucial point—and it's this: the truth-function for "and" has nothing to do
with manatees. The Stoics didn't even know about manatees! The conjunction function
works for any "and" sentence, no matter the content. So let's put a P here for the
left conjunct, and it doesn't represent anything in particular. It's just a
stand-in for whatever claim makes up this part of the conjunction. And the
same goes for Q here. It's just the right-hand conjunct whatever the exact
claim may be. Now we can see the full conjunction function. We already said
that a conjunction is true when each of its conjuncts are true and false when
one of them is false.
Now we just add that a conjunction is also false when both conjuncts are false.
This table represents the truth-function for conjunction. The only time a
conjunction is true is when both conjuncts are true. In any other case the
conjunction is false. Conjunction is only one of many truth-functions. We're just
getting started here! But now you have a pretty good sense of
what a truth-function is. And that's important because it's the fundamental
idea behind an entire system of logic.
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