# Determining the truth value of a statement

I am stuck with the following question,

Determine the truth value of each of the following statments(a statement is a sentence that evaluates to either true or false but you can not be indecisive).

If 2 is even then New York has a large population.

Now I don't get what does truth value means here. I would be thankful if someone could help me out.

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A simple Google search for "truth values if then" popped out this which should be of help: math.csusb.edu/notes/logic/lognot/node1.html –  JavaMan Sep 28 '11 at 21:04
Presumably you are meant to read this as $A$ implies $B$, where $A$ is "$2$ even" anf $B$ is "New York has a large population." But $A$ implies $B$ is true if $B$ is true or $A$ is false (or both). In this case, $B$ is true, so we don't even have to worry about whether $A$ is true. The statement is true, it has truth value "true." The possible truth values, as explained in the problem, are "true" and "false." Some people would use T for true and F for false, particularly in tables, to save space. –  André Nicolas Sep 28 '11 at 21:10

The truth value of a sentence is "true" or "false". A sentence of the form "If A then B" is true unless A is true and B is false. In this case A is "2 is even" and B is "New York has a large population." I would evaluate each of these as true, so the compound statement is true.

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"New York has a large population." Isn't this supposed to be indecisive as one can not surely say if it is true? –  Fahad Uddin Sep 28 '11 at 21:35
@Akito: translating natural language into math is often difficult and you have identified one problem it creates. I took "New York has a large population." as just as obviously true as "2 is even." Some logicians may disagree. –  Ross Millikan Sep 28 '11 at 21:41

If X Then Y is an implication. In other words, the truth of X implies the truth of Y. The "implies" operator is defined in exactly this manner. Google "implies operator truth table" to see the definition for every combination of values. Most importantly, think about why it's defined in this manner by substituting in place of X and Y statements that you know to be either true or false.

One easy way to summarise the definition, is that either X is false (in which case it doesn't matter what the second value is), or Y is true.

So applying this to your statement: 2 is indeed even (so now that X is true we only need to check that Y is true to conclude that the implication is indeed valid). NY does indeed have a large population, and so we conclude that the implication is valid, and the sentence as a whole is true!

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I would delete "only" in the last paragraph. Once you have noticed that 2 is even, you do have to check whether NY has a large population. If the sentence were "if 1 is even then NY has a large population", observing that 1 is not even would mean we have no need to check the population of NY. –  Ross Millikan Sep 28 '11 at 21:22
@RossMillikan, the "only" comes after the observation that X is true. I.e. Once we've found X to be true, we only need to check Y is true and then we can conclude the statement's truth value. Instead of deleting "only", I'll add this clarification. –  davin Sep 28 '11 at 21:24

When learning logic, it's important to keep the semantics separate from the syntax. This is one of the really big road blocks in getting a grasp on symbolic logic.

Semantics is what the sentence means. "2 is even." That means something different than "4 is even," even if they have similar structures (number is something).

Syntax is just the structure of the sentence. Think of it like the grammar, "Noun verb adjective". A lot of symbolic logic is seeing what can be done with just that grammar. In logic, we are looking at what is called truth-functional connectives. That means they only care about the truth value of statements that they work with, not what they actually mean.

Analogously, even if you don't know the actual words involved, you know that "Adjective Verb Noun" can't be a real sentence in English, because adjectives don't modify verbs.

So, back to your question. The truth value is simply whether a sentence is true or not. This has something to do with semantics, but generally you can look at a sentence and say whether it is true or not.

What trips people up about these questions is they are thinking of "If...then..." semantically. They say, "What does the fact that 2 is even have to do with New York? Nothing. So this is false."

Instead, you should ignore the actual meaning of the sentence and just look at the structure and truth-values. 2 is even, so that part is true. Write a big T over it. New York is a large city, that's true.. Write a big T over that, too. Look at the definition for "If...the...". It should say something like "A->B is true whenever "A and not B" is false." Basically, if A is true, then B better be true, too. Or it will be defined by the truth table. In that case, just look for the right row of the table and see what it tells you.

So, the statement "If 2 is even then New York has a large population" is true, not because there is any special connection between the number 2 and New York, but because of the way "If...then..." works.

Don't worry too much if this seems counter-intuitive. The conditionals we use in logic are probably different than the conditionals that we use in our day to day language. If that kind of thing sounds interesting to you, there is a good amount of research into how natural language conditionals work -> http://plato.stanford.edu/entries/conditionals/

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