Welcome back to Math Kos! Today's entry is the beginning of a series on logic, the winner of the poll in the previous installment.
Logic is a wonderful interdisciplinary field, and I hope those of you with some philosophy background will touch on some elements with which I may not be familiar. Applications of logic range from computer science and mathematics to philosophy and rhetoric. Just as mathematics and computers are tools for science, logic is largely a tool for mathematics, although there are certainly elements of it which are subjects of current research as fields in their own right. On the whole, however, because of its nature as a foundation on which the rest of the field builds, it has few elements which are clearly recognizable as "mathematical" and is therefore generally quite approachable for people with a liberal arts background.
Logic Part 1: Definitions
While in most realms of life we are accustomed to dealing with ambiguity, applications of formal logic demand unambiguous terminology. In order to kill the ambiguity, we start by formally defining certain terms.
Statement: A statement is a symbolic expression, in words or otherwise, that is either true or false. It must be sufficiently specific for the purposes for which it is being used; terms in it must be clearly defined so that they can be understood unambiguously by the audience. For purposes of distilling an argument down into its logical form, statements are often represented by single letters or single words. In the definitions below, when I use a single letter, it can be interpreted to mean "some generic statement, regardless of content."
Barack Obama is President of the United States.
There is a unicorn in my bathtub.
It might rain. (When? Where? How do you quantify "might"?)
Give me all your kittens. (This is an order, not a statement of fact.)
The Boolean operators are ways of stringing statements together to make bigger statements. They're named after George Boole, who invented Boolean algebra (the algebra of logic), which was later applied to electronic circuits because of its convenient natural binariness.
Logical AND: The logical AND has a very similar meaning to the word "and" in English. In general, you can interpret AND as "and." The equivalency usually works the other way too, although there are some rather contrived examples of ways that the English "and" can be used to really mean "or." There are several other English words that also usually mean AND (albeit with various other layers of meaning that must be stripped off in order to distill a sentence into its logical elements): "but," "however," "albeit," and others.
a AND b is true if a is true and b is true. If either is false, a AND b is false.
Logical OR: The logical OR is the inclusive or (and/or) and should usually be translated in a context-sensitive way as either "or" or "and/or.". Again, there are ways in English that "or" can mean AND, and there are a ton of situations (such as in the last sentence) where "or" means the exclusive or, so one should be careful with the meaning of a sentence when translating it into predicate logic.
a OR b is true if a is true, b is true, or a and b are both true. a OR b is only false if both a and b are false.
Logical NOT: NOT means...well, "not." It negates whatever comes after it.
NOT a is true if a is false and false if a is true.
Derived Operators: XOR (exclusive or), NAND (not AND), NOR (not OR/neither nor). These can be derived from the basic three described above.
a XOR b is true if a is true or if b is true but false if both are true or if both are false.
a NAND b is true if neither a nor b is true or if a is true or if b is true but false if both a and b are true.
a NOR b is true if neither a nor b is true, but false if either a or b is true or if both are true.
There are all number of annoying notations for the Boolean operators. Programmers have a couple, logicians have some, set theorists have their own (with different names), and there's even a really horrible graphical representation of them for circuit design. I'll spare you. If you want to learn more about Boolean algebra, I'll direct you to the Wiki intro page.
Conditionals have a subtly different meaning in logic than they do in ordinary English. In English, if-then constructions generally imply some sort of causality or connection between the two events; in logic, this is not necessary. All of the conditionals are logically equivalent to (can be translated to and from with no loss of information) expressions with Boolean operators; however, the Boolean expressions can be quite unwieldy and are somewhat foreign to the human way of reasoning, so unless we're actually building hardware circuits or programming in machine language, we prefer to use conditionals.
If-Then: The basic conditional, if a then b means if a is true then b is also true. It says nothing whatsoever about what happens if a is false; b could be false or it could still be true. If a then b is only false if a is true and b is false. If both are true or if a is false, then if a then b is true. Also remember that no causality is implied.
Example true if-then statements:
If Barack Obama is President, then Canada is north of the United States. (both true)
If Republicans controlled Congress, then the infamous FISA bill passed. (if false then true)
If Obama is a secret Muslim, then John McCain is President. (both false)
Example false if-then statement:
If Barack Obama is President, then racism is dead. (if true then false)
Only if: This one takes some thought to figure out. It's counterintuitive, but a only if b actually means if a then b. The difficulty comes in because in the translation, the causality seems to be reversed - but remember, there is no causality.
True only if statements:
Americans will only have full civil rights if gay and lesbian couples may legally marry. (means: If Americans have full civil rights, then gays and lesbians can marry. Does not mean that if gays and lesbians can marry, Americans have full civil rights; one can easily imagine a future in which same-sex marriage is allowed but transgendered persons are still denied their rights.)
Hillary Clinton will be President in 2009 only if Americans are willing to vote for a candidate who is not a white man.
False only if statement:
Barack Obama is only the President if he had his hand on the Bible when he was sworn in the second time.
If and only if: The biconditional, a if and only if b, means "if a then b AND if b then a." It is true if both a and b are true or if both are false, but false if one is true and the other false. "If and only if" is occasionally abbreviated "iff," which is one of the few conventions of notation in this field that I actually like, but I will refrain from using it here as it can be read as a typo.
True if and only if statements:
Barack Obama is President if and only if the Earth orbits the Sun.
John McCain is President if and only if the Sun orbits the Earth.
Arguments: An argument in logic is a series of statements that are asserted to be true (the premises) followed by a final statement asserted to follow from the premises (the conclusion). An argument is valid if and only if the conclusion does in fact follow from the premises. An argument can be logically valid whether or not anything in it is actually true; the important thing is the structure of the argument.
Example valid argument:
There can only be a unicorn in my bathtub if unicorns exist.
Unicorns do not exist.
Therefore, there is not a unicorn in my bathtub.
Example invalid argument:
If there is a unicorn in my bathtub, then unicorns exist.
There is not a unicorn in my bathtub.
Therefore, unicorns do not exist.
(note: this is the fallacy of the converse. We'll talk about both converses and fallacies thereof in the next installment.)
I'm sorry this part was a little dry; unfortunately, all the definitions are necessary so that when we start talking about more complicated arguments we can be certain that we're all speaking the same language. In Logic Part 2, I hope to discuss the rules for constructing and evaluating formal logical arguments, which should be much more interesting. I'll also try to touch on informal logic (the logic of language) and some of the fallacies unique to it. Thanks for reading!