Logical Form:
Logicians talk about something called the logical form of a statement.  I've mentioned this idea briefly in my attempts to explain the notion of validity.  That is, we've said that valid arguments have a good logical form, regardless of whether the component statements in those arguments are actually true or false.  We need to investigate further this notion of logical form.

The basic unit of logical form is a simple declarative sentence.  You may recall what a declarative sentence is from grammar school.  It's not a question, not a command, but a statement.  "The cat is on the mat."  "The sky is blue."  "I like oranges." would be examples.  Things like "Where is the cat?" or "Close the door, please." would not be counted as declarative sentences, as they are questions and imperatives respectively.  Interjections like "Ouch!" are also left out of a basic analysis of logical form.  These simple declarative sentences are often represented by single letters when we sketch the logical form of a longer passage.  It is important when adopting letters to use the same letter for the same sentence everywhere it occurs and always to use different letters for different sentences.  For instance, we might represent "The cat is on the mat." with: C.  And, we might represent "The sky is blue." with: B.
I called the basic unit of logical form a simple declarative sentence, but these basic units can actually be quite complex.  By calling it simple, I mean that it does not contain any of the logical words or phrases that I'll introduce shortly.  So a "complex" sentence like "President Carter felt that ceding control of the Panama Canal to Panama during his presidency was the best course of action for the United States at that time." is going to actually count as a simple sentence.  To avoid this confusion we will call simple declarative sentences that do not contain any further logical structure atomic sentences.  This is, in part, because they work like atoms in building up larger sentences with more detailed logical forms, which we might call molecular sentences.  These logical words or phrases (for starters) that do not appear in atomic sentences are: not, and, but, or, neither…nor, if, only if, and unless.. Let's look at each.
We can say, "The cat is not on the mat."  If we adopted C: to stand for "The cat is on the mat."  then the logical form of our more complex molecular sentence is: not C.  We could also say "The cat is on the mat and he is in the way."  If we continue to use C as before, and adopt W for "The cat is in the way." then the logical form is: C and W.  Notice here that the original sentence makes use of a pronoun "he" to refer back to "the cat".  In explaining the meaning of W we should spell out what "he" was referring to by defining W with "the cat" in place of "he".  (Other examples are similar...)

Truth Tables:

A	B	|	A	or	B	|	not	B	|	A
		|				|			|
T	T	|		T		|	F		|	T
T	F	|		T		|	T		|	T
F	T	|		T		|	F		|	F
F	F	|		F		|	T		|	F

There is no row on which all the premises are true and the conclusion false, therefore the above argument is valid.

A	B	|	A	or	B	|	B	|	not	A
		|				|		|
T	T	|		T		|	T	|	F
T	F	|		T		|	F	|	F
F	T	|		T		|	T	|	T
F	F	|		F		|	F	|	T

The first row is a row on which all the premises are true, but the conclusion false, therefore the above argument is invalid.