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 | and | B |
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T | T | | | T | ||
T | F | | | F | ||
F | T | | | F | ||
F | F | | | F |
A | B | | | A | or | B |
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T | T | | | T | ||
T | F | | | T | ||
F | T | | | T | ||
F | F | | | F |
A | B | | | A | => | B |
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T | T | | | T | ||
T | F | | | F | ||
F | T | | | T | ||
F | F | | | T |
A | | | not | A |
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T | | | F | |
F | | | T |
A | B | | | A | or | B | | | not | B | | | A |
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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 |
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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.
V/I | S/U | |
1. Michael Jackson was assassinated or was killed in an accident. Michael Jackson was not killed in an accident. Therefore, Michael Jackson was assassinated. | ||
2. Princess Diana was assassinated or was killed in an accident. Princess Diana was not assassinated. Therefore, Princess Diana was killed in an accident. | ||
3. Princess Diana was assassinated or was killed in an accident. Princess Diana was killed in an accident. Therefore, Princess Diana was assassinated. | ||
4. Michael Jackson was not shot by a lunatic. Princess Diana was not shot by a lunatic. Therefore, John Lennon was shot by a lunatic. | ||
5. If Princess Diana was assassinated then the world mourned her death. Princess Diana was assassinated. Therefore, the world mourned her death. | ||
6. Princess Diana was shot by a lunatic or Michael Jackson was shot by a lunatic. John Lennon was shot by a lunatic. Therefore, Michael Jackson was shot by a lunatic. | ||
7. Princess Diana was killed in an accident or Michael Jackson was killed in an accident. Michael Jackson was killed in an accident. Therefore, Princess Diana was killed in an accident. |