Are You Playing the Fool?
Part 1
posted almost 14 years ago
Formal Logic
Categorical Syllogism
First systematically studied by Aristotle.
Consists of 2 premises and a conclusion.
Each premise has one of 4 forms:
- All A are B
- No A are B
- Some A are B
- Some A are not B
The A's and B's above are algebraic constructs that represent actual classes of things. To use these forms in a syllogism is to construct a categorical syllogism.
Valid Syllogisms
All A are B.
All C are A.
Therefore, All C are B.
Example:
All Mammals are warm-blooded.
All dogs are mammals.
Therefore, all dogs are warm-blooded.
This argument consists of a true major and minor (the first and second, respectively) premise, leading to a true conclusion. The argument is therefore valid. However, not all valid arguments are sound. [the author is not at present able to determine whether a categorical syllogism may be valid but unsound]
Invalid Syllogisms
No A are B.
Some C are not B.
Therefore, some C are not A.
Example:
No stars are planets.
Some satellites are not planets.
Therefore, some satellites are not stars.
This is an interesting situation where the premises are true, the conclusion is true, but the argument is invalid. All formal arguments, whether valid or invalid, can be refuted by analogy, by replacing the placeholder symbols with different statements that then lead to an obviously false conclusion:
Like so:
No precious stones are cheap things. (true)
Some diamonds are not cheap things. (true)
Therefore, some diamonds are not precious stones. (false)
Because the syllogism fails to meet the requirement that the conclusion must be true if it follows from true premises, the argument is invalid.
Unsound Categorical Syllogisms (another example)
All A are B.
Some B are C.
Therefore, some A are C.
Example:
All dogs are mammals.
Some mammals are cats.
Therefore, some dogs are cats.
This is a specific fallacy known as the Undistributed Middle.
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Rules for Syllogisms
- Syllogisms with two negative premises are invalid
- Syllogisms with two positive premises must have a positive conclusion.
- Syllogisms with a negative and a positive premise must have a negative conclusion.
- Syllogisms have exactly 3 terms.
- The term that occurs in both premises must be modified by the words all or none at least once.
- A term that is modified by all or none in the conclusion must be modified by all or none in one of the premises.
Please note that all of the above refers to categorical syllogisms only. A syllogism is simply an argument with a major and minor premise and a conclusion. The categorical syllogism must contain premises of the form noted in the 4 examples at the top.
Feel free to attempt to disprove the laws of logic in order to prove it to yourself. It can get rather interesting applying the rules to real arguments people make.
(edited almost 14 years ago)
