Form and Validity
Validity of a categorical syllogism depends solely upon its logical form. The rules for deciding the validity of syllogism are:
- Rule 1: The middle term must be distributed at least once.
- Rule 2: If a term is distributed in the conclusion, then it must be distributed in a premise.
- Rule 3: Two negative premises are not allowed.
- Rule 4: A negative premise requires a negative conclusion, and a negative conclusion requires a negative premise
- Rule 5: If both premises are universal, the conclusion cannot be particular.
Fallacies arise when one of the rules are broken, for example, following fallacies arise when these rules are broken and the explanation given in the brackets
Undistributed middle (The middle term is what connects the major and the minor term. If the middle term is never distributed, then the major and minor terms might be related to different parts of the M class, thus giving no common ground to relate S and P.)
example-All sharks are fish
All dolphins are fish
Therefore All dolphins are sharks
Illicit major and illicit minor (When a term is distributed in the conclusion, let's say that P is distributed, then that term is saying something about every member of the P class. If that same term is NOT distributed in the major premise, then the major premise is saying something about only some members of the P class. Remember that the minor premise says nothing about the P class. Therefore, the conclusion contains information that is not contained in the premises, making the argument invalid.)
- All horses are animals
- Some dogs are not horses
- therefore, Some dogs are not animals
- and, All tigers are mammalians
- All mammalians are animals
- Therefore All animals are tigers
Exclusive premises (If the premises are both negative, then the relationship between S and P is denied. The conclusion cannot, therefore, say anything in a positive fashion. That information goes beyond what is contained in the premises.)
- No fish are mammals
- Some dogs are not fish
- Therefore Some dogs are not mammals
Drawing an affirmative conclusion from a negative premise, or drawing a negative conclusion from an affirmative premise (Two directions, here. Take a positive conclusion from one negative premise. The conclusion states that the S class is either wholly or partially contained in the P class. The only way that this can happen is if the S class is either partially or fully contained in the M class (remember, the middle term relates the two). And the M class fully contained in the P class. Negative statements cannot establish this relationship, so a valid conclusion cannot follow. Take a negative conclusion. It asserts that the S class is separated in whole or in part from the P class. If both premises are affirmative, no separation can be established, only connections. Thus, a negative conclusion cannot follow from positive premises. Note: These first four rules working together indicate that any syllogism with two particular premises is invalid.)
- All parrots are birds
- Some tigers are not parrots
- Therefore Some tigers are birds
Existential fallacy (On the Boolean model, Universal statements make no claims about existence while particular ones do. Thus, if the syllogism has universal premises, they necessarily say nothing about existence. Yet if the conclusion is particular, then it does say something about existence. In which case, the conclusion contains more information than the premises do, thereby making it invalid.)
- All mammals are animals
- All tigers are mammals
- Therefore Some tigers are animals