Question

In a valid standard form of categorical syllogism 'the middle term must be distributed in at least one premise'. Prove it.

Hint:

For a syllogism to be valid, the middle term must be distributed in at least one premise to ensure a logical connection between the premises and conclusion.

Answer 1

Proof: In a standard form of categorical syllogism, we have the following structure: \[ \text{Major Premise: } \text{All A are B} \] \[ \text{Minor Premise: } \text{All C are A} \] \[ \text{Conclusion: } \text{All C are B} \] In this syllogism, "A" is the middle term, which appears in both the major and minor premises but not in the conclusion. The rule that the middle term must be distributed in at least one of the premises can be proven as follows:
Step 1: The middle term must appear in both premises but not in the conclusion. To connect the major and minor terms logically, the middle term must be involved in such a way that it links the two premises.
Step 2: If the middle term were not distributed in at least one of the premises, then the premises would fail to establish a connection between the major and minor terms. This would result in a fallacy of ambiguity or an invalid syllogism.
Step 3: In logical terms, distribution refers to the extent to which a term refers to all members of the class it represents. If the middle term is not distributed, the argument cannot guarantee a valid conclusion, as there would be no universal link between the premises.
Conclusion: Thus, for a syllogism to be valid, the middle term must be distributed in at least one of the premises to establish a logical connection and ensure the conclusion follows from the premises.