# In categorical logic, why must any argument with two negative premises be invalid?

Why's 1 true please? I prefer an intuitive explanation, not one with Truth Tables or Formal Deduction.

Here is the best short answer I can offer. When there are two negative premises, one of two things generally happens: either the premises define two sets that are mutually exclusive, or the two premises define sets that are irrelevant to each other.

In either case, the content of the premises prevents any connection between them; so the reasoning fails.

This answer can be tested by preparing Venn diagrams for syllogisms where there are two negative premises: e.g., EEA, OEI, EOO, and so forth. The diagram will give you a picture of what the premises actually say, and illustrate why the conclusion must fail.

A syllogism needs three terms and three propositions (two premises and a conclusion). The predicate term of the conclusion, 'P', is the major term, the subject term, 'S', of the conclusion is the minor term, and the term common to both premises, 'M', the middle term. But let's put all that into an example :

All [human beings] (middle term) are [fallible persons] (minor term)

Some [philosophers] (major term) are [human beings] (middle term)

Some [fallible persons] (minor term) are philosophers (major term)

That's as formal as the answer gets !

Lacking a middle term there is no way in which a syllogism with two negative premises can link the minor and major terms together as subject and predicate in the conclusion - or indeed get any valid conclusion at all.

E.g.

No cats are vegetables

No vegetables are fruits

? - no conclusion is deducible

Or try :

No vegetable is a fruit

No fruit is a mouse

Conclusion ? - no conclusion is deducible

Someone might think that from 'No vegetable is a fruit' and 'No fruit is a mouse' we can deduce that 'No vegetable is a mouse'. But we can't : it doesn't follow, because no vegetable is a fruit and no fruit is a mouse, that being a non-fruit makes a mouse a non-vegetable.

The first para. of Mark Andrew's reply gives you the theoretical explanation in terms of sets. It is excellent. My answer might help - no more than that - as an illustration of the problem of two negative premises. I hope it counts as an intuitive explanation. At least it contains no truth tables.