# How should an argument containing an exceptive proposition be tested?

IX. Exceptive Propositions in 7.3 Translating Categorical Propositions into Standard Form in Copi's Introduction to Logic says:

Because exceptive propositions are not categorical propositions but conjunctions, arguments containing them are not syllogistic arguments as we are using that term. But they may nevertheless be susceptible to syllogistic analysis and appraisal. How an argument containing an exceptive proposition should be tested depends on the exceptive proposition's position in the argument.

If it is a premise, then the argument may have to be given two separate tests. For example, consider the argument:

Everyone who saw the game was at the dance.
Not quite all the students were at the dance.
So some students didn't see the game.

Its first premise and its conclusion are categorical propositions, which are easily translated into standard form. Its second premise, however, is an exceptive proposition, not simple but compound. To discover whether its premises imply its conclusion, first test the syllogism composed of the first premise of the given argument, the first half of its second premise, and its conclusion. In standard form, we have

All persons who saw the game are persons who were at the dance.
Some students are persons who were at the dance.
Therefore some students are not persons who saw the game.

The standard-form categorical syllogism is of form AIO-2 and commits the fallacy of the undistributed middle, violating Rule 2. However, the original argument is not yet proved to be invalid, because the syllogism just tested contains only part of the premises of the original argument. We now have to test the categorical syllogism composed of the first premise and the conclusion of the original argument together with the second half of the second premise. In standard form we then get a very different argument:

All persons who saw the game are persons who were at the dance.
Some students are not persons who were at the dance.
Therefore some students are not persons who saw the game.

This is a standard-form categorical syllogism in Baroko , AOO-2 , and it is easily shown to be valid. Hence the original argument is valid, because the conclusion is the same, and the premises of the original argument include the premises of this valid standard-form syllogism. Thus, to test the validity of an argument, one of whose premises is an exceptive proposition, may require testing two different standard-form categorical syllogisms.

Is the above testing method, when a premise is an exceptive proposition, based on the following inference rule?

If
P, Q1 |-> S,  or
P, Q2 |-> S,
then
P, Q1 & Q2 |-> S

If the premises of an argument are both categorical propositions, and its conclusion is exceptive, then we know it to be invalid, for although the two categorical premises may imply one or the other half of the compound conclusion, they cannot imply them both.

In the second case above, why is it invalid? What does "although the two categorical premises may imply one or the other half of the compound conclusion, they cannot imply them both" mean? Isn't it valid, because it is based on the following inference rule?

If
P |-> Q1,  and
S |-> Q2,
then
P, S |-> Q1 & Q2

Finally, if an argument contains exceptive propositions as both premises and conclusion, all possible syllogisms constructible out of the original argument may have to be tested to determine its validity.

Could you elaborate what "all possible syllogisms constructible out of the original argument" are?

Why isn't it the same as the second case, where only the conclusion is an exceptive statement?

Thanks.