Chapter 17:Missing Premises and

Conclusions

Enthymemes (p. 168)

• An enthymeme is an argument with an unstated premise or conclusion.

• There are systematic ways to determine whether, and if so what, conclusion follows from a set of premises.

• There are systematic ways to determine whether, and if so what, premise will yield a valid syllogism given a premise and the conclusion.

Rules and a Missing Conclusion (p. 169)

• Make sure the middle term is distributed exactly once. If not, the syllogism is invalid.

• Make sure you do not have two negative premises. If you do, the syllogism is invalid.

• Make sure you do not have two particular premises. If you do, the argument is invalid.

• If the argument is invalid stop.• If there is one particular premise, the

conclusion is particular.• If there is one negative premise, the

conclusion is negative.• Match the distribution of the major and minor

terms.• Make sure there are exactly three terms.

Missing Conclusion: Example (p. 169)

• What follows?All mammalsD are vertebratesU.

All aardvarksD are mammalsU.

So, ???

• The middle term is distributed once.• There are no negative or particular premises.• So the conclusion has to be universal and

affirmative.• Considerations of distribution show it has to be

“All aardvarks are vertebrates.”

Missing Conclusion: Example (p. 169)

• What follows?No aardvarksD are mastodonsD.Some mastodonsU are elephantine animalsU. So, ???

• The middle term is distributed once.• There is one negative premise and one

particular premise.• So the conclusion has to be a particular

negative.• Distribution shows it has to be “Some

elephantine animals are not mastodons.”

Missing Conclusion: Example (p. 169)

• What follows?Some aardvarksU are not mastodonsD.

No mastodonsD are spidersD.

So, ???

• The middle term is distributed twice. So, nothing follows: the argument is invalid.

• Also, there are two negative premises. So, nothing follows: the argument is invalid.

• Noting that it breaks either rule would be sufficient to show that it is invalid.

Rules and a Missing Premise (pp. 169-172)

• Make sure there are exactly three terms (Rule 1).

• Make sure you don’t have a particular premise and a universal conclusion (Rule 6).

• Make sure you don’t have a negative premise and an affirmative conclusion (Rule 5).

• Make sure the major or minor term in the conclusion has the same distribution in the premise (Rule 3 or 4).

• If any of these is violated, the argument is invalid: cite the rule.

• If none of the rules were violated by the given:– If the conclusion is particular, the premise must be

particular.– If the conclusion is negative, the premise must be

negative.– The distribution of the major or minor term in the

conclusion must be the same in the premise.

• Once you have found the premise, you should ask whether it is true.

Rules and a Missing Premise (pp. 169-172)

Missing Premise: Example (pp. 169-172)

• What is the missing premise?All mammalsD are vertebratesU.So, no spidersD are mammalsD.

• Rule 1 seems to be unviolated.• Neither rule 5 nor rule 6 is violated, that is, we have

neither a negative premise and an affirmative conclusion nor a particular premise and a universal conclusion.

• The major term, ‘mammals’, is distributed in both the premise and the conclusion.

• So, there is a premise that will yield a valid syllogism.

Missing Premise: Example (pp. 169-172)

• Rule 6: The conclusion is universal, so the premise must be universal.

• Rule 5: The conclusion is negative. The given premise is affirmative. So, the missing premise must be negative.

• So, the conclusion must be a universal negative proposition. It is either “No spiders are vertebrates” or “No vertebrates are spiders” — take your pick, they’re logically equivalent.

• Checking against Rule 2: The middle term, ‘vertebrates’, is distributed exactly once.

• The missing premise is true.

Missing Premise: Example (pp. 169-172)

• What is the missing premise?All spidersD are arachnidsU.Some water buffaloesU are not spidersD.

• Rule 1 seems to be unviolated.• Neither rule 5 nor rule 6 is violated, that is, we have

neither a negative premise and an affirmative conclusion nor a particular premise and a universal conclusion.

• The major term, ‘spiders’, is distributed in both the premise and the conclusion.

• So, there is a premise that will yield a valid syllogism.

Missing Premise: Example (pp. 169-172)

• Rule 6: The conclusion is particular. So, the missing premise must be particular.

• Rule 5: The conclusion is negative, so the missing premise must be negative.

• So, the missing premise is a particular negative. • Rule 4: The minor term, ‘water buffaloes’, is

undistributed in the conclusion, so it must be undistributed in the premise.

• So the missing premise must be, “Some water buffaloes are not arachnids.”

• Rule 2 check: The proposed premise yields the correct distribution of the middle term: It is undistributed in the given premise, so it must be distributed in the missing premise, as it is.

• The premise is true.

Missing Premise: Example (pp. 169-172)

• What is the missing premise?Some anteatersU are aardvarksU.All anteatersD are mammalsU.

• Rule 1 seems to be unviolated.• Rule 6 is violated: There is a particular premise and a

universal conclusion. No premise will yield a valid syllogism. Just indicate that it violates rule 6.

• Rule 4 is also violated. ‘Anteaters’ is distributed in the conclusion but not in the premise. Of course, breaking one rule is sufficient to show that no premise will yield a valid syllogism, so this is extra evidence.

Missing Premise: Example (pp. 169-172)

• What is the missing premise?No anteatersD are spidersD.Some anteatersU are aardvarksU.

• Rule 1 is not violated.• Rule 6 is not violated.• Rule 5 is violated. There is an affirmative conclusion

from a negative premise. No premise will yield a valid syllogism. Indicate that rule 5 has been violated.

• Rule 4 is also violated. ‘Anteaters’ is distributed in the conclusion but not in the premise. Of course, breaking one rule is sufficient to show that no premise will yield a valid syllogism, so this is extra evidence.

Missing Premise: Example (pp. 169-172)

• What is the missing premise?All mammalsD are vertebratesU.All aardvarksD are mammalsU.

• Rule 1 is not violated.• Rules 5 and 6 are not violated.• Rule 3 is violated. ‘Mammals’ is

distributed in the premise but not in the conclusion. So, no premise will yield a valid syllogism.

Venns and a Missing Conclusion (p. 172)

• If you have the premises and are looking for a conclusion, all you have to do is construct a Venn diagram and read off what, if anything, follows from the premises.

Venns and a Missing Conclusion (p. 172)

• What follows?All mammals are vertebrates.

All aardvarks are mammals.

So, ???

“All aardvarks (A) are vertebrates (V)” follows.

The conclusion is true.

Venns and a Missing Conclusion (p. 172)

• If you’d have constructed the diagram on the assumption that ‘aardvarks’ was the major term and ‘vertebrates’ was the minor term, that is, you’d represented the terms on the diagram with V on the left and A on the right, you would have constructed the diagram to look like this:

• It still would show that “All A are V” follows from the premises, so there would be no problem. Just read off the diagram what follows!

Venns and a Missing Conclusion (p. 172)

• What follows? No aardvarks are mastodons.

Some mastodons are elephantine animals.

So, ???

• You proceed in the same way. Diagram the premises. The diagram will look like one of the following:

• In either case, the conclusion is “Some elephantine animals are not aardvarks.”

Venns and a Missing Conclusion (p. 172)

• What follows?Some aardvarks are not mastodons.No mastodons are spiders.So, ???

• Diagram the premises. The diagram will look like one of the following:

• Regardless which you take to be the major and minor term, nothing follows from the premises.

Venns and a Missing Premise (p. 172)

• When looking for a missing premise, there are several procedures:– Diagram the premise and the conclusion.– If either the premise or the conclusion is a particular

proposition, place an X in every (unshaded) portion of the diagram warranted by the statement: never place an X on the line.

– Interpret the diagram.– If you were able to interpret the diagram in such a

way that the syllogism appears to be valid, check your interpretation by constructing a Venn diagram for the given premise and the proposed premise.

Venns and a Missing Premise (p. 173)

• There are several rules for interpreting the diagram:– If you have a universal premise and a universal conclusion, then

if you have shaded one area of the diagram twice, determine which premise containing the middle and the major or minor term, as is relevant, has been partially diagrammed.

– If both the premise and the conclusion are particulars, appeal to the diagrams for universal propositions involving the terms in the missing premise to see which of them, if diagrammed, would reduce the number of sections in which Xs are found to two.

– If the conclusion is particular and the premise is universal, consult the diagrams for the particular propositions involving the terms in the missing premise to see which of them, if diagrammed, would introduce one of the two Xs diagrammed in diagramming the conclusion.

– No premise will yield a valid conclusion if (1) the conclusion is a universal statement and you shade either more or fewer than three interior regions of the diagram, or (2) the conclusion is particular, and you either place an X in only one region of the diagram or you place Xs in four regions of the diagram.

• Is the premise true?

Missing Premise: Example (pp. 172-178)

• To find a missing premise, you diagram the premise and the conclusion and read off what the premise must be. If the conclusion is a universal and there is a premise that yields a valid syllogism, you will have diagrammed half of the premise.

• What is the missing premise?All mammals are vertebrates.So, no spiders are mammals.

• Notice that ‘mammals’ is the major term and ‘spiders’ is the minor term.

Missing Premise: Example (pp. 172-178)

• Diagram the premise and the conclusion:

• By diagramming the premise and the conclusion area S M not-V is shaded twice. This is as it should be. The missing premise concerns the terms S and V. The fact that the area is shaded, tells you that the missing premise must be “No S are V” or “No V are S” — the two statements are logically equivalent since you have diagrammed half of what should be diagrammed to diagram that premise.

Missing Premise: Example (pp. 172-178)

• If you have any question, plug in the premise and see if you diagram the conclusion:

• Voilà! And, of course, the conclusion, “No spiders are vertebrates” or “No vertebrates are spiders” is true.

Missing Premise: Example (pp. 172-178)

• Things are similar when you have a particular conclusion, but you put an X in every area where it could be placed. Consider the following premise and conclusion.

• What is the missing premise?

All spiders are arachnids.

Some water buffaloes are not spiders.

Missing Premise: Example (pp. 172-178)

• You construct the diagram as follows:

• You have two Xs. When you diagram the premise you can have only one X. Is the premise “Some W are A” or “Some A are W” — the two statements are logically equivalent — or is it “Some W are not A”? If you diagrammed the particular affirmative, you’d place an X on the line — that does not yield a valid syllogism. “Some W are not A” yields exactly one X in W and outside S and A, so it has to be the missing premise.

Missing Premise: Example (pp. 172-178)

• If you have any doubts, construct a diagram for the premises and see if you’ve diagrammed the conclusion:

• “Some water buffaloes are not arachnids” is true, of course.

Missing Premise: Example (pp. 172-178)

• What is the missing premise?Some houseflies are elephants.Some houseflies are aardvarks.

• Diagram the premise and conclusion, placing an X in every area in which you can do so.

• Notice that you have two Xs in area HEA, one introduced by diagramming the premise, and one introduced by diagramming the conclusion. The other X you introduced when diagramming the conclusion is in area H not-E A. The area that has two Xs will have one X when you diagram the premises. The X in H not-E A will “disappear” when diagramming the premises. You may place a circle around the X that is introduced when diagramming the conclusion and is alone in an area.

Missing Premise: Example (pp. 172-178)

The X in H E not-A must be “disposed of” by the missing premise. We’re concerned with A and E. The premise that will shade area H E not-A is “All elephants are aardvarks,” so that’s the missing premise. You have questions about that? Notice what happens when you diagram the premises:

And, of course, the missing premise, like both the given premise and the conclusion, is false.

Missing Premise: Example (pp. 172-178)

• What is the missing premise?

Some anteaters (AE) are aardvarks (AA).

All anteaters (AE) are mammals (M).• Diagram the premise and the conclusion:

• When only one X is introduced, no premise will yield a valid syllogism.

Missing Premise: Example (pp. 172-178)

• What is the missing premise?Some spiders (S) are not anteaters (AE).Some anteaters (AE) are not aardvarks (AA).

• Construct a diagram for the premise and conclusion:

• When diagramming the premise and conclusion yields Xs in four distinct areas, the argument is invalid.

Missing Premise: Example (pp. 172-178)

• Consider the following premise and conclusion:

All spiders are vertebrates.

No mammals are vertebrates.

• When diagramming the premise and conclusion results in having four distinct areas shaded, no premise will yield a valid syllogism.