Chapter 17: Missing Premises and Conclusions

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Chapter 17Missing Premises and Conclusions
2
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.

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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.

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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.

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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.

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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.

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

• Make sure there are exactly three terms (Rule 1).
• Make sure you dont have a particular premise and
  a universal conclusion (Rule 6).
• Make sure you dont 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.

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

• 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.

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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.

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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, theyre logically equivalent.
• Checking against Rule 2 The middle term,
  vertebrates, is distributed exactly once.
• The missing premise is true.

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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.

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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.

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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.

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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.

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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.

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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.

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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.
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Venns and a Missing Conclusion (p. 172)

• If youd have constructed the diagram on the
  assumption that aardvarks was the major term
  and vertebrates was the minor term, that is,
  youd 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!

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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.

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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.

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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.

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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?

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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.

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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.

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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.

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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.

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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, youd 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.

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Missing Premise Example (pp. 172-178)

• If you have any doubts, construct a diagram for
  the premises and see if youve diagrammed the
  conclusion
• Some water buffaloes are not arachnids is true,
  of course.

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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.

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Missing Premise Example (pp. 172-178)
The X in H E not-A must be disposed of by the
missing premise. Were concerned with A and E.
The premise that will shade area H E not-A is
All elephants are aardvarks, so thats 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.
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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.

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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.

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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.