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1. Lecture 5
The Categorical Syllogism (II)

2. Dictum de Omni et Nullo
(Maxim of All and None)
●Whatever is affirmed (or denied)
universally of any subject is thereby
affirmed (or denied) of every logical part
of that subject.
●The Scholastics thought that this was
the fundamental principle of reasoning

3. Dictum Continued
●Example: All men are elephants
●Socrates is a man
●Therefore, Socrates is an elephant
●What is affirmed universally of men is
affirmed of a part of the class of men, i.
e., Socrates

4. Still More Dictum
●How do we know that the dictum is true?
●We directly grasp its truth
●If it is denied, there is a violation of the Law of
Contradiction.
●You would affirm both an A proposition and
an O proposition about the same subject, e.
g., “All men are elephants” and “Some man is
not an elephant”.

5. More of Guess What?
●In the dictum, the universal subject is
taken as a whole, not as a collection of
individuals.
●The A proposition is not the result of an
enumeration

6. “Reasoning” By Enumeration
●Suppose we argue this way:
●All students in this course are interested in
logic
●Andreas is a student in this course
●Therefore, Andreas is interested in logic
●One could know that the first premise is true
only by checking to see whether every
student in this course was interested in logic.
But if we did this, we would already know that
Andreas was interested in logic.
●The argument would result in nothing new.

7. Fallacy of Division
●The dictum de omni shouldn’t be
confused with the fallacy of division
●In this fallacy, what applies to a
conjunction is wrongly taken to apply to
the parts of the conjunction
●The conjunction can be implicit

8. Diamond-Water Paradox
●The famous diamond-water paradox is an
example of the fallacy of division
●Before the “marginalist revolution”, most
economists thought that subjective values
couldn’t explain prices.
●The reason they thought that was this
argument:
●Water is more valuable than diamonds
●If subjective value explains prices, then water
should have a higher price than diamonds
●Diamonds have a higher price than water.

9. Structure of the Paradox
●We have derived by a valid argument
that water has a higher price than
diamonds.
●But this conclusion is false
●Therefore, at least one of the premises
must be false
●This is similar to a reductio argument,
but is not the same. “Water has a higher
price than diamonds” is false but not a

10. So What’s Wrong With the
Paradox?
●The paradox commits the fallacy of division
●“Water is more valuable than diamonds” is
true only for the total quantities of water and
diamonds
●There is thus an implicit conjunction in this
premise
●It doesn’t then follow that any particular
quantity of water will be more valuable than
any particular quantity of diamonds

11. Fallacy of Composition
●In this fallacy, what applies to a part of a
conjunction is wrongly applied to the
whole conjunction
●Mill’s proof of the principle of utility
appears to commit this fallacy

12. Mill’s Proof
●“No reason can be given why the general
happiness is desirable except that each
person. . .desires his own happiness. This,
however, being a fact, we have not only all
the proof which the case admits of, but all
which is possible to require, that happiness is
a good: that each person’s happiness is a
good to that person, and the general
happiness, therefore, a good to the aggregate
of all persons.” (Mill, Utilitarianism. Chapter 4)

13. Finding the Fallacy
●The fallacy is here: “and the general
happiness, therefore, a good to the
aggregate of all persons”
●This doesn’t follow from each person’s
taking his own happiness to be a good
●You don’t have to hold that there is a
good to the “aggregate of all persons”.

14. Another Fallacy in Mill’s
Passage
●The same paragraph from Mill also
says: “The only proof capable of being
given that an object is visible, is that
people actually see it. The only proof
that a sound is audible is that people
hear it. . .in like manner, I apprehend,
the sole evidence it is possible to prove
that anything is desirable, is that people

15. Equivocation
●Mill’s argument by analogy can be used to
show that if something is actually desired, this
shows it is capable of being desired.
●Mill uses the analogy to show that if
something is desired, then it ought to be
desired.
●Mill thus equivocates on “desirable”
●There have been defenses of Mill that acquit
him of both fallacies. There is a vast literature

16. Paradox of Thrift
●A famous example that uses ( not commits)
the fallacy of composition is the paradox of
thrift.
●If a single person tries to save, his savings
will increase.
●It doesn’t follow that if everyone tries to save,
total savings will go up.
●Keynes used this argument but it goes back
to Bernard Mandeville.
●Note that from the fact that it doesn’t follow
that total savings will go up that it is false that

17. Back to the Dictum
●The Scholastics held that the First Figure
shows the dictum most clearly
●In the first figure, the major premise has the
middle term as the subject in the major
premise and the middle term as predicate in
the minor premise.
●No Misesians are Keynesians
●All faculty at the Mises Institute are Misesians
●No faculty at the Mises Institute are
Keynesians
●This is a valid mood (EAE) of the first figure. It

18. Immediate Inference
●The second, third, and fourth figures of
the syllogism can be reduced to the first
figure
●We aren’t going to show how to do this,
but the process depends on immediate
inference.
●Without introducing a new term, we can
derive a new proposition from a given
proposition.

19. Some Types of Immediate
Inference
●Conversion
●We interchange the subject and predicate.
From “All men are mortal”, we get “Some
mortals are men.”
●Two rules for conversion
●Both propositions must have the same
quality, i.e., both affirmative or both negative
●If a term isn’t distributed in the original
proposition, it can’t be distributed in the new
proposition.

20. More on Conversion
●In the example, we can’t convert to “All
mortals are men” because “mortals” isn’
t distributed in the original
●We can convert “No Misesians to
Keynesians” to “No Keynesians to
Misesians”. An E proposition excludes
two classes from each other, so it works
either way.

21. Obversion
●In obversion, we keep the same subject but
change the predicate into its contradictory.
●To do this, we have to change the quality of
the proposition, i.e., affirmative to negative or
negative to affirmative
●The obverse of “All men are mortal” is “No
men are not-mortal”.
●There are many other types of immediate
inference, but we won’t go into them here.

22. Mill’s Criticism of the
Syllogism
●Mill thought that the major premise of
the syllogism begs the question. We
couldn’t know “All men are mortal’
unless we had examined each person
and found that he was mortal. But then
we would already know Socrates was
mortal and wouldn’t need the syllogism
to prove it. We couldn’t learn anything

23. The Attack on Mill Continued
●We have have already seen what is wrong
with Mill’s argument. He is taking the
syllogism to be an argument from
enumeration.
●The major premise of the syllogism isn’t an
enumeration. It is about the necessary
features of the concept of the middle term
●Mill didn’t believe in conceptual necessities.
He thought that “2 + 2 = 4” is an empirical
generalization.

24. Progressive Syllogisms
●A Progressive Syllogism reasons from
cause to effect
●Rent control will produce a shortage of
apartments
●The City Council has just imposed rent
control
●We will have a shortage of apartments.

25. Regressive Syllogisms
●A regressive syllogism reasons from effect to
cause
●A business cycle, in certain conditions, must
result from an expansion of bank credit
●We are now in a business cycle in these
conditions.
●This business cycle has resulted from an
expansion of bank credit

26. Example of a Fallacy
●“This dichotomy [defended by free
market economists] between markets
and states---between trade and rules---
is false and hides more than it reveals.
Market exchange, and especially long-
distance trade, cannot exist without
rules imposed from somewhere.” Dani
Rodrik, The Globalization Paradox (

27. The Fallacy Revealed
●Market exchange, and especially long-
distance trade, cannot exist without
rules imposed from somewhere.”
●It doesn’t follow from the necessity of
rules that these rules have to be
imposed or that we must have a state.