Syllogisms
Fun with Deductive
Reasoning
What is a syllogism?
» A syllogism is a deductive argument comprising
three categorical propositions: a major premise,
a minor premise and the conclusion. Categorical
propositions have four standard forms:
»
»
»
»
A = All S are P
E = No S are P
I = Some S are P
O = Some S are not P
In the mood
» The mood of a syllogism is defined by which of
the forms appear and where. So, for example:
All M are P
Some S are M
Therefore, All S are P
has the mood: AIA.
Syllogism overview
» A categorical syllogism contains only three
categorical terms: a major term, minor term and
middle term.
» The major term appears as the predicate in the
conclusion, and only once in the major premise
(i.e., the first premise).
» The minor term appears as the subject in the
conclusion, and only once in the minor premise
(i.e,. the second premise).
» The middle term appears once in the major
premise, once in the minor premise, and once in
the conclusion.
Distribution of terms
» A term is said to be distributed when all members
of the class denoted by the term are affected by a
proposition.
– All S are P
» S is distributed; P is not distributed
» Example “All cows are mammals” tells us
something about cows but nothing about
mammals
Syllogism Examples
Correct Syllogism:
• Major Premise: All mammals are warm-blooded animals.
• Minor Premise: No lizards are warm-blooded animals.
• Conclusion: Therefore, no lizards are mammals.
Correct Syllogism:
• Major Premise: All humans are mortal.
• Minor Premise: All Greeks are human.
• Conclusion: Therefore, all Greeks are mortal.
Descartes’ Syllogism (correct)
• Major Premise: Existence has to be true if one is thinking.
• Minor Premise: I am thinking.
• Conclusion: I think, therefore, I am.
Syllogisms can be
• Valid or Invalid (reasoning in
incorrect order)
AND
• True or False (reasoning from a
faulty major premise)
• If a syllogism is both true and valid
then it is said to be sound
Examples of Faulty Syllogisms
FALSE Syllogism (not TRUE -- false major premise)
– Major Premise: Blondes have more fun
– Minor Premise: Mary is blonde; Jane is brunette
– Conclusion: Mary has more fun than Jane.
INVALID Syllogism (not VALID – order of reasoning is
incorrect):
– Major Premise: All dogs eat meat
– Minor Premise: Bob (a human) eats meat
– Conclusion: Bob is a dog.
Corrections
Syllogism One:
The first faulty syllogism proceeds from a FALSE
major premise and therefore can be thrown out
entirely.
Syllogism Two:
– Major Premise: All dogs eat meat
– Minor Premise: Rover is a dog.
– Conclusion: Therefore, Rover eats meat.
Valid or invalid? True or False?
Example One:
• Major Premise: When it snows the streets get wet.
• Minor Premise: The streets are getting wet.
• Conclusion: Therefore, it is snowing.
Example Two:
• Major Premise: If you buy a Ferrari, you will instantly be popular.
• Minor Premise: Ed just bought a Ferrari.
• Conclusion: Ed will achieve instant popularity.
Example Three:
• Major Premise: When the battery is dead, the car will not start.
• Minor Premise: The car will not start.
• Conclusion: Therefore, the battery is dead.
Corrections: Valid and True
Example One:
• Major Premise: When it snows, the streets get wet.
• Minor Premise: It is snowing.
• Conclusion: Therefore, the streets are getting wet.
Example Two:
• Example Two proceeds from the beginning from a FALSE major
premise (Ferraris give instant popularity) and therefore can be
thrown out entirely.
Example Three:
• Major Premise: When the battery is dead, the car will not start.
• Minor Premise: The battery is dead.
• Conclusion: Therefore, the car will not start.
Types of valid syllogisms
•
•
•
•
Modus Ponens (Affirming the
antecedent)
Modus Tollens (Denying the
consequent)
Hypothetical Syllogism (Chain
argument)
Disjunctive Syllogism
Modus Ponens
1. If A then B
2. A
3. Therefore, B
Examples:
• If it’s spring, then the birds are chirping
• It’s spring.
• The birds are chirping.
•
•
•
If a world government doesn’t evolve soon, then wars will continue to occur
A world government isn’t going to evolve soon.
Wars will continue to occur
Modus Tollens
1. If A then B
2. Not B
3. Not A
Example:
1. If it’s spring then the birds are chirping
2. The birds aren’t chirping
3. Therefore, it isn’t spring.
Hypothetical Syllogism
1. If A then B
2. If B then C
3. If A then C
Example:
1. If we successfully develop nuclear fusion power, then power will become
plentiful and cheap.
2. If power becomes cheap and plentiful, then the economy will flourish.
3. If we successfully develop nuclear fusion power, then the economy will
flourish.
Disjunctive Syllogism
1. A or B
2. Not A
3. B
Example:
1. Either Romney won in 2012 or Obama did.
2. Romney didn’t win.
3. Obama did win.
Valid or invalid?
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Syllogism no-no’s
» Syllogisms need to follow 6 rules in order
to be valid. If they violate one of these
rules then that syllogism commits a formal
fallacy and is invalid
Rule 1
» There needs to be three categorical terms and those
terms cannot vary in how they are used
» A fallacy of equivocation occurs when a term is used in
a different way within the course of an argument. So,
for example:
– The priest told me I should have faith.
– I have faith that my son will do well in school this
year.
– Therefore, the priest should be happy with me.
» “faith” is being used in two different ways in this
argument
Rule 2
» The middle term of a valid syllogism is distributed in at
least one of the premises. The fallacy of the
undistributed middle occurs when this doesn't happen.
For instance, the middle term (furry animals) in this
syllogism
– All dogs are furry animals
– Some cats are furry animals
– Therefore,some dogs are cats
isn't distributed, and the argument is clearly fallacious
Rule 3
» If a term is distributed in the conclusion it must be
distributed in at least one of the premises
– All Protestants are Christians
– No Catholics are Protestants
– Therefore, no Catholics are Christians
doesn't work, because the term "Christians" is
distributed in the conclusion, but not in the (major)
premise.
Rule 3
» The fallacy of illicit major occurs (as above)
when the major term is distributed in the
conclusion, but not in the (major) premise.
»
» The fallacy of illicit minor occurs when the
minor term is distributed in the conclusion, but
not in the (minor) premise
Rule 4
» A valid syllogism can't have two negative
premises
» The fallacy of exclusive premises occurs when
a syllogism has two premises that are negative.
» A negative premise is either an "E" statement
("No S are P") or an "O" statement ("Some S
are not P"), and if you've got two of them in
your premises, your syllogism isn't valid.
Rule 5
» The conclusion of a syllogism must be negative, if
either premise is negative
» The fallacy of drawing an affirmative conclusion
from a negative premise occurs if this rule is
violated. Similarly, if a conclusion is negative, then
one of the premises must be negative (which rule,
if broken, constitutes the fallacy of drawing a
negative conclusion from an affirmative premise).
Rule 6
» No particular conclusion can be drawn from
two universal premises
» This is arguably the most counterintuitive of
the rules for validity. An existential fallacy
occurs whenever a particular conclusion
appears with two universal premises (for
example, All M are P, All S are M, Therefore,
some S are P).