Disjunctive & Hypothetical Syllogisms

Alternatives, conditions, and validly choosing.

The Three Principal Kinds of SyllogismsCategorical Syllogisms

Disjunctive Syllogisms

• Pure hypothetical• Mixed hypothetical (modus ponens, modus

tollens)

Hypothetical Syllogisms

Review: Categorical Syllogism

Categorical

• All M is P• All S is M.• Therefore all S is

P.

Disjunctive• Either P is true

or Q is true.• P is not true.• Therefore Q is

true.

Hypothetical

• If P is true, then Q is true.

• If Q is true, then R is true.

• Therefore if P is true, then R is true.

Differences

Categorical

• Contain categorical propositions only

• Simple propositions

Disjunctive & Hypothetical

• contain disjunctive or conditional propositions

• Compound propositions

Disjunctive Syllogisms• The case of the logical dog

PATH A

PATH CPATH B

Disjunctive SyllogismsSTRUCTUREEither P is true or Q is true.P is not true.Therefore Q is true.

*disjunctional proposition, disjuncts

A Valid Disjunctive Syllogism1. The disjuncts are not both false or both

true.The meaning of “or” [inclusive and exclusive]• Example:

He was captured either dead or alive.He was not captured dead.Therefore, he was captured alive.

A Valid Disjunctive Syllogism2. The alternatives must be mutually exhaustive.• Example:You are either a Catholic or an Aglipayan.You are not a Catholic.Therefore, you are an Aglipayan.

= the fallacy of possibilities not exhaustive

A Valid Disjunctive Syllogism3. The alternatives must be mutually exclusive.• ExampleJuan is either stupid or dishonest.Juan is not stupid.Therefore, Juan is dishonest.

= the fallacy of alternatives not mutually exclusive

An objection• Either Smith is in New York or Smith is in

Paris.• Smith is in New York.• Therefore Smith is not in Paris.

The disjunction plays no role in the argument. The conclusion is based from the second categorical premise with the unexpressed additional premise being the obviously true proposition that “Smith cannot be both in New York and in Paris.” In disjunctive form, it can be stated as, =>Either Smith is not in New York or Smith is not in Paris.

Disjunctive

• Either P is true or Q is true.

• P is not true.

• Therefore Q is true.

P is true.

Therefore Q is not true.

Hypothetical Syllogisms- contain one or more compound,

hypothetical (or conditional) propositions- affirms that if one of its components (the

antecedent) is true=> then the other of its components (the consequent) is true

Conditional (or hypothetical) Proposition • has two component propositions:1. the antecedent

– follows the “if”– states the condition or limitation

2. the consequent– comes after the “then”– states the result– asserts something as true on condition

that the “if clause” is true.

Conditional (or hypothetical) Proposition • This kind of compound proposition can be

further categorized according to the nature of the propositions they contain.

1. Pure hypothetical2. Mixed hypothetical

Pure Hypothetical Syllogism• contains conditional propositions only• The first premise and the conclusion

have the same antecedent, the second premise and the conclusion have the same consequent, and the consequent of the first premise is the same as the antecedent of the second premise.– If P is true, then Q is true.– If Q is true, then R is true.– Therefore if P is true, then R is true.

Same antecedent

Same consequent

Pure Hypothetical SyllogismForm

• If P is true, then Q is true.

• If Q is true, then R is true.

• Therefore if P is true, then R is true.

Example• If I do not wake up,

then I cannot go to work.

• If I cannot go to work, then I will not get paid.

• Therefore, if I do not wake up, then I will not get paid.

Mixed Hypothetical Syllogism• 1 conditional premise + 1 categorical

premise • two valid forms: Modus Ponens & Modus

Tollens1. Modus Ponens• in the affirmative mood• first premise is a statement of

alternatives. In this form, the categorical premise affirms the antecedent of the conditional premise and the conclusion affirms its consequent.If P is true, then Q is true.P is true.Therefore Q is true

Mixed Hypothetical Syllogism

Form• If P is true,

then Q is true.• P is true.• Therefore Q is

true

Example• If the cake is made with

sugar, then the cake is sweet.

• The cake is made with sugar.• Therefore, the cake is sweet.

Mixed Hypothetical Syllogism• invalid form:

– If Bacon wrote Hamlet, then Bacon was a great writer.

– Bacon was a great writer.– Therefore Bacon wrote Hamlet.its categorical premise affirms the consequent, rather than the antecedent, of the conditional premise.=> the fallacy of affirming the consequent.

Form• If P is

true, then Q is true.

• P is true.• Therefore

Q is true

Q is true.

P is true.

Mixed Hypothetical Syllogism

Þ the fallacy of affirming the consequent.

Another example:• If I have the flu, then I have a sore throat.• I have a sore throat.• Therefore, I have the flu.

Mixed Hypothetical Syllogism

2. Modus Tollens– categorical premise denies the

consequent of the conditional premise, and the conclusion denies its antecedent

If P is true, then Q is true.Q is false.Therefore P is false.

Mixed Hypothetical SyllogismForm

• If P is true, then Q is true.

• Q is false.• Therefore P is

false.

Example• If she walks,

then she moves.• She does not

move.• Therefore she

does not walk.

Mixed Hypothetical Syllogism

– Invalid form:• If Pedro marries Juana, she will be happy.• Pedro will not marry Juana.• Therefore, Juana will not be happy.Þ its categorical premise denies the

antecedent, rather than the consequent, or the conditional premise.

Þ the fallacy of denying the antecedent.

4-Square

1.  Affirming the Antecedent (AA) If Tweety is a bird, then Tweety flies.Tweety is a bird.Tweety flies

2.  Denying the Antecedent (DA) If Tweety is a bird, then Tweety flies.Tweety is not a bird.Tweety doesn’t fly

3.  Affirming the Consequent (AC) If Tweety is a bird, then Tweety flies.Tweety flies.Tweety is a bird

4.  Denying the Consequent (DC) If Tweety is a bird, then Tweety flies.Tweety doesn’t fly.Tweety is not a bird.

http://faculty.unlv.edu/beisecker/Courses/Phi-102/HypotheticalSyllogisms.htm

*Conjunctive Syllogism• Contains a CONJUNCTIVE PROPOSITION (and)NEGATIVE CONJUNCTIVE SYLLOGISM- Denies the possibility of 2 alternatives

– If not "P and Q"– Then, either "not P" or "not Q" or "not P and

not Q"– Ex: You cannot serve God and the devil at the

same time. You cannot both be in two place at the same time.Valid Invalid

• You cannot be single and married at the same time.

• But Samson is married.• Therefore, Samson is not

single

• You cannot be in Cotabato City and Davao City at the same time.

• But you are not in Davao City now;

• Therefore, you are in Cotabato City now.

“Both and…”• "P and Q" is true if and only if "P" is true, and "Q"

is true.- the only kind that yield two conclusions (or more) from only one premise. - we can separate them and affirm each as a separate conclusion. If both terms together are true, then each one separately is true also.1. Roses are red and violets are blue.2. Roses are red.3. Violets are blue.