Traditional Logic II - Memoria PressA syllogism is either valid or invalid. If it is not invalid, then it must be valid. 11 The Principle Behind ... sub-prae (1st Figure) An Example

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Traditional Logic II

Martin Cothran

Instructor

Indirect Reduction of Syllogisms

Review of ReductionExplanation of Indirect ReductionIndirect Reduction of BAROCOThe Three Steps of Reduction

A Review of Contradictory StatementsIndirect Reduction of BOCARDO

The Three Steps for Reducing a BOCARDO Review of Chapter

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Reduction of Syllogisms



The Three Steps for Reducing a BOCARDO Review of Chapter

What is Reduction?

Reduction of a syllogism is the process of changing a 2nd, 3rd, or 4th Figure syllogism into a

syllogism of the 1st Figure

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Two Methods for Reducing Syllogisms

Direct Reduction

Indirect Reduction

BARBARA, CELARENT, DARII, FERIOqueprioris;

CESARE, CAMESTRES, FESTINO, BAROCO secundae;

Tertia; DARAPTI, DISAMIS, DATISI, FELAPTON, BOCARDO, FERISON habet;

Quarta insuper addit; BRAMANTIP, CAMENES, DIMARIS, FESAPO, FRESISON

The Mnemonic Verse

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CESARE, CAMESTRES, FESTINO, BAROCOsecundae;



The Mnemonic Verse





The Mnemonic Verse

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The Mnemonic Verse





The Mnemonic Verse

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The Mnemonic Verse





The Mnemonic Verse

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The Mnemonic Verse





The Mnemonic Verse

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Review of ReductionExplanation of Indirect ReductionIndirect Reduction of BAROCOThe Three steps of Reduction


The Three Steps for Reducing a BOCARDOReview of Chapter

S: Simple conversion of the proposition signified by the preceding vowel

P: Per accidens, or partial conversion of the proposition signified by the preceding vowel

M: Mutatio, or transposition of the premises; make the minor premise the major, and the major the minor

C: Reduction by contradiction. This is the indirect method of reduction through (rather than to) BARBARA. It is signified by a non-initial c and is applied only to BAROCO AND BOCARDO

Rules for Reduction

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S: Simple conversion of the proposition signified by the preceding vowel

P: Per accidens, or partial conversion of the proposition signified by the preceding vowel

M: Mutatio, or transposition of the premises; make the minor premise the major, and the major the minor

C: Reduction by contradiction. This is the indirect method of reduction through (rather than to) BARBARA. It is signified by a non-initial c and is applied only to BAROCO AND BOCARDO

Rules for Reduction





The Mnemonic Verse

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What is Indirect Reduction?

Indirect reduction is the process of showing that an argument form

is valid by showing that it is impossible for it to be invalid

What does this mean?

A syllogism is either valid or invalid. If it is not invalid, then it

must be valid.

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The Principle Behind Indirect Reduction

In a valid syllogism, if the conclusion is false, then at least one of the

premises must be false

An Example of the Principle

All plants are animalsThe daisy is a plant

The daisy is an animal

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sub-prae (1st Figure)




sub-prae (1st Figure)

A:A:A:

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The Mnemonic Verse





The Mnemonic Verse

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The Mnemonic Verse





The Mnemonic Verse

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BARBARA




BARBARA

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The Principle Behind Indirect Reduction






BARBARA

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BARBARA




BARBARA

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Indirect Reduction of BAROCO

All collies are dogsSome animals are not dogsSome animals are not collies



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The Mnemonic Verse





The Mnemonic Verse

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A:O:O:

prae-prae (2nd Figure)





The Mnemonic Verse

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BAROCO

The Principle



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The PrincipleApplied

If the syllogism is valid, then when we deny the conclusion we should end up with a

denial of one of the premises.

Indirect Reduction of Syllogisms

Review of ReductionExplanation of Indirect ReductionIndirect Reduction of BAROCO

The Three Steps for Reducing a BAROCOA Review of Contradictory Statements

Indirect Reduction of BOCARDO The Three Steps for Reducing a BOCARDO

Review of Chapter

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The Three Steps for Reducing a BAROCO

STEP #1: Retain the original major premise

STEP #2: Use the contradiction of the original conclusion as the new minor

premiseSTEP #3: Derive the new conclusion, which, if the original syllogism is valid,

should be the contradiction of the original minor premise




premiseSTEP #3: Derive the new conclusion, which, if the original syllogism is valid,

should be the contradiction of the original minor premise

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premiseSTEP #3: Derive the new conclusion,

which, if the original syllogism is valid, should be the contradiction of the original

minor premise




premiseSTEP #3: Derive the new conclusion,

which, if the original syllogism is valid, should be the contradiction of the original

minor premise

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BAROCO



BAROCO

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BAROCO



BAROCO

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The Three Steps of Indirect Reduction

Retain the original major premiseUse the contradiction of the original

conclusion as the new minor premiseDerive the new conclusion, which, if

the original syllogism is valid, should be the contradiction of the original

minor premise

Four Kinds of Opposition

ContradictoryContrary

SubcontrarySubaltern

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Four Kinds of Opposition

ContradictoryContrary

SubcontrarySubaltern

Rule of Contradiction

Contradictory statements are statements that differ in both

quality and quantity.

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Quantity and Quality of the Four Categorical Statements

Quality

Affirmative Negative

Universal A E

Quantity

Particular I O

The First Law of Opposition

Contradictories cannot at the same time be true nor at the

same time false.

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Quality


Universal A E

Quantity

Particular I O

Contradictory Statements

A: All men are mortalO: Some men are not

mortal

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Quality


Universal A E

Quantity

Particular I O

Contradictory Statements

E: No men are godsI: Some men are gods

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minor premise



BAROCO

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What statement is the contradictory of this

statement?

Some animals are not collies

We first change the quantity


No animals are collies

FROM:

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No animals are colliesTO:

We then change the quality


All animals are collies

FROM:

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All animals are colliesTO:



BAROCO

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All collies are dogsAll animals are collies






minor premise

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All collies are dogsAll animals are collies

?


All collies are dogsAll animals are colliesAll animals are dogs

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BAROCO

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minor premise

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Is the new conclusion contradictory to the original

minor premise?

All animals are dogs

Some animals are not dogs


minor premise?

All animals are dogs

Some animals are not dogs

A:

O:

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O: Some animals are not dogsvs.

A: All animals are dogs



BARBARA

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Review of Chapter

Indirect Reduction of BOCARDO

Some dogs are not collies All dogs are animals


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The Mnemonic Verse

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The Mnemonic Verse




O:A:O:

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The Mnemonic Verse




BOCARDO

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The Principle






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The Three Steps of ReductionReview of Chapter

The Three Steps for Reducinga BOCARDO

Retain the original minor premiseUse the contradiction of the original

conclusion as the new major premiseDerive the new conclusion, which, if


major premise

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The Three Steps for Reducinga BOCARDO




major premise




BOCARDO

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BOCARDO




BOCARDO

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BOCARDO





major premise

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BOCARDO

What statement is the contradictory of this

statement?


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FROM:



No animals are colliesTO:

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All animals are collies

FROM:



All animals are colliesTO:

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Some dogs are not collies All dogs are animalsAll animals are collies


All animals are colliesAll dogs are animalsAll animals are collies

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major premise


All animalsM are colliesP

All dogsS are animalsM

?

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All dogsS are __________P

BARBARA




All dogsS are colliesP

BARBARA

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minor premise


minor premise?

All dogs are collies

Some dogs are not collies

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minor premise?



A:

O:



A:

O:


minor premise?

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minor premise


All animals are colliesAll dogs are animalsAll dogs are collies

BARBARA

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Review of Chapter

Indirect ReductionIndirect Reduction (IR) shows the validity of syllogisms

by showing that it is impossible for it to be invalid.

IR of BAROCO involves keeping the original majorpremise and replacing the minor premise with the

contradiction of the conclusion, which should yield the contradiction of the original minor premise.

IR of BOCARDO involves keeping the original minorpremise and replacing the major premise with the

contradiction of the conclusion, which should yield the contradiction of the original major premise.

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Review of Exercises

How to create a syllogism

A review of Direct Reduction

Exercises 25-43

Creating your own syllogisms

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Creating a Syllogism

Step #1: Construct your conclusion 1st. Step #2: Provide the correct quantifiers for the premises.Step #3: Determine the location of the middle terms.Step #4: Identify major & minor terms.Step #5: Fill in the middle terms.

Exercises 26

Creating your own CELARENT

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Step #1: Construct your conclusion 1st.Step #2: Provide the correct quantifiers for the premises.Step #3: Determine the location of the middle terms.Step #4: Identify major & minor terms.Step #5: Fill in the middle terms.

Step #1: Begin with a Conclusion

Make sure it is the correct quantity and quality

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Exercises 26


Exercises 26


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No _________ is a ____________


No _________ is a ____________

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No _________ is ____________


No _________ is ____________

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No Sneetch is a Lorax



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Exercises 26


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Exercises 26


Exercises 26


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Exercises 26


Step #2: Provide Correct Quantifiers for Premises

No ________ is ____________All ________ is ____________


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The Mnemonic Verse

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The Mnemonic Verse

Step #3: Determine Location of Middle Terms

No ________ is ____________All ________ is ____________


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No ________M is ____________All ________ is ____________



No ________M is ____________All ________ is ____________M


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Step #4: Identify Major & Minor Terms

No ________M is ____________All ________ is ____________M

No SneetchS is a LoraxP

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No ________M is ____________All ________ is ____________M



No ________M is ____________All ________ is ____________M


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No ________M is a LoraxP

All ________ is ____________M




All ________ is ____________M


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All SneetchesS are ____________M






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Step #1: Construct your conclusion 1st. Step #2: Provide the correct quantifiers for the premises.Step #3: Determine the location of the middle terms.Step #4: Identify major & minor terms.Step #5: Fill in the middle term.

Rule #1 for Determining the Middle Term

If the quality of the conclusion is affirmative, then pick something the things signified by the major

and minor terms have in common.

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If the quality of the conclusion is negative, then pick something the things signified by the major and

minor terms do NOT have in common.

Step #5: Fill in theMiddle Term




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No feathered thingM is a LoraxP





All SneetchesS are feathered thingsM


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All SneetchesS are feathered thingsM


Exercises 33

Creating your own DARAPTI

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Step #1: Construct your conclusion 1st.Step #2: Provide the correct quantifiers for the premises.Step #3: Determine the location of the middle terms.Step #4: Identify major & minor terms.Step #5: Fill in the middle terms.

Exercises 33


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Some _________ are ____________


Some things are things that can sing

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Exercises 33


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Step #2: Provide Correct Quantifiers for Premises

All ________ is ____________All ________ is ____________



Step #1: Construct your conclusion 1st. Step #2: Provide the correct quantifiers for the premises.Step #3: Determine the location of the middle term.Step #4: Identify major & minor terms.Step #5: Fill in the middle terms.

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Exercises 33






The Mnemonic Verse

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Step #3: Determine Location of Middle Term

All ________ is ____________All ________ is ____________


Step #3: Determine Location of Middle Term

All ________M is ____________All ________M is ____________


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Step #4: Identify Major and Minor Terms

All ________M is ____________All ________M is ____________

Some things are things that can singP

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All ________M is ____________All ________M is ____________

Some thingsS are things that can singP


All ________M is ____________All ________M is ____________


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All ________M are things that can singP

All ________M is ____________Some thingsS are things that can singP



All ________M are thingsS


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All ________M are thingsS



Step #1: Construct your conclusion 1st. Step #2: Provide the correct quantifiers for the premises.Step #3: Determine the location of the middle term.Step #4: Identify major & minor terms.Step #5: Fill in the middle term.

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If the quality of the conclusion is affirmative, then pick something the things signified by the major

and minor terms have in common.


All YingsM are things that can singP

All YingsM are thingsS


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All YingsM are things that can singP

All YingsM are thingsS


Reducing a BRAMANTIP

All dogs are mammalsAll mammals are animalsSome animals are dogs

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All mammals are animalsSome animals are dogs

All dogs are mammals

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All dogs are mammalsSome animals are dogs



All dogs are mammalsSome animals are dogs


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All mammals are animalsAll dogs are mammalsSome animals are dogs



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The Mnemonic Verse



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Review of Partial Conversion

Change the quantity

Switch the terms


All men are mortal

Some mortal things are men

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Some men are vicious

All vicious things are men



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All mammals are animalsAll dogs are mammalsAll animals are dogs

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All mammals are animalsAll dogs are mammalsAll animals are dogs


All mammals are animalsAll dogs are mammalsAll dogs are animals

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Traditional Logic II

Martin Cothran

Instructor

Traditional Logic II - Memoria PressA syllogism is either valid or invalid. If it is not invalid, then it must be valid. 11 The Principle Behind ... sub-prae (1st Figure) An Example

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