4 categorical propositions

4 Categorical Propositions

4.4. Conversion, Obversion, and Contraposition

Conversion, Obversion, and Contraposition

These are 3 operations that can be used to change standard form statements or change a statement into standard form.

Conversion is the simplest: simply switch the subject and predicate terms.

No S are P -----conversion----> No P are S

Conversion

Compare the Venn Diagrams of each statement:

No S are P No P are S

S P PS

For E propositions, there is no change in diagram, and so no change in truth value.

Conversion

Compare the Venn Diagrams of each statement:

All S are P All P are S

PS

For A propositions, there IS a change in diagram, and so a change in truth value.

S P

Conversion

Compare the Venn Diagrams of each statement:

Some S are P Some P are S

PS

For I propositions, there is no change in diagram, and so no change in truth value.

S P

X X

Conversion

Compare the Venn Diagrams of each statement:

Some S are not P Some P are not S

For O propositions, there IS a change in diagram, and so a change in truth value.

S P

X

PS

X

ConversionSo, what do we know?

Immediate inferences (inferences with one premise and one conclusion) are valid for the converses of E and I propositions, but not for A and O propositions.

All puppies are evil.Therefore, all evil things are puppies.

Since this is conversion of an A proposition, it is an illicit inference … an illicit conversion.

Some puppies are evil.Therefore, some evil things are puppies.

No problem.

Obversion

Obversion (2 steps):1)Change the quality2)Replace Predicate term with its complement

All S are P ------- obversion----->____________?No S are P -------obversion ------> _____________?Some S are P ----- obversion -----> _______________?Some S are not P ----obversion---> _____________?

Obversion

No S are P All S are non-P

S P PS

S PS P

All S are PNo S are non-P

S P

X

S P

X

S P

X

Some S are P

Some S are not P

Some S are not non-P

Some S are non-P

S P

X

Contraposition

Contraposition (2 steps):1)Switch Subject and Predicate terms2)Replace both Subject and Predicate terms

with their complements

All S are P ------- contraposition----->____________?No S are P -------contraposition ------> _____________?Some S are P ----- contraposition -----> _______________?Some S are not P ----contraposition---> _____________?

Contraposition

S PS P

All S are PAll non-P are non-S

S P

X

Some S are P

S P

XSome non-P are non-S

S P

X

Some S are not P

Some non-P are not non-S

S P

X

No S are P No non-P are non-S

S P PS

1) Place an X in each distinct area of subject class

2) Remove Xs based on what the proposition says using shading for universal propositions and eraser for particular ones

3) Remove Xs based on Boolean interpretation using your eraser

Method to Drawing Venn Diagrams

Helpful Wording

All non-P are non-SEvery single non-P is also a non-S

If there were any Xs inside the left part of the S circle, the statement would allow a non-P that was an S, which is denied in the statement. If there are any Xs, they would be outside the two circles (they don’t appear because this is the Boolean interpretation)

S P

Helpful Wording

No non-P are non-SNot even one of the non-Ps is also a non-S(an X outside both circles says, here is a non-P that

is also a non-S)

So, to prohibit allowing an X outside those circles, that area must be shaded.

PS