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The Traditional Square of Opposition

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The Traditional Square of Opposition. Lecture X – 03/10/11. The Aristotelian Standpoint. Now that we have examined categorical logic and immediate inferences from the Boolean standpoint, we will proceed to examine it from the Aristotelian standpoint as well.

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### The Traditional Square of Opposition

Lecture X – 03/10/11

The Aristotelian Standpoint

Now that we have examined categorical logic and immediate inferences from the Boolean standpoint, we will proceed to examine it from the Aristotelian standpoint as well.

Recall that the two views differ insofar as the Aristotelian standpoint recognizes that universal propositions about existing things do indeed imply that said things exist.

This distinction makes a huge difference to the square of opposition and the amount of valid immediate inferences that are possible.

The Traditional Square

All cats are animals.

The contrary relation is one of partial opposition, that is, at least one contrary must be false (but both might be).

The subcontrary relation is also one of partial opposition. At least one subcontrary must be true, both both might be.

The subalternation relation is one in which inferences about truth value can be made according to the arrows illustrating the direction that truth or falsity can “flow.”

Contrary

A

E

Contradictory

Contradictory

Truth

Falsity

Subalternation

Subalternation

Contradictory

Contradictory

O

I

Subcontrary

Illicit Aristotelian Inferences

It is false that no monsters are scary creatures.

Therefore, all monsters are scary creatures.

Some pants are not dirty clothes.

Therefore some pants are dirty clothes.

It is false that all musical instruments are tubas.

Therefore, it is false that some musical instruments are tubas.

Illicit Aristotelian Inferences

It is false that no monsters are scary creatures.

Therefore, all monsters are scary creatures.

Illicit contrary

Some pants are not dirty clothes.

Therefore some pants are dirty clothes.

Illicit subcontrary

It is false that all musical instruments are tubas.

Therefore, it is false that some musical instruments are tubas.

Illicit subalternation

More on the Existential Fallacy

The existential fallacy occurs, from the Aristotelian standpoint, when an otherwise valid inference is drawn from a premise about something that doesn’t exist.

All unicorns are animals.

Therefore, some unicorns are animals.

Boolean – Invalid, existential fallacy

Aristotelian - Invalid, existential fallacy

All cats are animals.

Therefore, some cats are animals.

Boolean – Invalid, existential fallacy

Aristotelian – Conditionally Valid

Conditional Validity

Conditional validity is a property of inferences viewed from the Aristotelian standpoint. It applies to (1) all inference forms that depend on valid applications of the contrary, subcontrary or subalternation relations or to (2) particular inferences in which we are not certain the premise denotes actually existing things.

No A are B

Therefore, it is false that all A are B.

Boolean – Invalid

Aristotelian – Conditionally Valid

All students who failed the exam are on probation.

Therefore, some students who failed the exam are on probation

Boolean – Invalid

Aristotelian – Conditionally Valid

Using All Our Tools

All inanimate objects are non-biological life forms.

Therefore, some biological life forms are not inanimate objects.

Use the Aristotelian square of opposition and conversion, obversion and/or contraposition to prove the following inference valid.

Using All Our Tools

All non-A are non-B.

All B are A. contrposition

It is false that no B are A. contrary

It is false that all B are non-A. obversion

Some B are not non-A. contradictory