1. The Traditional Square of Opposition
Lecture X – 03/10/11

2. 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.

3. The Traditional Square
All cats are animals.
Contrary A E
Contradictory
Contradictory
Truth
Falsity
Subalternation
Subalternation
Contradictory
Contradictory O I
Subcontrary
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.”

4. 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.

5. 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

6. 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. Aristotelian – Conditionally Valid

7. 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

8. 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.

9. 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