Chapter 4 Categorical Propositions contd Philosophy 1504 Language

Chapter 4: Categorical Propositions, contd. Philosophy 1504: Language and Logic March 30, 2016

4. 3 Venn Diagrams and the Modern Square of Opposition “Existential Import” ◦ Does the statement imply the existence of the subject, or not? ◦ E. g. , All whales are mammals Universal (A and E) propositions can be interpreted in two different ways. An argument might be valid on one interpretation but invalid on the other

4. 3 Continued Aristotle held that universal propositions about existing things have existential import. All trees are things that have branches. ◦ Implies the existence of trees No VT students are UVA students. ◦ Implies the existence of VT students All mogwai are potential gremlins. ◦ Does not imply the existence of mogwai

4. 3 Continued George Boole held that no universal propositions have existential import. All trees are things that have branches. ◦ Does not imply the existence of trees No VT students are UVA students. ◦ Does not imply the existence of VT students All mogwai are potential gremlins. ◦ Does not imply the existence of mogwai

4. 3 Continued The Boolean and Aristotelian standpoints are identical in their treatment of particular (I and O) propositions: All particular propositions imply existence. Some alligators are reptiles. ◦ Implies that at least one alligator exists. Some unicorns are not reptiles. ◦ Implies that at least one unicorn exists.

4. 3 Continued All gods are immortals? No zebras are striped animals? Some computers are Macs? Some fairy tale creatures are not monsters?

4. 3 Continued John Venn (who perfected Boole’s theory) developed a system of diagrams to represent the information they express. ◦ ◦ All S are P = No members of S are outside P No S are P = No members of S are inside P Some S are P = At least one S exists that is a P Some S are not P = At least one S exists that is not a P

4. 3 Continued

4. 3 Continued Contradictory

4. 3 Continued � The Modern Square of Opposition: a relationship of mutually contradictory pairs of propositions.

4. 3 Continued “Immediate Inferences” have only one premise, which proceeds immediately to the conclusion. Some trade spies are not masters of bribery. Therefore it is false that all trade spies are masters of bribery. “Unconditionally Valid” arguments are valid from the Boolean standpoint, regardless of whether they refer to existing things.

4. 3 Continued Testing immediate inferences for validity: ◦ If the information expressed by the conclusion diagram is contained in the premise diagram, the argument is valid; if not, it is invalid. ◦ We diagram the following way: All A are B It is false that all A are B It is false that some A are B

4. 3 Continued 1. 2. 3. Some T are not M. Therefore, it is false that some T are M. It is false that all M are C. Therefore, no M are C. All S are W. Some S are W.

4. 3 Continued The Existential Fallacy is a formal fallacy that occurs whenever an argument is invalid merely because the premise lacks existential import. All A are B. Therefore some A are B. It is false that some A are not B. Therefore it is false that no A are B.

4. 4 Conversion, Obversion and Contraposition Conversion ◦ Subject and predicate switch places. No cats are dogs No dogs are cats. • Which types of statements are identical to their converse?

4. 4 Continued Obversion: ◦ Change quality (affirmative or negative) and replace the predicate with its term complement. All horses are animals No horses are non-animals. • Which types of statements are identical to their obverse?

4. 4 Continued Contraposition: ◦ Subject and predicate switch places and replace each with its term complement. All horses are animals all non-animals are nonhorses. • Which types of statements are identical to their contrapositive?

4. 4 Continued Exercise: Determine the converse, obverse, and contrapositive of each of the following categorical propositions. 1) 2) 3) 4) All humans are mammals. Some people are not students. Some candy bars are Snickers. No homo sapiens are reptiles.