Categorical Propositions PHIL 121: Methods of Reasoning February 20, 2013 Instructor:Karin Howe Binghamton University

Four Types of Categorical Propositions Form Example A All S are P All cats are mammals E No S are P No cats are reptiles I Some S are P Some cats are long-haired cats O Some S are not P Some cats are not long-haired cats

Boolean Square of Opposition

What does it mean for a term to be distributed? "A term is distributed if it refers to an entire class, otherwise it is undistributed" (Copi, p. 110) Okay, but what does that mean? A term is distributed in a proposition if and only if the proposition is telling you something about that whole class of things

"All cats are mammals" (All S are P) This is telling you something about all cats (they’re all mammals), so the subject term ("cats") is distributed in this proposition However, it is not telling you anything about all mammals, so the predicate term is undistributed in this proposition Thus, in A propositions the subject term is distributed, but the predicate term is not.

"No cats are reptiles" (No S are P) This is telling you something about all cats (they're not reptiles), so the subject term ("cats") is distributed in this proposition Also, it is telling you anything about reptiles (namely, that they're not cats), so the predicate term isalso distributed in this proposition Thus, in E propositions both the subject term and the predicate term are distributed.

"Some cats are orange colored things" (Some S are P) This it is not telling you anything about all cats (it only tells you that some of them are orange colored), so the subject term is undistributed in this proposition Likewise, it is not telling you anything about all orange colored things (it only tells you that some of the orange colored things are cast), so the predicate term is also undistributed in this proposition Thus, in I propositions neither the subject term nor the predicate term are distributed.

"Some cats are not long-haired creatures" (Some S are not P) This it is not telling you anything about all cats (it only tells you that some of them don't have long hair), so the subject term is undistributed in this proposition However, this proposition is telling you something about the class of all long-haired creatures; namely, that none of them are the cats that are referred to in the statement. Thus, the predicate term is distributed in this proposition. Thus, in O propositions the subject term is undistributed, but the predicate term are distributed.

Transforming Categorical Propositions into Logically Equivalent Forms There are three operations that we can perform on the different types of categorical propositions: Conversion Obversion Contraposition

Relations between (pairs of) statements Logical equivalence: Two statements are logically equivalent if and only if they always have the same truth value Contradictory statements: Two statements are contradictory if and only if they always have opposite truth values (whenever one is true the other is false, and vice versa)

Immediate Inferences Copi does not explicitly define what an immediate inference is, but he seems to be using it to refer to any inference that can be drawn using one step of any of the transformations (converse, obverse, contraposition) Executive decision: we will only be talking about immediate inferences that preserve logical equivalence (e.g., we will not be allowing "conversion by limitation")

Complements of classes Complement of a class: "The complement of a class is the collection of all of the objects that do not belong to the original class" (Copi, p. 125) The complement of the class P is designated as as non-P E.g., the complement of the class of cats is non-cats

Important distinction Important distinction between "not P" and "non-P" Although these mean the same thing, in a certain sense (there's really no way to describe the complement other than to say it's the set of all things that are not P), we will be treating them as if they are different "not" will be treated as part of the structure of the statement, whereas "non" will be seen as attached to the term (part of the term) Consider the following pair of statements: Some cats are not orange-colored things. Some cats are non-orange-colored things. S1 is an O proposition, and S2 is an I proposition. S1 can be converted into S2 (and vice versa) via obversion.

Conversion Basic Schema: Switch subject and predicate terms, leave quantity and quality alone Example: No cats are dogs. Converse: No dogs are cats. Only valid for E and I propositions (convErsIon) Why isn't conversion valid for A propositions or O propositions? Consider the following statements: All cats are mammals. Some mammals are not cats.

Obversion Basic schema: Change the quality (change affirmative propositions to negative, or change negative propositions to affirmative) of the proposition, leaving subject and predicate classes where they are. Change the predicate term to its complement (i.e., change P to non-P, or change non-P to P) Valid for all types of propositions (A, E, I and O)

Contraposition Contraposition isn't really an immediate inference, although we will be treating it as such. Really a series of immediate inferences: obvert, convert, obvert Wherever this sequence of inference would be valid, we can use the following basic schema to generate the contrapositive as an immediate inference: Change both the subject and predicate terms into their complements and switch places, leave everything else alone (don't change the quality or quantity) Example: Some cats are not orange-colored things Contrapositive: Some non-orange-colored things are not non-cats

More on contraposition Only valid for A and O propositions (contrApOsition) Why isn't contraposition valid for E propositions or I propositions? Consider the following pair of statements: No cats are scaly things. Obvert: All cats are non-scaly things (A) Convert: stuck! Conversion is not valid for A propositions Some cats are furry things. Obvert: Some cats are not non-furry things (O) Convert: stuck! Conversion is not valid for O propositions

Let's Practice!  For each of the following statements, complete the following steps: Identify which kind of proposition the statement is (A, E, I, or O) Pick a letter to represent each term and identify the term as distributed or undistributed in the proposition Draw a Venn diagram representing the proposition, making sure to label the circles Derive the converse, obverse and contrapositive of the statement (wherever possible)

No parrots are pythons. Some zodiac signs are lucky signs. All kangaroos are things that can fly. Some jokes are not funny things. All scholars are nondegenerates. No pigs are animals of French descent. All banana splits are healthy desserts. Some poets are dead people.

All pigs are fantastic pets. All teachers are underpaid workers. Some musicians are not pianists. Some reptiles are not warm-blooded animals. All triangles are objects that have three sides. All UFOs are unidentified flying objects. No organic compounds are metals. All geniuses are nonconformists.