Logic Chapter 7 Terms

Testing Validity... the Rules: Rule #1

1) Middle Term must be distributed (If a term is distributed, it says something about every member if a class. •Rule of Thumb: (USNP) -(U)niversal Statements: (S)ubject (position) is distributed. -(N)egative statements: (P)redicate (position) is distributed. Is the middle term distributed?: If NOT, then the syllogism is invalid, and commits the fallacy of the Undistributed middle.

Testing Validity... the Rules: Rule #5

5) Cannot have two negative premises. Ex.: No valid arguments are completely negative. No completely negative arguments are good. So, no valid arguments are good.

Testing Validity... the Rules: Rule #6 (Only applies sometimes)

6) Cannot derive a particular conclusion from universal premises. Ex.: All zombies are green. No green things are putrid. So, some zombies are not putrid. -This rule follows from the Modern Square of Opposition. -This means that it applies ONLY in those cases in which the subject term does not exist.

Rules (2 regarding distribution, 2 regarding negation)

1) If a subject or predicate is distributed in the conclusion, it must be distributed in the premise. a) Fallacy of illicit major: when the major term (predicate of the conclusion) is distributed in the conclusion and not the major premise b) Fallacy of illicit minor: when the minor term (subject of the conclusion) is distributed in the conclusion and not the minor premise

Rules of Validity

1) In a valid syllogism, the middle term must be distributed in at least one of the premises. 2) If either of the terms in the conclusion is distributed, it must be distributed in the premise in which it occurs. 3) No valid syllogism can have two negative premises. 4) If either premise of a valid syllogism is negative, the conclusion must be negative, and if the conclusion is negative, one premise must be negative.

3 Types of Enthymemes

1st order: missing the major premise 2nd order: missing the minor premise 3rd order: missing the conclusion

Syllogism

2 premises and a conclusion

Testing Validity... the Rules: Rule #2

2) If a term is distributed in the conclusion, it MUST be distributed at least once in the premises. Ex.: All dogs are mammals. Some mammals are badgers. Some dogs are not badgers

Testing Validity... the Rules: Rule #3

3) Negative conclusion requires a negative premise; a negative premise requires a negative conclusion. Ex.: All badgers are wild. Some wild things are not furry. Some badgers are furry.

Testing Validity... the Rules: Rule #4

4) Cannot have two particular premises (follows from other rules). Ex.: Some penguins are black and white. Some old TV shows are black and white. So, some penguins are old TV shows.

Validity

If every premise in an argument is true then the conclusion, absolutely must be true. Only certain combinations of sentences types are valid: EIO is always valid no matter what in any square of reason.

Figures

Nailing down exactly where all the terms are in the syllogism (specifically the middle term) to determine if it is valid.

Major Term

Occurs in one premise and is the predicate of the conclusion.

Categorical syllogism

If an argument has two categorical premises and a categorical conclusion, and the two categorical premises jointly support the conclusion, then this argument is a categorical syllogism. Example: All mammals are animals that breathe by means of lungs. All whales are mammals. Therefore, all whales are animals that breathe by means of lungs.

Minor Term

Occurs in one premise and is the subject of the conclusion.

Categorical proposition

a statement that makes a straightforward assertion with no "ifs," "ands," or "buts." Categorical propositions are typically expressed by simply structured sentences containing a subject and a predicate, but not conjunctions or the other grammatical devices involved in more complex sentences.

Enthymemes

• A syllogism that is missing a premise or conclusion, but implies its missing part -- so its not a syllogism, you have to make it into a syllogism with the rules of validity. Ex.: Suppose someone infers that Jane's new car gets poor gas mileage because it has a V-8 engine. In standard form: (Any car with a V-8 engine gets poor gas mileage.) Jane's car is a car with a V-8 engine. Therefore, Jane's car gets poor gas mileage. We use parentheses around the major premise to indicate that it is stated implicitly.

Middle Term

The common term between the two premises that makes a logical bridge for the two premises to be connected in the conclusion

Immediate Inference: Conversion

The converse of a proposition is the result of switching its subject and predicate terms. Converse is legitimate only for E and I propositions. -No cats are dogs. -No dogs are cats.

Mood

The order of sentence types (AEIO) when a categorical syllogism is in standard form.

Terms of the proposition

The parts of the proposition that refer to classes. Two terms: the subject and the predicate, symbolized by S and P. Subjects and predicates are not always single words.

Distribution: Distribution of Predicate Terms

The predicate term is distributed if the proposition is negative (E or O) and undistributed if the proposition is affirmative (A or I). For the predicate term, what matters is quality. Quantity is irrelevant.

Immediate Inference

The square of opposition expresses the relationships between the four standard forms of categorical propositions. These are relationships of compatibility or incompatibility. If two propositions are related in such a way that the truth of one implies the truth of the other, they are equivalent from a logical standpoint.

Existential Import

The statement is E.I. when its truth depends on evidence for the existence of things in a certain category--in the case of categorical propositions, the existence of things in the categories signified by its subject and predicate terms.

Distribution: Distribution of Subject Terms

The subject term is distributed if the proposition is universal (A or E) and undistributed if the proposition is particular (I or O). For the subject term, what matters is quantity. Quality is irrelevant.

Using Venn-Diagrams to test Arguments

- Every categorical syllogism has three terms: a middle term that occurs only in the premises, and two end terms that occur in the conclusion. We call the two end terms Subject and Predicate terms, based on their positions in the conclusion. - Remember that SHADING MEANS A SECTION OF THE DIAGRAM IS EMPTY. - Diagram the PREMISES, starting with any UNIVERSAL premises. - Note whether the conclusion is represented by diagramming the premises.

Valid Aristotelian

All figures •(A)tt(a)(i)n Hern(a)ndo's tr(a)j(e)ct(o)ry •Figure 1: (A)tt(a)(i)n Hern(a)ndo's (AAI, EAO) •Figure 2: Hern(a)ndo's tr(a)j(e)ct(o)ry (EAO, AEO) •Figure 3: (A)tt(a)(i)n Hern(a)ndo (AAI, EAO) •Figure 4: (A)tt(a)(i)n Hern(a)ndo's tr(a)j(e)ct(o)ry (AAI, EAO, AEO) •Universal sentence types for premises (AE) •Insert existential import •End (conclusion) with particular sentence types (IO) •To due insertion of existential import

Square of Opposition: Contraries

An A proposition and an E proposition that have the same subject and predicate terms cannot both be true, but they could both be false. We identify this relationship in logic by calling A and E contrary propositions. "All S is P" and "No S is P" are contraries.

Distribution

Any term that makes a claim about the entire class of Ss is said to be distributed. The concept of distribution applies to the terms in all categorical propositions -- A, E, I, and O. We need to learn the rules for telling whether a given term is distributed or undistributed. Two warnings about distribution: 1. Unlike quality and quantity, distribution is not a feature of the proposition as a whole. It is a feature of its terms. 2. It is only when a term is used as the subject or the predicate of a proposition that it acquires a distribution, for only then is it being used to make a statement about all or some members of a given class.

Standard Form

Because there are two possible qualities and two possible quantities, there are just four standard logical forms for categorical propositions, no matter how complex their subject and predicate terms may be. UNIVERSAL: A: All S are P (affirmative) E: No S is P (negative) PARTICULAR I: Some S are P (affirmative) O: Some S are not P (negative)

Square of Opposition: Subalternates

Both A and I are affirmative propositions; they differ only in quantity. A is the more sweeping statement, because it makes a claim about all Ss -- that they are P. I is more cautious. When we say that some S are P, we are not committing ourselves to any claim about the whole class of Ss. *The concept of subalternates with true universals and the concept of subalternates with false particulars.

Major Premise, Minor Permise/Conclusion

Categorical syllogism standard form.

Immediate Inference: Contraposition

Formed by two steps: 1. Switching the subject and predicate terms, as in taking the converse 2. Replacing both the subject and the predicate terms with their complements The quality and quantity of the proposition remain as they were. Contraposition is not a legitimate operation for I and E propositions. The O proposition is the only one, besides the A, that is equivalent to its contrapositive. -All dogs are non-cats. -All cats are non-dogs.

Copula

In addition to the subject and predicate, there is a second component of categorical propositions, indicated by the words "is" or "are." It links S and P.

Square of Opposition: Contradictories

In some cases, there are propositions that cannot both be true and both be false.

Quality

In terms of classes, we can make both the affirmative statement that S is included in P and the negative statement that S is excluded from P. The affirmative or negative character of a proposition is called its quality.

Square of Opposition

In the system of Aristotelian logic, the square of opposition is a diagram representing the different ways in which each of the four propositions of the system is logically related ('opposed') to each of the others.

Square of Opposition: Subcontraries

Let's look at the propositions, "Some S are P" (I) and "Some S are not P" (O). Propositions I and O cannot both be false. This relationship is called subcontrary. Can I and O both be true? Yes--that happens quite often. For instance, some animals are mammals, some are not. However, I and O can never both be false. Any given object in the class of Ss must either be P or not be P. If it is P, that makes the I proposition true. If it is not P, that makes the O proposition true. For instance, any given animal either is a mammal or is not.

Quantity

The 4th component. A proposition with the form "All S are P" is universal, as is the proposition "No S are P." A proposition with the form "Some S are P" is particular. A proposition with the form "Some S are not P" is also particular.

Dealing with Non-Standard Forms

Very often, we'll find categorical arguments expressed in language that gives them more than 3 terms. - This is poses a challenge, since standard-form categorical syllogisms MUST have 3 terms. - We can fix this using operations like CONVERSION, OBVERSION and CONTRAPOSITION. Ex.: All photographers are non-writers. Some editors are writers. So, some non-photographers are not non-editors.

Immediate Inference: Obversion

We arrive at the obverse of a proposition by making two changes: 1. Replace the predicate term with its complement 2. Change the quality of the proposition (affirmative to negative or negative to affirmative)