UConn Philosophy 1102: Rossberg - Final Exam

Categorical Statements

a statement that relates two classes or categories

Standard Forms of Categorical Statements

A, E, I, O

Standard form A

All S are P; universal affirmative

Standard form E

No S are P; universal negative

Standard form I

Some S are P; particular affirmative

Standard form O

Some S are not P; particular negative

Quality

Affirmative or negative nature of a statement

Quantity

Universal or particular nature of a statement

Putting Categorical Statements into Standard Form

1) When its predicate is an adjective, add an appropriate noun

"All humans are rational" → "All humans are rational animals/things"

2) Elements are present but not in right order, rearrange the elements

"Rubies are all gems" → "All rubes are gems"

3) Verb other than "are" shifted into predicate

"All fish swim" → "All fish are swimmers"

"All criminals should be punished" → "All criminals are people who should be punished"

"All workers were tired" → "All workers are people who were tired"

"No persons who confess will be prosecuted" → "No persons who confess are persons who will be prosecuted"

Immediate Inference

a conclusion is drawn from only one premise

Corresponding statements

categorical statements having the same subject term and the same predicate term

Contradictories

two statements that cannot both be true and they cannot both be false

Contraries

Two statements that cannot both be true, but can both be false

Necessary truth

If a statement is true and cannot be false under any possible circumstances

Subcontraries

two statements that cannot both be false but they can both be true

Necessary falsehood

it is false and it cannot be true in any possible circumstances

Subalternation

the logical relationship between a universal statement and its corresponding particular statement

Superaltern

universal statement (in regard to itself and it's particular statement)

Subaltern

particular statement (in regard to itself and its universal statement)

Converse

formed by interchanging its subject and predicate terms

Conversion

the inference from a categorical statement to its converse

Logically equivalent

two statements that each validly imply the other

Conversion by limitation

the inference in which we switch the subject and predicate terms of an A statement and change the quantity from universal to particular

Complement

the class containing all things that are not a member of X

Term-complement

the word or phrase that denotes the class complement

Obverse

formed by (a) changing its quality and (b) replacing the predicate with its term-complement

Obversion

the inference from a categorical statement to its obverse

Every standard-form categorical statement is logically equivalent to its this

Contrapositive

(a) replacing its subject term with the term-complement of its predicate term and

(b) replacing the predicate term with the term-complement of its subject term

Contraposition

the inference from a categorical statement to its contrapositive

Contraposition by limitation

(a) replace the subject term of an E statement with the term-complement of the predicate term

(b) replace the predicate term with the term-complement of the subject term

(c) change the quantity from universal to particular

General pattern of inference in contraposition by limitation

1) No S are P

So, 2) Some non-P are not non-S

Categorical syllogisms

Have exactly 3 terms: 2 premises and 1 conclusion

Middle term

the term that occurs once in each premise

Major term

the predicate term of the conclusion

Minor term

subject term of the conclusion

Standard form of Categorical Syllogisms

The premises and the conclusion are categorical statements in standard form ("All S are P," "No S are P", "Some S are P", or "Some S are not P").

The first premise contains the major term

The second premise contains the minor term

The conclusion is stated last

Major premise

contains the major term; in standard form, it's the first premise

Minor premise

contains the minor term; in standard form, it's the second premise

Logical form of a categorical syllogism

determined by its mood and figure

Shading

indicates no items in this area

Marking an X

indicates at least one ("some") items in area

Single premise arguments

1) Diagram the premise

2) Determine whether diagram infers conclusion is true

Let first circle (right) be subject term and second be (left) predicate

Syllogism contains both universal and particular premises

Diagram universal first in case of universal and particular statements

Existential import

categorical statements are these if they imply that their subject terms denote nonempty classes

Modern Square of Opposition

Only diagonal relationships (contradictories)

Enthymemes

unstated premises or conclusion

Evaluating enthymematic categorical syllogism for validity:

1. Identify the missing step (conclusion, major premise, minor premise, etc)

2. Put the syllogism into standard form

3. Apply the Venn method

Rule with removing term-complements (via conversion, obversion, or contraposition)

Can remove term-complements as long as the changes made to each statement produce a logically equivalent statement

Distributed

A term is ___ in a statement if the statement says something about every member of the class that the subject term denotes

Undistributed

A term is ____ in a statement if the statement does not say something about every member of the class the term denotes

Rules for evaluating syllogisms

Rule 1: A valid standard-form categorical syllogism must contain exactly three terms, and each term must be used with the same meaning throughout the argument

Rule 2: In a valid standard-form categorical syllogism, the middle must be distributed in at least one premise

Rule 3: In a valid standard-form categorical syllogism, a term must be distributed in the premises if it is distributed in the conclusion

Rule 4: In a valid standard-form, categorical syllogism, the number of negative premises must be equal to the number of negative conclusions

Rule 5: No valid standard-form categorical syllogism with a particular conclusion can have two universal premises

Atomic statement

one that does not have any other statement as a component

Compound statement

One that has at least one atomic statement as a component

Negations

Tilde (~)

Main logical operator

the one that governs the largest component or components of a compound statement

Minor logical operator

governs smaller components (in a compound statement)

Conjunctions

Dot (*)

Disjunctions

vee (v)

Conditionals

Arrow (-->)

Biconditionals

Double arrow (<->)

Well-formed formula (WFF)

a grammatically correct symbolic expression

Statement variable

a lowercase letter that serves as a placeholder for any statement (p, q, r, s)

Truth-functional

A compound statement whose truth value is completely determined by the truth value of the atomic statements that compose it

Negations (~)

Always the opposite

Conjunctions (dot)

True if both its constituents are true; otherwise false

Disjunctions (v)

False if both its constituents are false; otherwise it is true

Conditional

An English ____ is always false when its antecedent is true and its consequent is false

Material conditional

False only when its antecedent is true and its consequent is false; otherwise, it is true

Material biconditional

Conjunction of two material conditionals; it is true when its constituent statements have the same truth value and false when they differ in truth

Summary of truth-table method

1. Assign truth values mechanically

Place the capital letters of atomic statements in sequence from left to right in the order that they appear in our symbolization

The number of rows for atomic statements that you need is 2^n, where n is the number of atomic statements

Start assigning truth values to atomic statements in columns by first assigning truth values to the far right statement: alternate Ts and Fs in the column beneath it. The next column to the left: alternate pairs of Ts and Fs. The next column to the left: alternate quadruples of Ts and Fs. The next column to the left: alternate groups of eight, and so on (doubling)

2. Identify the main logical operator of each premise and the conclusion

3. In the case of complex compound statements, work out the truth values of simpler compounds first, then work your way "outward" to the main logical operator

4. Look for a row where all the premises are true and the conclusion is false. Assuming you've done everything correctly up to this point, if there is one, the argument is invalid; if not, it's valid

Abbreviated Truth tables method

Making all the premises true while the conclusion is false

Hypothesize that there is such a row and then to confirm the hypothesis, thereby showing that the argument is invalid, or disconfirm it, thereby showing that the argument is valid

Work backwards, main logical operators and conclusion

/ (have to assign both a true and false value) 326

Principles of inferring validity

Principle 1: If there is any assignment of values in which the premises are all true and the conclusion is false, then the argument is invalid

Principle 2: If more than one assignment of truth values will make the conclusion false, then consider each such assignment; if each assignment that makes the conclusion false makes at least one premise false, then the argument is valid

Summary of Abbreviated Truth table method

1. After placing the argument in a truth table, determine whether there are multiple ways in which the conclusion can be false

2. If there is just one way, place an F under the (main operator of the) conclusion and a T under (the main operator of) each premise

(a). To show invalidity, uniformly assign Ts and Fs to all of the components of the conclusion and the premises; write Ts and Fs under the atomic statements on the left of the table

(b). To show validity, uniformly assign Ts and Fs to all of the components of the conclusion and the premises; write a backslash under (the main operator of) the premise you were led to say was both true and false. Do not write Ts and Fs under the atomic statements on the left

3. If there is more than one way for the conclusion to be false, place one F under the (main operator of the) conclusion for each way it can be false, thereby creating as many rows as there are ways for the conclusion to be false. On each row, place a T under (the main operator of) each premise

To show invalidity, follow instruction 2a for at least one row

To show validity, follow instruction 2b for every row

Tautology

a statement is a ____ if and only if it is true on every assignment of truth values to its atomic components

Contradiction

a statement is a ____ if and only if it is false on every assignment of truth values to its atomic components

Contingent

A statement is ____ if and only if it is true on some assignments of truth values to its atomic components and false on others

Logically equivalent

Two statements if and only if they agree in truth value on every assignment of truth values to their atomic components

Logically contradictory

Two statements if and only if they disagree in truth value on every assignment of truth values to their atomic components

Logically consistent

two (or more) statements are _______ ______ if and only if they are both (all) true on some assignment of truth values to their atomic components

Logically inconsistent

two (or more) statements are _____ _____ if and only if they are never both (all) true on any assignment of truth values to their atomic components