Phil 210 - Final Exam Study Guide

Operators

Symbols used to connect simple propositions in propositional logic; truth-functional definitions of.

Symbols used for tilde, dot, wedge, horseshoe, and triple bar.

Connectives

Symbols used to connect or negate propositions in propositional logic.

The five logical operators symbolized

Tilde

This symbol is used to translate any negated simple proposition and is always placed in front of the proposition it negates. All of the other operators are placed between two propositions. This is the only operator that can immediately follow another operator.

Wedge

This symbol is used to translate “or” and “unless.” “Unless” is equivalent in meaning to “if not.”

Horseshoe

This symbol is used to translate “if…then..,” “only if,” and similar expressions that indicate a conditional statement. Also used to translate “implies.” Conditionals (Material Implications)

For truth table

Triple bar

This symbol is used to translate “if and only if.” Biconditional statement/Material equivalence. The triple-bar symbol is used to translate the expressions “if and only if” and “is a sufficient and necessary condition for”:

Examples:

JFK tightens security if and only if O’Hare does: J triple O.

JFK’s tightening security is a sufficient and necessary condition for O’Hare’s doing so: J triple bar O.

For truth table

Propositional Logic

A kind of logic in which the fundamental components are whole statements or propositions,

Simple Statements

Propositions that assert only one thing. A statement that does not contain any other statement as a component.

Examples:

Fast foods tend to be unhealthy.

James Joyce wrote Ulysses.

Parakeets are colorful birds.

The bluefin tuna is threatened with extinction.

It’s Friday!

Compound Statement

Contain at least one simple statement plus something else. A statement that contains at least one simple statement as a component.

Examples:

It is not the case that al-Qaeda is a humanitarian organization.

Translated: It is not the case that A.

Shorted: ~A

Dianne Reeves sings jazz, and Christina Aguilera sings pop.

Either people get serious about conservation or energy prices will skyrocket.

If nation spurn international law, then future wars are guaranteed.

The Broncos will win if and only if they run the ball.

Negation

Symbolized by the tilde operator. This symbol is used to translate “not” and “it is not the case that.”

Example:

It is not the case that A: ~A

For truth table

Conjunction

Symbolized by the dot as its main operator. Means “and, also, moreover.”

Example:

Tiffany sells jewelry and Gucci sells cologne: T dot G.

For truth table

Conjuncts

The component in a conjunctive statement on either side of the main operator.

Example:

D and C: D dot C

In a conjunctive statement, the components D and C are called this.

Disjunction

A statement having a wedge as its main operator.

Example:

Either Aspen sells snowboards or Teluride does: A v T.

For truth table

Disjuncts

The component in a disjunctive statement on either side of the main operator.

Example:

Either P or E: P v E.

This would be the P and E in the disjunctive statement.

Condtional Statement

Implication. Has a horseshoe as its main operator. Used to translate “if, then” and “only if”

Example:

If Virginia Tech raises tuition, then so does James Madison: T horseshoe J.

Material Implication

The relation expressed by a truth-functional conditional. A valid rule of inference that allows an implication sign to be replaced by a disjunction sign if and only if the Antecedent is negated. Conditional statement expresses the relation of this.

Antecedent

The statement that follows “if.”

Consequent

The statement that follows “only if.”

Biconditional

A statement having a triple bar as its main operator. Used to translate “if and only if.” Material equivalence.

Example:

JFK tightens security if and only if O’Hare does: J triple bar O.

Main Operator

The operator (connective) in a compound statement that has as its scope everything else in the statement.

Sufficient Condition

Event A is said to be a sufficient condition for event B whenever the occurrence of A is all that is required for the occurrence of B.

Examples:

Having the flu is a sufficient condition for feeling miserable.

Hilton’s opening a new hotel is a sufficient condition for Marriott’s doing so: H horseshoe M.

Necessary Condition

Event A is said to be a necessary condition for event B whenever B cannot occur without the occurrence of A.

Examples:

Having air to breathe is a necessary condition for survival.

Hilton’s opening a new hotel is a necessary condition for Mariott’s doing so: M horseshoe H.

Well-formed Formula

A syntactically correct arrangement of symbols.

1. Statements cannot be combined without having an operator between them.

   

2. A tilde cannot immediately follow a statement, but it can immediately precede one.

   Incorrect: (A v B)~ 

   Correct: ~(A v B)

   Correct: (A v B) ~ (C dot D)

3. A tilde cannot immediately precede another operator.

   

4. A sign for conjunction (dot), disjunction (wedge), implication (horseshoe), or Biconditional (triple bar) must go between conjoined statements.

   

5. Parentheses, brackets, and braces must be used to prevent ambiguity.

   

Truth Function

Any compound proposition whose truth value is completely determined by the truth values of its components.

Statement Variables

Lowercase letters that can stand for any statement (p, q, r, s) (p could stand for the statements A, A horseshoe B, B v C, and so on) (Uppercase letters stand for specific terms or statements, while lowercase letters are generic placeholders). Used to construct statement forms.

Statement Form

An arrangement of statement variables and operators such that the uniform substitution of statements in place of the variables results in a statement.

Example:

Not p and p horseshoe q are these because substituting the statements A and B in place of p and q, respectively, results in the statements notA and A horseshoe B.

Also, (p dot q) horseshoe (p v r) are these.

Truth Table

An arrangement of truth values that shows in every possible case how the truth values of a compound proposition is determined by the value of its simple components.

Inclusive Disjunction

The truth-fuinctional interpretation fo “or” is that of this. Cases in which the disjunction is true include the case when both disjuncts are true. This inclusive sense of “or” corresponds to many instances of ordinary usage, as the following example illustrates:

Either Steven King or Cate Blanchett is a novelist: S v C

The first statement is true because in each case at least one of the disjuncts is true.

Exclusive Disjunction

Examples:

The Orient Express is on either track A or track B.

You can have either soup or salad with this meal.

Tammy is either ten or eleven years old.

The sense of these statements excludes the possibility of both alternatives being true.

Thus, if these statements were translated using the wedge, a portion of their ordinary meaning would be lose.

Material Conditional

Conditonal. The truth table shows that a conditional statement is false when the Antecedent is true and the consequent false eand is true in all other cases. This truth-functional interpretation fo conditional statements conforms in part with the ordinary meaning of “if, then” and in part it diverges.

Material Equivalence

Biconditional. The truth table shows that the biconditional is true when its two components have the same truth values and that otherwise it is false.

p triple bar q is simply a shorter way of writing (p horseshoe q) dot (q horseshoe p).

If p and q are either both true or both false, then p horseshoe q and q horseshoe p are both true, making their conjunction true. But if p is true and q is false, then p horseshoe q is false, making the conjunction false. Similarly, if p is false and q is true, then q horseshoe p is false, again making the conjunction false. Thus, is true when p and q have the same truth value and false when they have opposite truth values.

Logically True

A logically true statement is true regardless of the truth values of its components.

Tautologous

If all of the lines under the main operator are filled with Ts, then the statement as whole is autologous. A tautologous statement is true regardless of the truth values of its components.

Logically False

False regardless of the truth values of its components.

Self-contradictory

If all the lines under the main operators are filled with Fs, then the statement as a whole is self-contradictory. A self-contradictory statement is false regardless of the truth values of its components.

Logically Contingent Statement

If the lines under the main operator contain both Ts and Fs, then the statement is “contingent.” Only contingent statements have truth values that are dependent on t eh truth values of their components.

Logically Equivalent Statements

Have the same truth values on each line under their main operators.

Logically Contradicatory Statements

Have the opposite truth truth on each line under their main operators.

Consistent Statements

Have at least one line on which the truth values of both statements are true. It is possible for both propositions to be true at the same time.

Inconsistent Statements

Have no line on which the truth values of both statements are true. It is impossible for both propositions to be true at the same time.

Know the four steps in constructing truth tables for arguments

1. Convert the argument to symbolic form using letters for simple propositions

2. Write out the symbolized argument on a single line, using a slash between premises and a double slash between the last premise and the conclusion.

3. Draw a truth table for the symbolized argument as if it were a long proposition. 

4. Look for a line where the premises are true and the conclusion false. If there is no such line, then the argument is valid. If there is even one such line, then it is invalid. 

Indirect Truth Table

A quicker way to test the validity of arguments and the consistency of sets of multiple propositions. They are “indirect” in that we are attempting to infer validity or consistency from our inability to find any inconsistency. They are quicker than traditional truth tables, but they require the ability to work backward from the truth value of the main operator to the truth values of the other components.

Steps in Testing Arguments for Validity

1. Write the argument out on a single line. 

2. Label the premises T and the conclusion F (thus making the argument invalid). 

3. Work backward to logically derive the truth values of the separate components. 

4. If you find a contradiction in any premise, then it is not possible for the premises to be true and the conclusion false, so the argument must be valid.

   Testing Statements for Consistency is similar to the method for testing arguments:

   • Write the statements on a single line, separated from each other by slash marks. 

   • Assign the value T to each main operator. 

   • Compute the truth value of the other components. 

   • If there is no contradiction, then the statements are consistent.

Disjunctive Syllogism (DS)

Valid argument. A syllogism that consists of a disjunctive statement and another statement that negates the left-hand disjunct.

Pure Hypothetical Syllogism (HS)

Valid argument. The premises link together like a chain.

Modus Ponens (MP) (“the way of affirmation”)

Valid argument.

Modus Tollens (MT) (“the way of denial”)

Valid argument.

Constructive Dilemma (CD)

A valid argument form that consists of a conjunctive premise made up of two conditional statements, a disjunctive premise that asserts the antecedents in the conjunctive premise (like modus ponens), and a disjunctive conclusion that asserts the consequents of the conjunctive premise. It is defined as follows:

Destructive Dilemma

A valid argument form. It is similar to the constructive dilemma in that it includes a conjunctive premise made up of two conditional statements and a disjunctive premise. However, the disjunctive premise denies the consequents of the conditionals (like modus tollens), and the conclusion denies the antecedents:

Grasping the horns of the dilemma

Prove the conjunctive premise false by proving either conjunct false.

Escape between the horns of the dilemma

Proving the disjunctive premise false.

Counter-dilemma

Typically done by changing either the antecedents or the consequents of the conjunctive premise while leaving the disjunctive premise as it is, so as to obtain a different conclusion.

Constructing a counterdilemma falls short of a refutation of a given dilemma because it merely shows that a different approach can be taken to a certain problem. It does not cast any doubt on the soundness of the original dilemma. Yet the strategy is often effective because it testifies to the cleverness of the debater who can accomplish it successfully. In the heat of debate the attending audience is often persuaded that the original argument has been thoroughly demolished.

Affirming a Disjunct

Invalid argument form.

p v q

q

~p

Affirming the Consequent

Invalid argument form.

Denying the Antecedent

Invalid argument form.