Logic Test 2

Constructive Dilemma

(p > q) • (r > s) p v r / q v s

What steps are you supposed to follow in order to construct a truth table for an argument?

1. Symbolize the arguments using letters to represent the simple propositions. 2. Write out the symbolized argument, placing a coma between the premises and slash between the last premise and the conclusion. 3. Draw a truth table for the symbolized argument as if it were a proposition broken into parts, outlining the columns representing the premises and conclusion. 4. Look for a line in which all of the premises are true and the conclusion is false. If such a line exists, the argument is..invalid; if not, it is valid.

Natural Deduction

A proof procedure by which the conclusion of an argument is validly derived from the premises through the use of rules of inference.

Noncontingent Statement

A statement such that the truth values in the main operator column do not depend on the truth values of the component parts. -Tautology -Self-contradiction

Self-contradiction

A statement that is necessarily false.

Tautology

A statement that is necessarily true

Statement Variable

A statement variable p, q, r, s .... can stand for any statement, simple or complex

Truth Value

A statement's truth value is either true or false

Definition of Ponens

Affirming

Argument Form

An arrangement of logical operators and statement variables such that a substitution instance of statement variables results in an argument

Implication rules 2

Constructive Dilemma Simplification Conjunction Addition

Definition of Tollens

Denying

What is L=2n used for?

Determining the number of lines in a truth table

Fallacy of Denying the Antecedent

If P then Q Not P Therefore, not Q

Fallacy of Affirming the Consequent

If P then Q Q Therefore P

Hypothetical Syllogism

If P, then Q If Q, then R If P then R

Modus Tollens

If P, then Q Not Q Therefore, not P

Modus Ponens

If P, then Q P Therefore, Q

Propositional logic

Its basic elements are statements rather than classes. Provides precise definitions for sentences containing and, or, if, and only if. Captures much more of ordinary language than is possible with categorical logic.-Its basic elements are statements rather than classes. Unlike A, E, I, O, contains an unlimited number of complex statements.

Rules of inference

Justify the steps of the proof

Strategy 4

Look at the conclusion (Try working backward from the conclusion by imagining what the next to last line of the proof might be. Use this to help determine a short-term strategy to derive that line.)

Strategy 3

Look for conditionals (MP, MT, HS)

Strategy 2

Look for negation (MT, DS)

Definition of Modus

Method

Implication rules 1

Modus Ponens Modus Tollens Hypothetically Syllogism Disjunctive Syllogism

What does L stand for?

Number of lines?

When will a conditional be false?

Only when the antecedent is true and the consequent is false

Disjunction indicators

Or Unless Otherwise Either... or

Addition

P / P v Q

Conjunction

P Q / P• Q

Disjunctive Syllogism

P or Q Not P Therefore, Q

Rule 4 WFF

Parentheses, brackets, and braces are required in order to eliminate ambiguity in a complex statement.

What are the two types of rules of inference

Rules of implication and rules of replacements

Proof

Sequence of steps in which each step is either a premise or follows from earlier steps in the sequence according to the rules of inference.

Truth Tables

Show every possible truth value for compound propositions. Provide definitions of logical operators. Truth tables can determine the validity of arguments.

Strategy 1

Simplify and isolate (MP, MT, DS)

Contingent Statement

Statements that are neither necessarily true nor necessarily false (they are sometimes true, sometimes false)

Inconsistent Statements

Statements that do not have even one line on their respective truth tables where the main operators are both true.

Consistent Statements

Statements that have at least one line on their respective truth tables where the main operators are both true.

Logical operators

Symbols used in translations of ordinary language statements

Rule 1 of WFF

The dot, wedge, horseshoe, and triple bar symbols must go between two statements (either simple or compound)

What does n stand for?

The number of different simple propositions in the statement

What does the 2 represent in L=2n?

The number of truth values (true and false)

Rule 3 WFF

The tilde (~) cannot, by itself, go between two statements

Rule 2 WFF

The tilde (~)....goes in front of the statement it is meant to negate.

Truth Function

The truth value of a compound proposition is a function of the truth values of its component statements and the logical operators

Tatics

The use of small-scale maneuvers or devices

Contradictory Statements

Two statements that have opposite truth values on every line of their respective truth tables.

Strategy

Typically understood as referring to a greater overall goal

WFF

Well formed formula. Compound statement forms that are grammatically correct.

When will a conjunction be true?

When both conjucts are true

When will a disjunction be false?

When both disjuncts are false

When is a biconditional true?

When p and q have the same truth value

Logically Equivalent

When two truth-functional statements appear different but have identical truth tables,

Statement Form

a pattern of statement variables and logical operators~ ( p v q )

Conjunction indicators

and but also Moreover Still While However Although Yet Nevertheless Whereas

When is a proof valid?

each step is either a premise OR is validly derived using the rules of inference

Negation indicators

not; it is not the case that It is false that It is not true that

Simplification

p • q / p

Justification

refers to the rule of inference that is applied to every validly derived step in a proof.

When does a proof end?

when the conclusion of the argument has been derived