PHIL 3 Midterm 2

Indirect Truth Tables: Steps

1. Assign F to the conclusion 2. Assign T to the premises and complete the table 3. If the argument allows for true premises and a false conclusion, then it is invalid. 4. If the argument is invalid, label it as invalid and place a check next to the line that proves it's invalid. If the argument is valid, label it as valid and indicate where a rule was broken for each line with a circle.

Truth Tables for Arguments: Steps

1. Set up the truth table 2. Follow order of operations and main logical operator procedures for each statement 3. Check for line(s) with true premises and a false conclusion. If no such lines exist, the argument is valid. 4. If the argument is invalid, place a checkmark on the line that has true premises and a false conclusion.

Statement Form

A pattern of statement variables and logical operators.

Argument Form

An arrangement of logical operators and statement variables in which a consistent replacement of the statement variables by statements results in an argument

Implication Rules II

Constructive Dilemma (CD) Simplification (Simp) Conjunction (Conj) Addition (Add)

Logical Properties of Statements

Contingent Noncontingent

Simplification

-Any uniform substitution using simple or compound statements of the argument form: -p . q p

Negation

-The truth table definition for negation shows that for any statement p, ~p will have the opposite truth value.

Implication Rules I

Modus Ponen (MP) Modus Tollens (MT) Hypothetical Syllogism (HS) Disjunctive Syllogism (DS)

Truth Functional Connectives

Negation Conjunction Disjunction Conditional Biconditional

Tautology

Statement that is necessarily true

Contingent Statements

Statements that are neither necessarily true, nor necessarily false

Rules of Replacement

Pairs of logically equivalent statement forms

Justification

Refers to a Rule of Inference that is applied to every validly derived step in a proof. Ensures that a line is a valid deduction of what comes before it.

Self-Contradiction

Statement that is necessarily false

Inconsistent Statements

Two (or more) statements that do not have even one line on their respective truth tables where the main operator for each statement is true at that line

Consistent Statements

Two (or more) statements that have at least one line on their respective tables where the main operator for each statement is true at that line

Contradictory

Two statements whose main operators have opposite truth values on every line

Truth Function

When the truth value of a compound proposition is determined by the truth values of its component statements and by the logical operators (connectives) Ex. "Obama is not president" • Whether this statement is true depends on the nature of the connective "not" and the truth value of statement "Obama is president" • This statement is said to be truth functional -In propositional logic, we'll only be concerned with truth-functional compound propositions • Where the truth value of the propositions are determined by the logical operator and the component statements

Logical Equivalence

When two (or more) truth-functional statements have identical truth tables (for the main operators), they are logically equivalent.

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 • Along with truth tables, natural deduction is another way of proving the validity of an argument • Any argument that can be proved valid in our system of natural deduction is indeed valid. -Proofs

Proof

-A 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 • A proof is valid if each step in the proof is either a premise or is correctly derived using the rules of inference -Justification -Rules of Inference

Truth Tables for Statements

-A truth table is a computation of the truth value of a statement or set of statements under each combination of truth values for its statement letters. -The main operator determines the truth value of the entire statement for each combination of truth values. -Box the main operator column. This column tells you the truth value of the statement for every possible assignment of truth values to statement letters.

Modus Ponen (MP)

-Any uniform substitution instance using simple or compound statements of an argument form -p u q p q

Addition

-Any uniform substitution instance using simple or compound statements of an argument form: -p p v q -Any statement can be added by a disjunction to a true simple statement and the resulting disjunction will be true.

Hypothetical Syllogism

-Any uniform substitution instance using simple or compound statements of the argument form: -p u q q u r p u r

Disjunctive Syllogism

-Any uniform substitution instance using simple or compound statements of the argument form: -p v q ~p q

Modus Tollens (MT)

-Any uniform substitution using simple or compound statements of the argument form. -p u q ~q ~p

Constructive Dilemma (CD)

-Any uniform substitution using simple or compound statements of the argument form: -(p u q) . (r u s) p v r q v s

Conjunction

-Any uniform substitution using simple or compound statements of the argument form: -p q p . q

Truth Tables

-Arrangement of truth values for a truth-functional compound statement -Shows every possible truth value for compound statement -Shows how truth value of a compound statement is determined by the truth values of its simple components -Provides definitions of logical operators

Inclusive Disjunction

-Both disjuncts can be true at the same time. -We deal mostly with inclusive disjunction. -Examples • Today is Monday or it is raining outside. • Joan went to the store or Jim went to store.

Exclusive Disjunction

-Both disjuncts cannot be true at the same time -Examples • Today is Monday, or today is Wednesday. • Triangles have exactly three sides or exactly four sides.

Truth Tables for Arguments

-Can determine the validity of arguments -If an argument is valid, its truth table will have no line on which the premises are true and the conclusion is false -If an argument is invalid, its truth table will have at least one line on which the premises are true and the conclusion is false

Natural Deduction and Validity

-For valid arguments, if the premises are true, the conclusion must be true -For natural deduction, whatever follows from a set of statements by means of valid inferences must be true if all the statements in the set are true

Rules of Inference

-Serve to justify the steps of a proof; ensures the validity of the steps they are used to justify -Rules of Implication -Rules of Replacement

Indirect Truth Tables: Strategies/Tips

-Start with the premise or conclusion that requires filling out the least number of lines. -Fill in the main-operator columns of the premises with Ts and the main operator column of the conclusion with Fs. -If you can't make the premises true and the conclusion false on a line without breaking a rule, move to the next line. If you can't make the premises true and the conclusion false on any line without breaking a rule, then the argument is valid. -If you come up with at least one line for which you can make the premises true and the conclusion false without breaking a rule, you've shown that the argument is invalid.

Noncontingent Statements

-Statements that are either necessarily true or necessarily false -Tautology -Self-Contradiction

Conjunction

-The truth table definition for conjunction shows that for any combination of truth values for p, q, "p . q" will have the following truth values. -A conjunction will be true only when both conjuncts is true (line 1)

Biconditional

-The truth table definition for the biconditional shows that for any combination of truth values for p, q, "p Ξ q" will have the following truth values: • A biconditional is true when p and q have the same truth value (lines 1 and 4).

Disjunction

-The truth table for (inclusive) disjunction shows that for any combination of truth values for p, q, "p u q" will have the following truth values: • A disjunction will be false only when both disjuncts are false (lines 1-3) -Inclusive Disjunction vs. Exclusive Disjunction

Conditional

-The truth table for the conditional shows that for any combination of truth values for p, q, "p u q" will have the following truth values: • A conditional will be false only when the antecedent is true and the consequent is false (line 2).

Rules of Implication

-Valid argument forms applied to an entire line -Implication Rules I and II

Indirect Truth Tables

-We do not complete the full truth table, but only those lines are necessary to determine if the argument is valid or invalid. -So we attempt to make the premises true and the conclusion false. If we succeed, the argument is invalid. If we don't, the argument is valid.

Statement Variables

p, q, r, s,... can stand for any statement (simple or complex)