LOGIC the 6th Chapter

Truth Functions

Hypothesis: in compound sentence the truth or falsity of the whole compound is a function (is uniquely determined by) the truth or falsity (truth values) of its components sentences. Propositional rests on this claim. (*see tables in notes)

Statement Variables

Lowercase letters (p,q,r,s) that can stand for any statement. Used to construct statement forms.

Constructing a Truth Table

Symbolize the arguments using letters to represent the simple prepositions. Write out the symbolized argument, placing a single slash between the premises and a double slash between the last premise and the conclusion. 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. Look for a line in which all the premises are true and the conclusion false. If such a line exists the argument is invalid, if not its valid. pg. 322 for samples.

Connectives

Symbols used to connect or negate propositions in propositional logic.AKA Operators (pg 291 for table)

Truth table for a biconditional statement

a biconditional is true when its two components have the same truth value and that is otherwise it is false.

Propositional Logic

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

Sentential Connective

a piece of english with blanks (_____&______) such that when the blanks are filled with sentences that result in a new sentence. ex: The u.s. invaded iraq and bagdad fell _____ and ________

Conjunctive Statement

a statement having a dot as its main operator

Negation

a statement having a tilde as its main operator.

Biconditional Statement

a statement having a triple bar as its main operator

Disjunctive Statement

a statement having a wedge as its main operator

Logically True Statement

a statement is necessarily true (a tautology)

Simple Statement

a statement that does not contain any other statement or component (Ex: fast foods tend to be unhealthy)

Logically False Statement

a statement that is necessarily false, a self contradictory statement.

Contigent Statement

a statement that is neither necessarily true nor necessarily false.

Validity

an argument form is truth-functionally valid iff every valuation which makes all the premises true makes the conclusion the conclusion true. there is no valuation in which the presmises are all true and the conclusion false.

Statement Forms

an arrangement of statement variables and operators such that the uniform substitution of statements in place of the variables result in a statement. (see pg. 303 for example)

Truth Table

an arrangement of truth values that shows in every possible case how the truth value of a compound proposition is determined by the truth values of its simple components (see pg. 303 for examples)

Formally Valid

if the truth of the premises guarantees the truth of the conclusion and this guarantee is due to the logical form of the argument. Every argument of this form if the premises are true, the conclusion will be true.

Valuation

is an assignment of truth values to a set of sentence letters.

Truth Function

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

Truth Table for Disjunction

is true when at least one of the disjuncts is true and that otherwise is false. Logical Disjunction p q p ∨ q T T T T F T F T T F F F

Argument Form

is valid if and only if for every argument sharing the dorm the truth of the premises guarantees the truth of a conclusion. A particular argument is valid if and only if it is an instance of a valid form.

Compound Statement

one that contains at least one simple statement as a component (Ex: Dianne Reeves sings jazz and Christina Aguilera sings Pop)

Truth Table for Conjunction

shows how any statement having the form of a conjunction (p*q) is determined by the truth values of its conjuncts (p,q) Logical Conjunction p q p ∧ q T T T T F F F T F F F F

Truth Table for Negation

shows how any statement having the form of negation is determined by the truth value of the statement that is negated. Logical Negation p ¬p T F F T

Consistent Statement

statements for which there is at least one line on their truth tables in which all of them are true.

Contradictory Statement

statements that necessarily have opposite truth values.

Logically equivalent statement

statements that necessarily have the same truth value. having the same truth value on each line under their main operators.

Symbolic Logic

study of formally valid argument

Truth table of a Conditional Statement

that a conditional statement is false when the antecedent is true and the consequent false is true in all other cases. p q p → q T T T T F F F T T F F T

Main Operator

that operator in a compound statement that governs the largest components in the statement. (pg. 292-293 for examples)

Sufficient Condition

the condition represented by the antecedent (what comes after the IF) in a conditional statement

Necessary Condition

the condition represented by the consequent (What comes after the then) in a conditional statement

Material Equivalence

the relation expressed by a truth-functional biconditional

Material implication

the relation expressed by a truth-functional biconditional.

Logical Form (of a compound Sentence)

the result of replacing its simple sentences with schematic letters called sentence letters. What's leftover is a sentential connective