Intermediate Logic Lessons 16-17, Intermediate Logic Unit 1, Intermediate Logic-Lessons 10-14

Proposition

A statement

9 rules of inference

1. Modus Ponens 6. Conjunction (Conj.) 2. Modus Tollens 7. Absorption (Abs.) 3. Hypothetical Syllogism 8. Simplification (Simp.) 4. Disjunctive Syllogism 9. Addition (Add.) 5. Constructive Dilemma

The Truth Table Method for Validity (Step Four)

Remove any unnecessary columns of T and F, leaving only the columns for the premises and conclusion.

A defining truth table is

a display of the truth values produced by a logical operator modifying a minimum number of variables.

formal proof of validity

a formal series of steps which deduce the argument's conclusion from its premise(s)...can ONLY prove validity

A truth table is

a listing of the possible truth values for a set of one or more propositions.

Conjunction ('and') is

a logical operator that joins two propositions and is true if and only if both the propositions (conjuncts) are true.

Disjunction (v 'or') is

a logical operator that joins two propositions and is true if and only if one or both of the propositions (disjuncts) are true.

A propositional variable is

a lowercase letter that represents any proposition.

A propositional constant or variable can represent

a simple or compound proposition.

The validity of most arguments can be determined with

a truth table having only one row.

Rule of inference

a valid argument form which can be used to justify steps in a proof.

Dilemma

a valid argument which presents a choice between two conditionals...often used to trap an opponent in debate.

Though in English grammar the word 'or' is called a conjunction, in logic it is

always called a disjunction.

Destructive dilemma

an argument that follows this form: works like "modus tollens."

Constructive dilemma

an argument that follows this form: works like an extended "modus ponens."

A propositional constant is

an uppercase letter that represents a single, given proposition.

Conditional Propositions are false if

and only if the antecedent is true and the consequent is false.

If the biconditional of two statements is a self- contradiction, then the statements

are contradictory.

To test equivalence using shorter truth tables...

assume the two propositions are not logically equivalent, then check to see if that leads to an unavoidable contradiction.

One proposition may be expressed

by many different sentances

The conditional includes many types of implications:

cause/effect, definition, promises, conditions, and so on.

After determining the truth values for negations,

complete the truth values for compound propositions within parentheses.

The biconditional can be used to test for

contradiction.

"go between the horns"

denying the disjunction and providing a third alternative

Assume that the conclusion is

false and the premises are true, then work backwards looking for unavoidable contradictions.

Finish the truth tables for a compound proposition by

finding the truth tables for all of its component parts and then putting them together.

Two Propositions are logically equivalent

if and only if they have identical truth values.

If p then q is equivalent to

if not q then not p

p unless q means

if not q then p

p is sufficient for q means

if p then q

p only if q means

if p then q

p is a necessary condition for q means

if q then p

For an argument to be valid,

if the premises are true, the conclusion must be also.

The logical operator for disjunction is always understood in the

inclusive sense : "this or that, or both." If you intend the exclusive, you must specify explicitly.

If a proposition has only one component part, then

it is a simple proposition. Otherwise, it's compound.

A self- contradiction is always false due to its

logical structure.

A tautology is a proposition that is always true due to its

logical structure.

Conditions can take

many other forms than the traditional if/then form.

Three fundamental logical operators are

negation, conjunction, and disjunction

Do not confuse

not both and both not. Use parentheses to distinguish between them.

Without parentheses, assume that negation

only attaches to the proposition it immediately precedes.

Addition

p Therefore p v q

Conjunction

p q p ^ q

Tautology (Taut.)

p = (p ∨ p) p = (p • p)

Double negation (DN)

p = ~~p

Absorption

p > q Therefore, p > (p ^ q)

Simplification

p ^ q Therefore, p

Disjunctive Syllogism

p v q ~p Therefore q

Distribution (dist.)

p • (q ∨ r) = (p • q) ∨ (p • r) p ∨ (q • r) = (p ∨ q) • (p ∨ r)

Association (Assoc.)

p ∨ (q ∨ r) = (p ∨ q) ∨ r p • (q • r) = (p • q) • r

Generally, in a series of three or more connected propositions,

parentheses should be used to avoid ambiguity.

"rebut the horns"

providing a counter-dilemma

"grasp it by the horns"

rejecting one of the conditionals in the conjunctive premise

The symbol (:.) means "therefore" and

signals the conclusion of the argument.

The Conditional Operator (> "if/then") asserts

that the antecedent implies the consequent.

The conditional is always true if

the antecedent is false.

If a truth table shows at least one row in which the premises of an argument are true but the conclusion is false,

the argument is invalid. No exceptions. Otherwise, it is valid.

Negation (~ 'not') is

the logical operator that denies or contradicts a proposition.

When completing a truth table, start with

the standard truth values for the variables (or constants), then find the truth values for the negated variables (or constants).

The biconditional can be used to test equivalence. If the biconditional of two statements is a tautology

the the statements are equivalent.

When a shorter truth table is completed for an invalid argument,

the truth values found for the variables (or constants)are the same truth values from a row showing the argument to be invalid on the longer truth table.

If the premises are true and the conclusion is false,

then the argument is invalid.

The Biconditional Operator (=, "if and only if") is true

when both component propositions have the same truth value, and is false when their truth values differ.

Logical Operators are

words which combine or modify simple propositions to make compound propositions.

The Truth Table Method for Validity (Step One)

write the argument in symbolic form on top of a line.

When you make a truth table for propositions that only use constants with known truth values,

you only need one row.

The conditional p>q is always equivalent to

~(p {conjunct} ~q) by definition.

De Morgan's Theorem (De M.)

~(p • q) = (~p ∨ ~q) ~(p ∨ q) = (~p • ~q)

Consistent propositions

Propositions are consistent when assuming them all to be true involves no contradiction.

Material Implication (lmpl.)

( p ⸧ q) = (~ p ∨ q)

Constructive Dilemma

(p > q) ^ (r > s) p v r Therefore, q v s

Commutation (com.)

(p ∨ q) = (q ∨ p) (p • q) = (q • p)

Material Equivalence (equiv.)

(p ≡ q) = [(p ⸧ q) • (q ⸧ p)] (p ≡ q) = [(p • q) ∨ (~p • ~q)]

Transposition (trans.)

(p ⸧ q) = (~q ⸧ ~p)

3 main ways to escape the horns of a dilemma (facing a dilemma.)

1. go between the horns 2. grasp it by the horns 3. rebut the horns

Shorter Truth Table Method of Validity (Step Two)

Assume the argument is invalid by assigning the premises the value T and the conclusion the value F

The Truth table Method for Validity (Step Three)

Determine the columns of T and F for the propositions following the defining truth tables.

The Truth Table Method for Validity (Step Five)

Examine the rows. If any row has all true premises and a false conclusion, the argument is invalid. Otherwise it is valid. Mark the row(s) showing validity or invalidity.

Rules of Replacement

Forms of equivalent propositions that can be used to justify steps in proofs (Can only be used with conjunctions and disjunctions)

Shorter Truth Table Method of Validity (Step Five)

If a contradiction is unavoidable, then the original assumption of invalidity is false, and the argument is valid.

Hypothetical Syllogism

If p then q If q then r therefore If p then r

Modus Ponens

If p then q p Therefore q

Modus Tollens

If p then q ~q Therefore ~p

Shorter Truth Table Method of Validity (Step Four)

If the truth values are completed without contradiction, then the argument is invalid as assumed.

A proposition is Truth-Functional when

Its truth value depends upon the truth values of its component parts.

The Truth Table Method for Validity (Step Two)

Under the variables, place the columns of T and F.

Shorter Truth Table Method of Validity (Step Three)

Work backwards along the argument, determining the remaining truth values to be T or F as necessary, avoiding contradiction if possible.

Shorter Truth Table Method of Validity (Step One)

Write the argument is symbolic form on a line

Exportation (exp.)

[(p • q) ⸧ r] = [p ⸧ (q ⸧ r)]

Propositional Logic

a branch of symbolic logic dealing with propositions as units and with their combinations and the connectives that relate them