PROPOSITIONAL LOGIC

LOGICAL CONNECTIVES

Logical connectives (also called “logical operators”) are symbols or words used to connect two or more sentences (of either a formal or a natural language) in a grammatically valid way, such that the value of the compound sentence produced depends only on that of the original sentences and on the meaning of the connective. There are five common logical connectives: and, or, not, if-then, if and only if.

Conjunction (and)

P & Q

Sentence letters are called “conjuncts”; the two conjuncts can be reversed and retain the original meaning.

Disjunction (or)

P v Q

Sentences letters are called “disjuncts”; the two disjuncts can be reversed and retain the original meaning. The disjunction is always understood inclusively as an or/and; that is, P might obtain, Q might obtain, or both P and Q might obtain. Thus, the disjunction is never taken to be exclusive, where either P obtains or Q obtains but both cannot obtain.

Negation (not)

~P

For clarity, the negation symbol is often read as “it is not the case that”.

Conditional (if-then)

P → Q

First sentence letter is called “antecedent” and second is called “consequent”. The antecedent and consequent cannot be reversed and still retain its original meaning.

Necessary-sufficient conditions

P (sufficient condition) → Q (necessary condition)

Example: if it rains, then the sidewalk will get wet

P is a sufficient condition for Q

Raining is a sufficient condition for the sidewalk being wet

P is sufficient for bringing about Q, but is not the only thing that brings about Q)

Q is a necessary condition for P

The sidewalk being wet is a necessary condition of it raining

Q is a necessary result of P in every case in which P arises

If / only if

P only if Q

It will be raining only if the sidewalk gets wet

P occurs only if, of necessity, Q also occurs

Q if P

The sidewalk will get wet if it rains

Q occurs if P first occurs

Biconditional (if and only if)

The biconditional combines a conditional relation between P and Q and its reverse. Sentence letters are called “biconditions”; the two biconditions can be reversed and retain the original meaning.

P ↔ Q

(P→Q) and (Q→P)

P is a necessary and sufficient condition for Q

Example: Bob’s being a bachelor is both a necessary and sufficient condition for Bob being an unmarried man

P is a sufficient condition for Q (Q is a necessary condition for P)

P is a necessary condition for Q (Q is a sufficient condition for P)

P if and only if Q

Example: Bob is a bachelor if and only if Bob is an unmarried man

P if Q (if Q then P)

P only if Q (if P then Q)

WELL FORMED FORMULAS (WFFS)

Vocabulary of the language of propositional logic

Sentence letters

Logical operators

Brackets

Well-formed formulas (wff): a meaningful symbolic proposition

Wff Formation rules

1. Any sentence letter is a wff

2. If P is a wff, then so is ~P

3. If P and Q are wwfs, then so are (P & Q), (P v Q), (P → Q), (P ↔ Q)

Atomic wwfs: sentence letters

Complex wwfs (compound, molecular): built up from simple ones by repeated application of the formation rules

Sub wff: a part of a wff that is a wff itself

TRUTH TABLES FOR LOGICAL OPERATORS

Terms

Truth-functional operators: logical connectives such as “and”, “or”, “not”, “if-then” that have a consistent truth value.

Truth value: the truth and falsity of a statement

Principle of bivalence: true and false are the only truth values and in every possible situation each statement has one and only one of them

Truth table: a summary of the truth value of a wff

Truth tables of logical operators

Negation: ~p

Hint: the truth value of the negated proposition is opposite that of the original proposition.

Example: “*it is not
the case that* Bob is here”

__p | ~ p__

T | F

F | T

Conjunction: p & q

Hint: for the conjunction to be true, both conjunts need to be true.

Example: “Bob is here *and*
Joe is here”

__p q |
p & q__

T T | T

T F | F

F T | F

F F | F

Disjunction: p v q

Hint: for the disjunction to be true, either disjunct or both disjuncts need to be true (remember that “or” is inclusive).

Example: “I will eat an
apple *or* I will eat a banana”

__p q |
p v q__

T T | T

T F | T

F T | T

F F | F

Conditional: p → q

Hint: there no easy hint here, but remember that the consequent is the necessary condition in the conditional.

Example: “*If* it
rains *then* the sidewalk will be wet”

__p q |
p → q__

T T | T

T F | F

F T | T

F F | T

Biconditional: p ↔ q

Hint: for the biconditional to be true, both simple propositions need to have the same truth value.

Example: “Bob is a
bachelor *if and only if (iff)* Bob is an unmarried man”

__p q |
p ↔ q__

T T | T

T F | F

F T | F

F F | T

TRUTH TABLES FOR COMPLEX WFFS

Rules for truth tables of complex wffs

The column for any wff or subwff is always written under its main operator;

Circle the column under the main operator of the entire wff to show that the entries in it are the truth values for the whole formula

Truth tables for complex wwfs:

Find the truth values for the smallest subwffs and then use the truth tables for the logical operators to calculate values for increasingly larger subffs, until you obtain the values for the whole wff

Example: (p & q) v r

__ __

__p q r |
p & q | (p & q) v r__

T T T | T | T

T T F | T | T

T F T | F | T

T F F | F | F

F T T | F | T

F T F | F | F

F F T | F | T

F F F | F | F

Example: (p & q) v (p & r)

__p q r |
p & q | p & r | (p & q) v (p & r)__

T T T | T | T | T

T T F | T | F | T

T F T | F | T | T

T F F | F | F | F

F T T | F | F | F

F T F | F | F | F

F F T | F | F | F

F F F | F | F | F

TAUTOLOGIES, INCONSISTENCIES, AND CONTINGENCIES

Tautology: each line is true under the main operator

Example: “Bob is here or it is not the case that Bob is here”

__p | ~ p |
p v ~p__

T | F | T

F | T | T

Inconsistency: each line is false under the main operator

Example” “Bob is here and it is not the case that Bob is here”

__p | ~ p |
p & ~p__

T | F | F

F | T | F

Contingency: some lines are true, others false, under the main operator

Example: “it is not the case that Bob is here”

__p | ~ p__

T | F

F | T

DEMONSTRATING LOGICAL EQUIVALENCE OF WFFS

If two different wffs have the same truth value in their truth-tables, then they are logically equivalent

Example: logical equivalence of “p → q” and “~q → ~p”

“p → q” (“if it rains, then the sidewalk is wet”)

__p q |
p → q__

T T | T

T F | F

F T | T

F F | T

“~q → ~p” (“if it is not the case that the sidewalk is wet then it is not the case that it is raining”)

__p q |
~q | ~p | ~q → ~p__

T T | F | F | T

T F | T | F | F

F T | F | T | T

F F | T | T | T

Example: logical equivalence of “~(p & ~p)” and “~p v q”

“~(p & ~p)” “It is not the case that (it is raining and it is not the case that the sidewalk is wet)”

__p q |
~q | p & ~q | ~(p & ~q)__

T T | F | F | T

T F | T | T | F

F T | F | F | T

F F | T | F | T

“~p v q” “it is not the case that it is raining or the sidewalk is wet”

__p q |
~p | ~p v q__

T T | F | T

T F | F | F

F T | T | T

F F | T | T

Example: logical equivalence of “p ↔ q” and “(p → q) & (q → p)” and “(p & q ) v (~p & ~q)”

“p ↔ q” “Bob is a bachelor iff Bob is an unmarried man”

__p q |
p ↔ q__

T T | T

T F | F

F T | F

F F | T

“(p → q) & (q → p)” “(If Bob is a bachelor then Bob is an unmarried man) and (If Bob is an unmarried man then Bob is a bachelor)”

__p q |
p→q | q→p | (p→q)&(q→p)__

T T | T | T | T

T F | F | T | F

F T | T | F | F

F F | T | T | T

“(p & q ) v (~p & ~q)” “(Bob is a bachelor and Bob is an unmarried man) or (it is not the case that Bob is a bachelor and it is not the case that Bob is an unmarried man)”

__p q |
~p | ~q | p&q | ~p&~q | (p&q)v(~p&~q)__

T T | F | F | T | F | T

T F | F | T | F | F | F

F T | T | F | F | F | F

F F | T | T | F | T | T

ARGUMENT FORMS

Terms

Informal logic: the study of particular arguments in natural language and the contexts in which they occur (argument diagrams, fallacies)

Formal logic: the study of argument forms, abstract patterns common to many different arguments

Valid deductive argument: an argument whose conclusion cannot be false while the premises are all true

Invalid deductive arguments are arguments which purport to be deductive but in fact are not.

Soundness: a valid deductive argument with all true premises

Propositional logic: the study of the logical concept of validity insofar as this is due to the truth-functional operators (i.e., logical connectives such as “and”, “or”, “not”, “if-then” “if and only if” that have a consistent truth value

Sentence letters: P, Q, R, place holders for declarative sentences

Four common deductively valid argument forms

Modus ponens

p → q

p

:. q

Modus tollens

p → q

~q

:. ~p

Disjunctive syllogism

p v q

~p

:. Q

Hypothetical syllogism

p → q

q → r

:. p → r

Three fallacies

Fallacious modus ponens (fallacy of affirming the consequent)

p → q

q

:. p

Fallacious modus tollens (fallacy of denying the antecedent)

p → q

~p

:. ~q

Fallacious disjunctive syllogism (fallacy of asserting an alternative

p v q

p

:. ~q

TRUTH TABLES FOR ARGUMENT FORMS

Rules:

Display each premise and conclusion as a separate wff on the top

The argument is valid when, for every row where all premises are true, the conclusion in all of those rows are also true. But if a conclusion in any one of those selected rows is false, the argument is invalid.

Example: disjunctive syllogism

__p q |
p v q, ~p ˫q__

T T | T F T

T F | T F F

F T | T T T

F F | F T F

This is valid since only row 3 has true premises, and the conclusion of row 3 is also true.

Example: fallacious disjunctive syllogism

__p q |
p v q, p ˫~q__

T T | T T F

T F | T T T

F T | T F F

F F | F F T

This is invalid since both rows 1 and 2 have true premises, but the conclusion in row 1 is false. It does not matter that the conclusion in row 2 is true; this is an all or nothing situation where one defective row makes the entire argument form invalid.