Propositional Logic

Propositional logic is the relationships between sentences that are either true or false. (e.g. Socrates is a man. The sun is hot.)

Sentences in propositional logic are represented by letters. The letter representation of a statement is called its logical form. Frequently, logicians simply use “P” and “Q” as representations of a logical statement, but any symbol or representation is acceptable as long as the schema being used is communicated.

Connective: Connectives are the symbols in logic which represent words in natural language that describe a relationship between two statements. These are words and phrases such as, “if,” “either,” “if and only if,” “and,” “both,” and “or.”

*technically “not” is considered a connective, although it does not operate as a connector between two expressions.

Symbols:

~	The negation symbol stands for the English word “not”
& (sometimes represented as ˄)	The conjunction symbol stands for the English word “and”
˅	The disjunction symbol stands for the English word “or”
→ (sometimes represented as ⊃)	The conditional symbol stands for the English phrase “if…then”
↔	The biconditional symbol stands for the English word “if and only if” and sometimes abbreviated as “iff”
˫	The turnstile symbol communicates that a statement can be derived from another

Statement Categories:

Conjunctive Statement:

A conjunctive statement asserts that two or more expression are true. Conjunctives are represented by the conjunction symbol (&, ˄).

The statement, “Ryan is a man and trees are plants” is a conjunctive statement and can be formalized as P & Q.
Each individual component of a conjunctive statement is called a conjunct. In the example provided, “Ryan is a man,” is the first conjunct, which is represented by P. The statement “trees are plants” is the second conjunct and was represented as Q.

P & Q & R

The conjunctive statement above is composed of three conjuncts P, Q, and R. What this statement is declaring, is that all three conjuncts are true.

Disjunctive statement:

A disjunctive statement is an expression that posits the possibility of two or more expressions. Disjunctives are represented by the disjunction symbol (˅).

The statement “Ryan is a man or trees are plants” is a disjunctive statement and can be formalized as P ˅ Q.
Each individual component of a disjunctive statement is called a disjunct. In the example provided, “Ryan is a man” is the first disjunct, which is represented by P. The statement “trees are plants is the second disjunct and was represented as Q.

It is important to note that for a disjunctive statement to be true at least one of its disjuncts must be true. All the disjuncts could be true, but do not need to be as long as at least one is.

P ˅ Q ˅ R

The disjunctive statement above is composed of three disjuncts P, Q, and R. What this statement is declaring, is that at least one of the three disjuncts is true. It could be that more than one is true, as the “or” statements do not entail exclusivity as it often does in regular English.

Conditional Statement:

A conditional statement is an expression that follows the “if…then” structure. Conditionals are represented by the conditional symbol (→).

The statement, “If Ryan’s car works, then he will go to school” is a conditional statement and can be formalized as P → Q.

The first part of a conditional statement is called the antecedent. In the example above, the antecedent of the conditional is the statement “Ryan’s car works,” or its representation, “P.” The second part of a conditional is called the consequent. In the example, the consequent of the conditional is the statement “he will go to school,” or its representation, “Q.”

P → Q

The conditional statement above asserts that if P is true, then Q is true. It is important to know that it is possible for Q to be true, but P be false, but not for P to be true and Q be false. To see why that makes sense, imagine the following conditional: If Ryan takes a plane to New York, then he will arrive in New York (P → Q). Certainly, if Ryan took a plane to New York, it follows that he is in New York, but the reverse isn’t necessarily true
~(Q → P). Just because Ryan is in New York, doesn’t mean that he got there by plane. He could have taken the bus, drove a car, or maybe even walked there.

Biconditional Statement:

A biconditional statement is an expression that follows the “if and only if” structure. Biconditionals are represented by the biconditional symbol (↔). In writing, biconditionals are frequently abbreviated as “iff”

The statement, “The soldier will live, if and only if he has surgery” is a biconditional statement and can be formalized as P ↔ Q.

The names of the parts in a biconditional is the same as that of a conditional; the first member is the antecedent and the second member is the consequent.

Unlike a conditional, in a biconditional the truth of the consequent entails the truth of the antecedent.

Logical Equivalencies

Any statement in formal logic can be represented in a variety of ways. Frequently this can clarify what logical concomitants a statement or idea commits one to. Here is a table of some common equivalencies you may encounter:

Statement	Logical Equivalencies
P & Q	~ (~P ˅ ~Q)
P ˅ Q	~ (~P & ~Q) ~P → Q
P → Q	~Q → ~P ~P ˅ Q
P ↔ Q	(P → Q) & (Q → P) (P & Q) ˅ (~P & ~Q)

*There are an infinite number of possible logical equivalencies for any given statement (i.e. one could always add double negation signs to an entire statement).

Logical Statements and English Equivalencies

Logical Statement	English Equivalencies
P → Q	If P, Q Q if P P only if Q In case P, Q. Whenever P, Q P is sufficient for Q Provided that P, Q Q provided that P Q is necessary for P Given that P, Q Q on the condition that P
P & Q	P and Q P as well as Q P in addition to Q P, but Q. Both P and Q P, also Q
P ˅ Q	P or Q P unless Q Either P or Q
P ↔ Q	P if and only if Q P just in case Q P is the same as Q P is necessary and sufficient for Q

*The English Sentence, “neither P nor Q” can be formalized as:
~(P ˅ Q)
or
~P & ~Q

Propositional Logic Axioms

Logical axioms are statements which are necessarily true. They are infallibly true in all possible worlds, since their veracity is not contingent on anything but themselves.

There are technically an infinite number of axioms, but they can be reduced to fundamental axioms which build on each other to form more complex ones. We will call these fundamental axioms, primitive axioms, because they are primitive in that they are the simplest form of axiomatic truths.

Primitive Axioms

Primitive Axiom Name	Description
Conjunction Introduction	Given two expressions, a conjunction may be formed. E.G. Given P and Q, one can form the statement P&Q
Conjunction Elimination	Given a conjunction statement, one may conclude either conjunct. E.G. Given P&Q, one can conclude P.
Disjunctive Introduction	Given an expression, one can conclude any disjunctive statement with the given expression as one of the disjuncts. E.G. give P, one can conclude P ˅ Q
Disjunctive Elimination (Also known as Modus Tollendo Ponens)	Given a disjunctive expression with two disjuncts, given an expression which is the denial of one of the disjuncts, one may conclude the other disjunct. E.G. Given P ˅ Q, and given ~ Q, one can conclude P
Conditional Introduction	Given an expression, one may conclude a conditional with the expression as the consequent and any statement as the antecedent. E.G. Given P, one can conclude Q → P
Conditional Elimination (Also known as Modus Ponens, or Modus Ponendo Pones)	Given a conditional and given its antecedent, one may conclude its consequent. E.G. Given P → Q, and P, one can conclude Q
Biconditional Introduction	Given two conditionals with the form P → Q and Q → P, one can conclude P ↔ Q
Biconditional Elimination	Given a biconditional, P ↔ Q, one can conclude P → Q or Q → P

Derived Axioms

Derived Axioms are provable by the primitive axioms. These statements are true no matter what statements the letters represent.

Axiom Name	Axiom Form
Identity	P = P
Excluded Middle	P ˅ ~P
Non-Contradiction	~ (P & ~P)
Double Negation	P ↔ ~ ~ P
DeMorgan’s Law	~ (P ˅ Q) = (~P & ~Q) ~ (P & Q) = (~P ˅ ~Q)
Negated Biconditional	~ (P ↔ Q) = ~P ↔ Q
Transposition	P → Q = ~Q → ~P
Paradox of Material Implication	(P → Q) ˅ (Q → P)