Logical Techniques with Examples

Table of Logical Symbols

Symbol	Meaning
$$\forall$$	For all
$$\exists$$	There exists
$$\neg$$	Not
$$\land$$	And (conjunction)
$$\lor$$	Or (disjunction)
$$\to$$	Implies
$$\leftrightarrow$$	If and only if (biconditional)
$$\vdash$$	Therefore (logical consequence)
$$=$$	Equals
$$\subseteq$$	Subset of

Modus Ponens: $$P \to Q, P \vdash Q$$ If P is true, and P implies Q, then Q must be true. Learn More

Modus Tollens: $$P \to Q, \neg Q \vdash \neg P$$ If Q is false, and P implies Q, then P must be false. Learn More

Contrapositive: $$P \to Q \equiv \neg Q \to \neg P$$ If P implies Q, then not Q implies not P. Learn More

Disjunction Introduction: $$P \vdash P \lor Q$$ If P is true, then P or Q is true. Learn More

Disjunction Elimination: $$P \lor Q, \neg P \vdash Q$$ If either P or Q is true, and P is false, then Q must be true. Learn More

Conjunction Introduction: $$P, Q \vdash P \land Q$$ If both P and Q are true, then P and Q is true. Learn More

Conjunction Elimination: $$P \land Q \vdash P$$ and $$P \land Q \vdash Q$$ If P and Q is true, then P is true and Q is true. Learn More

Negation Introduction: $$P \vdash \neg P$$ Assume P, show it leads to a contradiction, conclude ¬P. Learn More

Double Negation Elimination: $$\neg \neg P \vdash P$$ If it is not the case that not P, then P is true. Learn More

Universal Instantiation: $$\forall x (P(x)) \vdash P(c)$$ If P(x) holds for all x, then it holds for a specific c. Learn More

Existential Instantiation: $$\exists x (P(x)) \vdash P(c)$$ If there exists an x such that P(x) holds, then P(c) holds for some c. Learn More

Existential Generalization: $$P(c) \vdash \exists x (P(x))$$ If P(c) is true for a specific c, then P(x) is true for some x. Learn More

De Morgan's Laws: $$\neg (P \land Q) \equiv \neg P \lor \neg Q$$ and $$\neg (P \lor Q) \equiv \neg P \land \neg Q$$ The negation of a conjunction is the disjunction of the negations. Learn More

Biconditional Introduction: $$P \to Q, Q \to P \vdash P \leftrightarrow Q$$ If P implies Q and Q implies P, then P is equivalent to Q. Learn More

Biconditional Elimination: $$P \leftrightarrow Q \vdash P \to Q$$ and $$P \leftrightarrow Q \vdash Q \to P$$ If P is equivalent to Q, then P implies Q and Q implies P. Learn More

Modus Ponens Example: If it rains, the ground will be wet. It is raining, so the ground is wet.

Modus Tollens Example: If the light is on, the room is bright. The room is not bright, so the light is not on.

Contrapositive Example: If it’s snowing, it’s cold. If it’s not cold, it’s not snowing.

Disjunction Introduction Example: If it's sunny, then it’s either sunny or cloudy.

Negation Introduction Example: Assume there is a smallest positive rational number. Then divide it by 2, getting a smaller number, which contradicts the assumption. Therefore, no smallest positive rational number exists.

Double Negation Elimination Example: If it is not true that it is not raining, then it must be raining.

Universal Instantiation Example: All humans are mortal. Socrates is a human, so Socrates is mortal.

Existential Instantiation Example: There exists someone who is a doctor. Dr. Smith is a doctor.

Existential Generalization Example: Dr. Smith is a doctor. Therefore, there exists someone who is a doctor.

De Morgan's Laws Example: The negation of "It is hot and sunny" is "It is not hot or it is not sunny."

Biconditional Introduction Example: If you win, you get a prize. If you get a prize, you must have won. Therefore, winning and getting a prize are logically equivalent.

Biconditional Elimination Example: If you win, you get a prize, and if you get a prize, you must have won. Therefore, if you won, you get a prize, and if you get a prize, you must have won.