2.3 Valid and Invalid Arguments, Schemes and Mind Maps of Discrete Mathematics

Download 2.3 Valid and Invalid Arguments and more Schemes and Mind Maps Discrete Mathematics in PDF only on Docsity!

2.3 Valid and Invalid Arguments

Valid Arguments

Definition

(^1) An argument (argument form) is a sequence of statements (statement forms). (^2) All statements in an argument, except the final one, are called premises (or assumptions or hypothesis). (^3) The final statement is called the conclusion.

Valid Arguments

Definition

(^1) An argument (argument form) is a sequence of statements (statement forms). (^2) All statements in an argument, except the final one, are called premises (or assumptions or hypothesis). (^3) The final statement is called the conclusion. (^4) An argument form is valid if, no matter what particular statements are substituted for the statement variables in its premises, whenever the resulting premises are all true, the conclusion is also true. (Hint: If any premises are false, then the argument is

Valid Arguments

Definition

(^1) An argument (argument form) is a sequence of statements (statement forms). (^2) All statements in an argument, except the final one, are called premises (or assumptions or hypothesis). (^3) The final statement is called the conclusion. (^4) An argument form is valid if, no matter what particular statements are substituted for the statement variables in its premises, whenever the resulting premises are all true, the conclusion is also true. (Hint: If any premises are false, then the argument is vacuously true.)

Testing an Argument Form for Validity

Fact

(^1) Identify the premises and conclusion of the argument form. (^2) Construct a truth table showing the truth values of all the premises and the conclusion. (^3) A row of the truth table in which all the premises are true is called a critical row.

Testing an Argument Form for Validity

Fact

(^1) Identify the premises and conclusion of the argument form. (^2) Construct a truth table showing the truth values of all the premises and the conclusion. (^3) A row of the truth table in which all the premises are true is called a critical row. (^1) If there is a critical row in which the conclusion is false, then the argument is invalid. (^2) If the conclusion in every critical row is true, then the argument form is valid.

Modus Ponens

Fact

An argument form consisting of two premises and a conclusion is called a syllogism. The most famous example is modus ponens (“mood that affirms”): If p → q, p ∴ q

p q p → q p q T T T T T T F F T F T T F F F T F

Example

Example

Here is a modus ponens argument: If it snows more than 2” then the Naval Academy closes. It snowed more than 2”.

Modus Tollens

Fact

Modus tollens (“mood that denies”) has the form If p → q. ∼ q ∴ ∼ p.

Example

If it snows more than 2” then the Naval Academy closes. The Naval Academy did not close.

Modus Tollens

Fact

Modus tollens (“mood that denies”) has the form If p → q. ∼ q ∴ ∼ p.

Example

If it snows more than 2” then the Naval Academy closes. The Naval Academy did not close. ∴ It snowed less than 2”.

Additional rules of inference

Definition

(^1) Generalization: p ∴ p ∨ q or q ∴ p ∨ q. (^2) Specialization: p ∧ q ∴ p or

Additional rules of inference

Definition

(^1) Generalization: p ∴ p ∨ q or q ∴ p ∨ q. (^2) Specialization: p ∧ q ∴ p or p ∧ q ∴ q. (^3) Elimination: p ∨ q, ∼ q ∴ p or

Additional rules of inference

Definition

(^1) Generalization: p ∴ p ∨ q or q ∴ p ∨ q. (^2) Specialization: p ∧ q ∴ p or p ∧ q ∴ q. (^3) Elimination: p ∨ q, ∼ q ∴ p or p ∨ q. ∼ p ∴ q. (^4) Transitivity: p → q, q → r ∴ p → r. (^5) Division into Cases: p ∨ q, p → r , q → r

Additional rules of inference

Definition

(^1) Generalization: p ∴ p ∨ q or q ∴ p ∨ q. (^2) Specialization: p ∧ q ∴ p or p ∧ q ∴ q. (^3) Elimination: p ∨ q, ∼ q ∴ p or p ∨ q. ∼ p ∴ q. (^4) Transitivity: p → q, q → r ∴ p → r. (^5) Division into Cases: p ∨ q, p → r , q → r ∴ r.