Guide to Logic: Conditions and Arguments

Necessary condition

Something that must be true for another statement to be true.

Example of necessary condition

Having a valid driver's license is a necessary condition for legally driving a car.

Sufficient condition

Guarantees the truth of another statement.

Example of sufficient condition

Being a square is a sufficient condition for being a rectangle, as all squares meet the criteria for rectangles.

Conditional statements

Statements that include an antecedent and a consequent.

Antecedent

The 'if' part of a conditional statement, representing the condition or premise.

Consequent

The 'then' part of a conditional statement, representing the outcome or result that follows from the antecedent.

Logical reasoning

A process where a condition can be both necessary and sufficient, meaning the presence of one guarantees the presence of the other, and vice versa.

If and only if

A phrase often used in logical statements to express that a condition is both necessary and sufficient.

Modus Ponens

A form of argument where, given a conditional statement 'If A, then B' (A → B), and the affirmation of A, one can conclude B.

Example of Modus Ponens

If it is stated that 'If it rains, the ground will be wet,' and it is raining, then one can conclude that the ground is wet.

Modus Tollens

Involves a conditional statement 'If A, then B' (A → B), but instead of affirming A, it denies B to conclude the negation of A.

Example of Modus Tollens

If 'If it rains, the ground will be wet' is true, and the ground is not wet, then one can conclude that it did not rain.

Affirming the Consequent

Occurs when one argues from the premises 'If A, then B' and 'B' to the conclusion 'A.' This form of reasoning is invalid.

Example of Affirming the Consequent

'If John is a bachelor, then he is male. John is male. Therefore, John is a bachelor' is a fallacy because being male does not necessarily mean John is a bachelor.

Denying the Antecedent

Involves arguing from the premises 'If A, then B' and 'not-A' to the conclusion 'not-B.' This reasoning is invalid.

Example of Denying the Antecedent

'If Othello is a bachelor, then he is male. Othello is not a bachelor. Therefore, Othello is not male' is a fallacy because not being a bachelor does not mean Othello is not male.

Deductive Arguments

These are arguments where the conclusion necessarily follows from the premises. If the premises are true, the conclusion must also be true.

Example of Deductive Argument

In a syllogism, if 'All men are mortal' and 'Socrates is a man,' then 'Socrates is mortal' is a deductive conclusion.

Inductive Arguments

Inductive reasoning involves drawing general conclusions from specific observations. The conclusions of inductive arguments are probable rather than certain.

Example of Inductive Argument

Observing that the sun has risen every day in recorded history leads to the inductive conclusion that it will rise again tomorrow.

Indicative Conditionals

Deal with real or probable situations.

Subjunctive Conditionals

Address hypothetical or unreal scenarios.

Indicative Conditionals

Used to describe situations that are factual or likely to happen.

Subjunctive Conditionals

Used to express hypothetical, unreal, or wished-for situations.

Logical Conjunction (∧)

Represents logical conjunction, meaning 'and.' It is used to combine two propositions that are both true.

Logical Disjunction (∨)

Represents logical disjunction, meaning 'or.' It signifies that at least one of the propositions is true.

Negation (¬ or ~)

Denotes negation, meaning 'not.' It is used to invert the truth value of a proposition.

Implication (→ or ⊃)

Represents implication, meaning 'if...then.' It indicates that if the first proposition is true, then the second one is also true.

Biconditional (↔ or ≡)

Denotes biconditional, meaning 'if and only if.' It signifies that both propositions are either true or false together.

Universal Quantifier (∀)

The universal quantifier, meaning 'for all.' It is used to indicate that a proposition applies to all elements in a domain.

Existential Quantifier (∃)

The existential quantifier, meaning 'there exists.' It indicates that there is at least one element in the domain for which the proposition is true.

Conjunction

A conjunction (and) is true if both propositions are true.

Disjunction

A disjunction (or) is true if at least one proposition is true.

Negation

Negation (not) inverts the truth value of a proposition, making a true proposition false and vice versa.

Conditional Statement

A conditional statement (if...then) is true unless a true proposition leads to a false one.

Biconditional

A biconditional (if and only if) is true when both propositions have the same truth value.

Implication

This is a relationship where one proposition logically follows from another.

Material Implication

Material implication (if A, then B) is true except when A is true and B is false.

Strict Implication

Strict implication considers the meanings of propositions, not just their truth values.

Logical Equivalence

Propositions can be equivalent if they always have the same truth value.

Square of Opposition

The square of opposition illustrates relationships like contrariety and contradiction, where propositions cannot both be true or false simultaneously.

Fallacy

In logic, erroneous reasoning that has the appearance of soundness.

Reductio ad Absurdum

A logical argument technique used to refute a proposition by demonstrating that its denial leads to a contradiction or an absurd conclusion.

Indirect Proof

This form of argument is also known as indirect proof or reductio ad impossibile.