Unit 1.5

LOGIC
& PROOFS

Geometry is not just about shapes; it is about absolute truth. We use Deductive Reasoning to chain simple facts into undeniable conclusions.

IF P THEN Q

Logic is the operating system
of the universe.

Truth Tables

A Conditional Statement (p → q) is only false when a True Hypothesis leads to a False Conclusion. This is the bedrock of valid arguments.

Inputs

Variables

Toggle the variables to see how they affect the logic gates below.

Conjunction (AND)

p ∧ q

TRUE

Disjunction (OR)

p ∨ q

TRUE

Conditional

p → q

TRUE

Contrapositive

~q → ~p

TRUE

Original

p → q

Logically Equivalent

Contrapositive

~q → ~p

Laws of Deduction

Law of Detachment

[(p → q) ∧ p] → q

If p → q is true and p is true, then q is true.

Law of Syllogism

[(p → q) ∧ (q → r)] → (p → r)

If p → q and q → r are true, then p → r is true.

The Vocabulary

Inductive Reasoning

Pattern

Making a generalization based on observed patterns or examples. (e.g., "The sun rose every day before, so it will rise tomorrow.")

Deductive Reasoning

Law

Using facts, rules, definitions, or properties to reach a logical conclusion. (e.g., "All men are mortal. Socrates is a man. Therefore, Socrates is mortal.")

Conditional Statement

Structure

A logical statement written in "If-Then" form. Symbolized as p → q.

Contrapositive

Transformation

Negating and switching the hypothesis and conclusion (~q → ~p). It is always logically equivalent to the original statement.