Introduction
Mathematics is not only a collection of formulas and computations, but also a discipline built upon precise reasoning. Before engaging with advanced mathematical structures, it is essential to understand the logical foundations that govern how mathematical statements are formed, interpreted, and justified. This section is devoted to developing that foundational perspective.
We begin with the classical roots of logic, tracing the basic principles of reasoning back to Aristotle’s formulation of logical laws. From there, we study propositional logic: logical connectives, compound statements, and truth tables, which provide a concrete and systematic way to analyze the validity of logical expressions. These tools allow us to determine when statements are true, false, or logically equivalent, independent of any particular mathematical context.
Building on this framework, we turn to methods of proof, which form the backbone of mathematical practice. Direct proofs, proof by contradiction, contrapositive arguments.
Some Historical Background
Aristotle (384–322 BCE) is widely regarded as the founder of formal logic. While earlier philosophers had reflected on reasoning and argumentation, Aristotle was the first to develop a systematic theory that analyzed the structure of valid arguments independently of their specific content. His work on logic laid the groundwork for deductive reasoning and shaped mathematical and philosophical thought for over two millennia.
1. Aristotle’s Laws of Logic
Classical logic is traditionally grounded in three fundamental principles attributed to Aristotle. These laws form the basis of rational thought and are implicitly assumed in mathematical reasoning, philosophical argumentation, and formal proof techniques.
1.1 Law of Identity
Statement: A thing is identical to itself.
“A is A.”
The Law of Identity asserts that every object, concept, or proposition remains the same as itself. This principle ensures consistency in meaning and allows definitions to be fixed throughout a logical or mathematical discussion. Without this law, precise reasoning would be impossible, since symbols and statements could not retain stable interpretations.
1.2 Law of Non-Contradiction
Statement: A proposition cannot be both true and false at the same time and in the same respect.
“It is impossible for the same thing to belong and not belong to the same thing at
the same time and in the same respect.”
This law is considered by Aristotle to be the most fundamental principle of logic. Historically, this law also played a central role in metaphysical and theological discussions, as it was used to rule out the possibility of mutually contradictory attributes.
1.3 Law of Excluded Middle
Statement: Every proposition is either true or false.
“Either A is true, or its negation is true.”
The Law of Excluded Middle asserts that there is no third option between truth and falsity. Every meaningful statement must fall into one of these two categories.