Logic

At its core, logic is the systematic study of valid reasoning - how we can move from what we know to be true to reach reliable conclusions about what else must be true. It’s the foundation of rational thought.

Truth Values

The simplest logical concept is that of truth values - statements can be either true or false. This binary nature forms the basis of logical reasoning. For example, “This apple is red” must be either true or false.

Propositions

Moving up one level, we have propositions - statements that can be evaluated as true or false. These are the atoms of logical reasoning. A proposition like “All humans are mortal” can be analyzed for its truth value.

Logical Operators

To connect these basic propositions, we use logical operators - AND, OR, NOT, IF-THEN. These allow us to build more complex statements while preserving truth relationships. For instance, “IF it is raining, THEN the ground is wet” connects two propositions in a way that maintains logical consistency.

Rules of Inference

Building further, we establish rules of inference - patterns of reasoning that preserve truth. The most fundamental is modus ponens: if we know “IF A THEN B” is true, and we know “A” is true, we can conclude “B” must be true.

Validity vs. Truth

A crucial distinction emerges here: logical validity (the form of the argument) differs from truth (the content). A logically valid argument with true premises must lead to a true conclusion, but valid reasoning can still lead to false conclusions if it starts from false premises.

Logical Systems

From these foundations, we can construct entire systems of logic - frameworks for determining what conclusions necessarily follow from given premises. Classical logic, modal logic, and mathematical logic all build upon these basic principles in different ways.