What Is Mathematical Logic? The Science of Proof

Logic doesn't care what you're talking about.

"All mammals are warm-blooded. Whales are mammals. Therefore, whales are warm-blooded."

"All snorgs are blibble. Twerk is a snorg. Therefore, twerk is blibble."

Both arguments have exactly the same logical structure: All A are B. x is an A. Therefore, x is B. The first argument is about biology. The second is about nonsense. Both are logically valid.

This is the key insight: logic studies the structure of arguments, not their content. A valid argument is one where the conclusion must be true if the premises are true — regardless of what the premises are about.

Valid vs. True

The most important distinction in logic: validity is not truth.

Valid argument, true conclusion: All humans are mortal. Socrates is human. Therefore, Socrates is mortal.

Valid argument, false conclusion (from false premises): All fish can fly. Salmon are fish. Therefore, salmon can fly.

The second argument is perfectly valid — the conclusion follows inevitably from the premises. The premises just happen to be false.

Invalid argument, true conclusion: Most birds can fly. Penguins are birds. Therefore, penguins can fly.

The conclusion is actually false, but even if it were true, the argument would still be invalid. "Most" doesn't guarantee "all."

Logic tells you: if your premises are true, is your conclusion guaranteed? That's validity. Whether your premises are actually true is a separate question.

The Structure of Arguments

An argument has:

• Premises: statements assumed to be true
• Conclusion: the statement claimed to follow from the premises

Logic extracts the structure:

"If it's raining, the ground is wet. It's raining. Therefore, the ground is wet."

Structure: If P, then Q. P. Therefore, Q.

This structure — called modus ponens — is valid regardless of what P and Q mean. Any argument with this structure has a guaranteed conclusion if its premises are true.

Why Formalize?

Natural language is ambiguous. "I saw the man with the telescope" — who has the telescope?

Formal logic removes ambiguity by using precise symbols:

• ∧ for "and"
• ∨ for "or"
• ¬ for "not"
• → for "if-then"
• ∀ for "for all"
• ∃ for "there exists"

The formalism isn't the point. The formalism is a tool that makes reasoning checkable. Once an argument is translated into symbols, you can verify its validity mechanically.

Deductive vs. Inductive

Deductive reasoning: If the premises are true, the conclusion must be true.

All men are mortal. Socrates is a man. Therefore, Socrates is mortal.

If the premises are true, there's no possible world where the conclusion is false.

Inductive reasoning: The premises support but don't guarantee the conclusion.

The sun has risen every day for billions of years. Therefore, the sun will rise tomorrow.

Strong evidence, but not certainty. The conclusion could be false even if the premises are true.

Logic, in the formal sense, studies deduction. It's the science of what must follow, not what probably follows.

The Three Laws

Classical logic rests on three principles:

Law of Identity: A is A. Everything is what it is.

Law of Non-Contradiction: Nothing can be both A and not-A. A statement can't be true and false simultaneously.

Law of Excluded Middle: Everything is either A or not-A. Every statement is either true or false — no middle ground.

These seem obvious. But some logics reject them. Intuitionistic logic questions excluded middle. Paraconsistent logic questions non-contradiction. Classical logic keeps all three.

Logic and Mathematics

Logic is the foundation of mathematical proof.

Every theorem follows from axioms via logical rules. When you prove something, you're showing that the conclusion must be true if the axioms are true.

The relationship goes deeper: in the early 20th century, logicians tried to reduce all of mathematics to logic. The program failed (Gödel showed there are limits), but the attempt revealed how deeply logic and mathematics intertwine.

What Logic Can't Do

Logic won't tell you:

• What to believe (only what follows from beliefs)
• Whether your premises are true (that requires observation, evidence, or other means)
• How to discover good premises
• What matters or what to value

Logic is a tool for checking reasoning, not for generating beliefs or values. You supply the premises. Logic tells you what you're committed to if you accept those premises.

The Roadmap

This series covers:

Propositional Logic: Reasoning with whole statements and connectives (and, or, not, if-then)

Predicate Logic: Reasoning with internal structure (variables, predicates, quantifiers)

Meta-Logic: What formal systems can and cannot do (soundness, completeness, limits)

We start with the simplest atoms — propositions — and build up.

Part 1 of the Logic series.

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