Synthesis: Logic as the Foundation of Mathematical Reasoning

Logic is a technology for thinking clearly.

Not a set of beliefs. Not a method for discovering truth. A set of tools for checking whether conclusions follow from premises — and for building arguments that can't be honestly rejected if their premises are accepted.

This is the machinery that makes mathematics possible. Every theorem is a chain of logical inferences. Every proof is a demonstration that the conclusion must follow from the axioms. Without logic, mathematics would be a collection of hunches.

But logic extends far beyond mathematics. Programming languages are logical systems. Legal arguments have logical structure. Scientific hypotheses make logical predictions that can be tested.

What We've Built

Propositional Logic:

• Propositions: statements that are true or false
• Connectives: ¬, ∧, ∨, →, ↔
• Truth tables: mechanical computation of truth values
• Equivalences: different expressions with identical truth conditions
• Valid forms: modus ponens, modus tollens, etc.

Predicate Logic:

• Predicates: properties and relations
• Quantifiers: ∀ (for all) and ∃ (there exists)
• Variables and scope
• Translation between English and symbolic form

Meta-Logic:

• Soundness: proofs don't produce falsehoods
• Completeness: everything valid is provable
• The limits of formal systems

The Two Layers

Object language: The statements we reason about. P → Q, ∀x(Human(x) → Mortal(x)), etc.

Metalanguage: Statements about the object language. "Propositional logic is complete." "Modus ponens is a valid argument form."

This distinction matters because:

• We can prove things in the object language using logical rules
• We can prove things about the object language using metalogic

Gödel's theorems are metalogical results — statements about what formal systems can and cannot do.

The Central Achievement

Classical logic provides:

A universal vocabulary: Connectives and quantifiers that work regardless of subject matter.

Mechanical verification: Given an argument in logical form, we can check validity algorithmically (in propositional logic) or systematically (in predicate logic).

Compositional semantics: The truth value of complex expressions depends only on the truth values of components and how they're combined.

Proof systems: Rules for deriving conclusions that are sound (never invalid) and complete (capture all valid inferences).

This is infrastructure for all rigorous thought.

What Logic Doesn't Do

Logic won't tell you:

• What's actually true about the world (only what follows from assumptions)
• Which premises to accept
• How to formalize natural language arguments
• What matters or what to value

Logic is a tool for conditional reasoning: IF these premises are true, THEN this conclusion follows. The premises themselves come from elsewhere — observation, intuition, authority, choice.

The Translation Problem

Natural language is messy:

"John and Mary got married" — to each other? To other people? "Everyone loves someone" — same person or different? "If you're interested, call me" — is the interest necessary or polite?

Translating English to logic requires interpretation. The same sentence can map to different logical forms depending on context.

This isn't a flaw in logic — it's a feature of natural language. Logic provides precise tools for reasoning. Using those tools requires first deciding what, precisely, we mean.

Beyond Classical Logic

Classical logic assumes:

• Every proposition is true or false (bivalence)
• No proposition is both true and false (non-contradiction)
• Every proposition is either true or not true (excluded middle)

Alternative logics relax these:

Intuitionistic logic: Rejects excluded middle. "P or not-P" isn't automatically true — you need constructive proof of one or the other.

Paraconsistent logic: Tolerates contradictions without explosion. Useful when dealing with inconsistent data.

Modal logic: Adds necessity (□) and possibility (◇). "Necessarily P" vs. "Possibly P."

Fuzzy logic: Allows degrees of truth. P might be 0.7 true.

Classical logic remains the default, but knowing alternatives exist prevents confusing "logic" with "one particular logical system."

Logic and Computation

The connection is deep:

Boolean algebra is propositional logic in circuit form. AND gates, OR gates, NOT gates.

Programming languages are logical systems. Type systems are proofs. Programs are structured logical arguments.

Database queries (SQL) are predicate logic. SELECT WHERE is essentially ∃x with conditions.

Proof assistants mechanically verify mathematical proofs by checking logical derivations.

The same structures that govern valid reasoning govern computation.

The Hierarchy of Expressiveness

More expressiveness comes with costs: second-order logic isn't complete. There's always a tradeoff.

First-order logic hits a sweet spot: expressive enough for most of mathematics, still complete.

How to Use Logic

Analyzing arguments:

1. Identify premises and conclusion
2. Translate to logical form
3. Check structure against valid patterns

Building arguments:

1. Make premises explicit
2. Chain valid inferences
3. Reach conclusion

Debugging reasoning:

1. Look for hidden premises
2. Check for fallacious patterns
3. Test with counterexamples

Logic doesn't replace thinking. It disciplines thinking.

The Core Insight

Logic separates the form of reasoning from its content.

Valid arguments are valid because of their structure, not because of what they're about. "All A are B, all B are C, therefore all A are C" works whether we're talking about cats, numbers, or legal principles.

This abstraction is logic's power. Once you identify the structure, you can evaluate the reasoning independently of the subject matter. You can transfer argument patterns between domains. You can spot fallacies regardless of how persuasive the rhetoric.

Logic makes thinking visible and checkable. That's its gift.

Part 12 of the Logic series.

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