Logic

Core symbolic logic for theorem reading, formal reasoning, and research-facing math in CS, AI, and engineering.

Modified

April 26, 2026

Keywords

logic, propositions, predicates, negation, formal reasoning

1 Why This Module Matters

Logic is where mathematical language becomes explicit enough to inspect.

Proofs use logic all the time, but often indirectly. This module slows that down and makes the structure visible: what counts as a statement, how compound statements are built, how negation changes meaning, and how quantifiers control the scope of a claim.

That matters far beyond pure math. Logic sits under specification, verification, SAT solving, constraint systems, database queries, knowledge representation, and theorem proving. It also makes theorem statements in analysis, probability, and optimization much easier to parse correctly.

The first-pass logic module is now live as five pages: Propositional Logic, Logical Equivalence and Negation, Predicate Logic, Sets, Functions, and Relations, and Translation Between English and Symbols.

Prerequisites Proof-reading habits, notation comfort, and patience with symbolic language

Unlocks Discrete math, formal verification, theorem proving, cleaner theorem reading

Research Use Specifications, SAT and automated reasoning, quantified theorem statements, formal models

2 First Pass Through This Module

These five pages now form a complete first pass through the logic module: from propositional form to equivalence, quantifiers, mathematical objects, and practical translation.

4 Core Concepts

Propositional Logic: introduces statements, connectives, truth tables, satisfiability, and validity.
Logical Equivalence and Negation: teaches how to rewrite statements without changing truth conditions and how to negate claims precisely.
Predicate Logic: adds variables, domains, predicates, and quantified statements.
Sets, Functions, and Relations: connects logical form to mathematical collections, mappings, and structured pairs.
Translation Between English and Symbols: turns the rest of the module into a practical theorem-reading and symbolization workflow.

5 Proof Patterns In This Module

Separate syntax from semantics: a well-formed formula is not the same thing as a true formula.
Negate carefully: most beginner mistakes in logic come from sloppy negation.
Track the bad row: argument validity becomes clearer once you ask whether any assignment makes the premises true and the conclusion false.
Make hidden quantifiers explicit: theorem prose often becomes readable only after you write down the missing for every and there exists.

6 Applications

6.1 Formal Verification and SAT

Propositional formulas encode constraints, hardware behavior, transition guards, and proof obligations. SAT solvers and model checkers begin from this logical layer before moving to richer systems.

6.2 Reading Quantified Mathematics

Quantified theorems in analysis, optimization, probability, and learning theory become much easier to read when you can distinguish implication, equivalence, negation, and quantifier scope cleanly.