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Formal Language and Natural Language Semantics

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First-Order Logic Semantics and StructuresFirst-Order Logic Syntax+3 more→→Temporal Semantics and Linguistic TenseTwo-Dimensional Semantics
formal-logic natural-language semantics

Core Idea

Natural language differs from formal logic in crucial ways: it is ambiguous, context-dependent, imprecise, and contains many non-truth-functional expressions. Formal semantic methods apply to natural language, but require adapting logical tools to preserve both accuracy and applicability.

Explainer

You already know how formal languages work from your study of first-order logic: a formal language has a fixed syntax, an explicit semantics defined over models, and no ambiguity — every well-formed formula has exactly one meaning relative to an interpretation. When you learned model theory, you saw how a model assigns objects to constants, extensions to predicates, and truth conditions to sentences in a fully determined, mechanical way. Natural language — the English, French, or Swahili you grew up speaking — operates very differently, and the gap between the two is where most of the philosophical action in semantics lives.

The most immediate difference is ambiguity. In first-order logic, "bank" simply does not appear — you would introduce a predicate BANK and specify what it applies to. In English, "She went to the bank" is genuinely ambiguous between a financial institution and a riverside, and listeners resolve the ambiguity using context, prior discourse, and world knowledge. Formal systems eliminate ambiguity by design; natural language lives with it and relies on pragmatic inference to recover the intended meaning. This means that a naïve translation of natural language into logic — treating each English sentence as having a single logical form — would misrepresent the phenomenon.

A second gap is context-dependence. You know that truth conditions specify what would make a sentence true or false. But many natural language sentences cannot be assigned truth conditions without knowing the context of utterance. "I am tired" is true in some contexts and false in others — the word "I" shifts referent with each speaker. "It's raining" needs a location. "That is tall" requires a comparison class — tall for a building, a person, or a blade of grass? Formal semantics handles this through indexicals (expressions whose reference is fixed by context) and context parameters (a context providing speaker, time, location, etc.) that supplement the model. The compositional machinery you learned — how complex meanings are built from parts — must be extended to take these parameters into account.

The deeper challenge is that natural language contains constructions that resist direct translation into first-order logic. Ordinary conditionals ("If it rains, the game is canceled") seem to work differently from material conditionals. Attitude reports ("Mary believes the president is corrupt") create contexts where substituting co-referring names can change truth value — a phenomenon that violates the substitutivity you expect from standard logic. Tense, aspect, modality, generics ("Tigers are striped") and questions all require extensions of the basic first-order toolkit. The project of formal semantics for natural language — pursued through tools like type theory, possible-worlds semantics, and dynamic logic — is precisely to find a systematic, compositional treatment of these phenomena that preserves the precision of formal methods while respecting the actual behavior of the language. The lesson is not that formal tools fail but that applying them to natural language is an ongoing, fine-grained empirical and theoretical enterprise, not a simple translation.

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Prerequisite Chain

Understanding Zero → The Number Zero → Counting to Five → Counting to 10 → Counting to 20 → Counting a Set of Objects Up to 20 → Cardinality: The Last Number Counted → Matching Numerals to Quantities → Subitizing Small Quantities → Addition Within 10 → Number Bonds to 10 → Addition Within 20 → Doubles and Near Doubles → Doubles Facts Within 10 → Near Doubles Facts Within 20 → Mental Math Strategies for Addition → Mental Math: Adding and Subtracting Tens → Addition Within 100 → Repeated Addition as Multiplication → Multiplication as Equal Groups → Multiplication: Arrays → Basic Multiplication Facts (0s, 1s, 2s, 5s, 10s) → Multiplication Facts Within 100 → Division as Equal Sharing → Division as Grouping (Measurement Division) → Division: Grouping (Repeated Subtraction) Model → Division: Fair Sharing Model → Division as Equal Sharing → Division as Grouping → Basic Division Facts → Division Facts Within 100 → Multiplication and Division Fact Families → Relationship Between Multiplication and Division → Division Facts as Inverse of Multiplication → Remainders and Quotients in Division → Division Word Problems → Multi-Step Word Problems → Solving Multi-Step Word Problems → Multiplication Word Problems → Division Word Problems → Introduction to Long Division → Factors and Multiples → Prime and Composite Numbers → Equivalent Fractions → Relating Fractions and Decimals → Decimal Place Value → Integers and the Number Line → Comparing and Ordering Integers → Absolute Value → Adding Integers → Subtracting Integers → Multiplying Integers → Introduction to Exponents → Order of Operations → Integer Order of Operations → Variable Expressions → The Distributive Property → Variables and Expressions Review → Introduction to Polynomials → Adding and Subtracting Polynomials → Multiplying Polynomials → Factorial → Permutations → Combinations → Counting Principles: Addition and Multiplication Rules → Introduction to Graph Theory → Propositional Logic Foundations → Logical Equivalences → Boolean Algebra → Introduction to Propositional Logic → Introduction to Predicate Logic (First-Order Logic) → First-Order Logic Syntax → Terms and Atomic Formulas in FOL → Variable Binding and Scope → Open and Closed Formulas in First-Order Logic → Variable Substitution and Capture-Avoidance in First-Order Logic → Quantifier Instantiation Rules in First-Order Proof Systems → Universal Quantification: Meaning and Scope → Free Variables and Bound Variables → Substitution and Instantiation in Predicate Logic → Terms and Atomic Formulas → Formulas and Well-Formed Expressions → Structures and Interpretations → Model Interpretation and Satisfaction → Interpretation, Truth, and Satisfaction of Formulas → Logical Consequence and Entailment → Soundness Theorem and Validity of Proof Systems → Deductive Reasoning and Formal Proof Systems → First-Order Resolution → Propositional Resolution → Semantic Tableaux (Propositional) → Semantic Tableaux (First-Order) → Decidable Fragments of First-Order Logic → Gödel's Completeness Theorem for First-Order Logic → Gödel's Incompleteness Theorems → Introduction to Intuitionistic Logic → Introduction to Modal Logic → A Priori and A Posteriori Knowledge → Rationalism vs. Empiricism → The Problem of Induction → Popper's Falsificationism → Falsifiability as the Criterion of Demarcation → The Falsifiability Criterion and Its Problems → Kuhn's Paradigm Theory → Normal Science and Anomalies → Thomas Kuhn and Paradigm Shifts → Scientific Progress and Convergence to Truth → Scientific Realism → Naturalism About Semantic Facts → Propositions and Semantic Content → Truth Conditions and Meaning → Formal Language and Natural Language Semantics

Longest path: 112 steps · 582 total prerequisite topics

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