Discrete mathematicsLesson 2 of 5 in the course Mathematical logic and proof

First-Order Logic: Syntax, Semantics, Proof, and Limits

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

First-order logic (FOL), also called predicate logic, lets us describe objects, their properties, and relationships among them. It extends propositional logic by adding variables and quantifiers.

Propositional logic treats statements such as “Socrates is a philosopher” as indivisible units that are true or false. First-order logic can analyze the statement into a predicate applied to an object: Phil(Socrates)Phil(Socrates).

All flashcards

What is a predicate?
A property or relation that, under an interpretation and suitable object assignments, evaluates as true or false.
What is a domain of discourse?
The collection of objects over which variables and quantifiers range.
What is a first-order structure?
A nonempty domain together with interpretations of the language’s non-logical symbols.
What is logical consequence?
A conclusion is a logical consequence of premises if every model of the premises makes the conclusion true.
What is variable capture?
The accidental binding of a free variable when substituting a term; renaming bound variables can prevent it.

Quiz questions

  1. What does a signature specify in a first-order language?Answer: The available non-logical symbols and their arities.
  2. What makes a structure a model of a first-order theory?Answer: It satisfies every sentence in the theory’s axioms.
  3. What does soundness mean for a deductive system?Answer: Every formula it proves is logically valid.
  4. What does the compactness theorem imply about a set of first-order sentences?Answer: If every finite subset has a model, the whole set has a model.
  5. What does semidecidability of first-order consequence guarantee?Answer: Proof search eventually finds a proof when the consequence holds.

Based on the Wikipedia article First-order logic, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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