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7 - Predicate Logic Syntax and Semantics
from Part II - Predicate Logic
Yannai A. Gonczarowski, Harvard University, Massachusetts, Noam Nisan, Hebrew University of Jerusalem
Book: Mathematical Logic through Python
Published online: 25 August 2022
Print publication: 15 September 2022, pp 109-128
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Summary

Introduction to first-order Predicate Logic. Syntax and semantics, models and quantification. Implementation of “Term,” “Formula,” and “Model” objects for Predicate Logic in Python.

Frontmatter
Yannai A. Gonczarowski, Harvard University, Massachusetts, Noam Nisan, Hebrew University of Jerusalem
Book: Mathematical Logic through Python
Published online: 25 August 2022
Print publication: 15 September 2022, pp i-iv
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Part I - Propositional Logic
Yannai A. Gonczarowski, Harvard University, Massachusetts, Noam Nisan, Hebrew University of Jerusalem
Book: Mathematical Logic through Python
Published online: 25 August 2022
Print publication: 15 September 2022, pp 11-12
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13 - Sneak Peek at Mathematical Logic II: Gödel’s Incompleteness Theorem
from Part II - Predicate Logic
Yannai A. Gonczarowski, Harvard University, Massachusetts, Noam Nisan, Hebrew University of Jerusalem
Book: Mathematical Logic through Python
Published online: 25 August 2022
Print publication: 15 September 2022, pp 256-265
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Gödel numbering and Gödel’s Incompleteness Theorem. Undecidability of the halting problem and how it can be used to prove Gödel’s Incompleteness Theorem.

0 - Introduction and Overview
Yannai A. Gonczarowski, Harvard University, Massachusetts, Noam Nisan, Hebrew University of Jerusalem
Book: Mathematical Logic through Python
Published online: 25 August 2022
Print publication: 15 September 2022, pp 1-10
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An introduction to formal Logic and to the contrast and connections between syntax and semantics. The unique pedagogical approach of the book of capturing logic in a computer program and of proving by programming.

12 - The Completeness Theorem
from Part II - Predicate Logic
Yannai A. Gonczarowski, Harvard University, Massachusetts, Noam Nisan, Hebrew University of Jerusalem
Book: Mathematical Logic through Python
Published online: 25 August 2022
Print publication: 15 September 2022, pp 231-255
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The Completeness Theorem and the Compactness Theorem for Predicate Logic. Henkin constants. The Löwenheim–Skolem Theorem.

Preface
Yannai A. Gonczarowski, Harvard University, Massachusetts, Noam Nisan, Hebrew University of Jerusalem
Book: Mathematical Logic through Python
Published online: 25 August 2022
Print publication: 15 September 2022, pp xi-xii
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Introduction, synopsis, and acknowledgments.

Dedication
Yannai A. Gonczarowski, Harvard University, Massachusetts, Noam Nisan, Hebrew University of Jerusalem
Book: Mathematical Logic through Python
Published online: 25 August 2022
Print publication: 15 September 2022, pp v-vi
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10 - Working with Predicate Logic Proofs
from Part II - Predicate Logic
Yannai A. Gonczarowski, Harvard University, Massachusetts, Noam Nisan, Hebrew University of Jerusalem
Book: Mathematical Logic through Python
Published online: 25 August 2022
Print publication: 15 September 2022, pp 178-210
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Summary

In this chapter we fix a specific axiomatic system and provide several examples of its formal use. These include logical inferences such as syllogisms, proofs of mathematical theorems in structures such as groups and fields, and foundations of mathematics, specifically Peano Arithmetic and Set Theory.

Contents
Yannai A. Gonczarowski, Harvard University, Massachusetts, Noam Nisan, Hebrew University of Jerusalem
Book: Mathematical Logic through Python
Published online: 25 August 2022
Print publication: 15 September 2022, pp vii-x
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4 - Proof by Deduction
from Part I - Propositional Logic
Yannai A. Gonczarowski, Harvard University, Massachusetts, Noam Nisan, Hebrew University of Jerusalem
Book: Mathematical Logic through Python
Published online: 25 August 2022
Print publication: 15 September 2022, pp 53-68
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An introduction to deductive proofs. Inference rules and their specializations. Implementation of a “Proof” object in Python. The Soundness Theorem.

2 - Propositional Logic Semantics
from Part I - Propositional Logic
Yannai A. Gonczarowski, Harvard University, Massachusetts, Noam Nisan, Hebrew University of Jerusalem
Book: Mathematical Logic through Python
Published online: 25 August 2022
Print publication: 15 September 2022, pp 24-40
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The semantics of propositional formulas: models and truth tables, tautologies and contradictions. Disjunctive Normal Form and the DNF Theorem. Optional: Conjunctive Normal Form and the CNF Theorem, computational search problems, satisfiability, P and NP.

Cheatsheet - Axioms and Axiomatic Inference Rules Used in This Book
Yannai A. Gonczarowski, Harvard University, Massachusetts, Noam Nisan, Hebrew University of Jerusalem
Book: Mathematical Logic through Python
Published online: 25 August 2022
Print publication: 15 September 2022, pp 266-267
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11 - The Deduction Theorem and Prenex Normal Form
from Part II - Predicate Logic
Yannai A. Gonczarowski, Harvard University, Massachusetts, Noam Nisan, Hebrew University of Jerusalem
Book: Mathematical Logic through Python
Published online: 25 August 2022
Print publication: 15 September 2022, pp 211-230
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Summary

Converting formulas to Prenex Normal Form and the Deduction Theorem for Predicate Logic.

3 - Logical Operators
from Part I - Propositional Logic
Yannai A. Gonczarowski, Harvard University, Massachusetts, Noam Nisan, Hebrew University of Jerusalem
Book: Mathematical Logic through Python
Published online: 25 August 2022
Print publication: 15 September 2022, pp 41-52
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Summary

Logical operators and replacing an operator by others in a formula via substitution. Complete and incomplete sets of operators.

5 - Working with Proofs
from Part I - Propositional Logic
Yannai A. Gonczarowski, Harvard University, Massachusetts, Noam Nisan, Hebrew University of Jerusalem
Book: Mathematical Logic through Python
Published online: 25 August 2022
Print publication: 15 September 2022, pp 69-83
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Summary

Lemmas and their use in proofs and the “Lemma” Theorem. Modus Ponens, the Deduction Theorem, and proofs by way of contradiction.

Part II - Predicate Logic
Yannai A. Gonczarowski, Harvard University, Massachusetts, Noam Nisan, Hebrew University of Jerusalem
Book: Mathematical Logic through Python
Published online: 25 August 2022
Print publication: 15 September 2022, pp 107-108
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9 - Deductive Proofs of Predicate Logic Formulas
from Part II - Predicate Logic
Yannai A. Gonczarowski, Harvard University, Massachusetts, Noam Nisan, Hebrew University of Jerusalem
Book: Mathematical Logic through Python
Published online: 25 August 2022
Print publication: 15 September 2022, pp 143-177
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Summary

Proofs by deduction in Predicate Logic. Axiom schemas and their instances. Implementation of “Schema” and “Proof” objects for Predicate Logic in Python. Use of tautologies in proofs. The Soundness Theorem for predicate logic.

1 - Propositional Logic Syntax
from Part I - Propositional Logic
Yannai A. Gonczarowski, Harvard University, Massachusetts, Noam Nisan, Hebrew University of Jerusalem
Book: Mathematical Logic through Python
Published online: 25 August 2022
Print publication: 15 September 2022, pp 13-23
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Summary

An introduction to Propositional Logic, focusing on the syntax. The unique readability theorem. Implementation of a “Formula” Python object and parsing formulas.

6 - The Tautology Theorem and the Completeness of Propositional Logic
from Part I - Propositional Logic
Yannai A. Gonczarowski, Harvard University, Massachusetts, Noam Nisan, Hebrew University of Jerusalem
Book: Mathematical Logic through Python
Published online: 25 August 2022
Print publication: 15 September 2022, pp 84-106
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Summary

A culmination of the first part of the book: the Tautology Theorem. The Completeness and Compactness Theorems for Propositional Logic.

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