# Personal Handbook of Logic

This is a personal collection of definitions and results from first-order logic.

https://doi.org/10.31219/osf.io/8wck9

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 252, 2021 [JX]

# Two non-isomorphic structures with the same number of elements

[white paper: pedagogical]

For pedagogical purposes, we define a simple language to show that two different structures with the same number of elements in their universes are not isomorphic.

https://doi.org/10.31219/osf.io/ytcru

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 234, 2021 [JE]

# The isomorphism between structures is an equivalence relation

[white paper: pedagogical]

We present a pedagogical proof that the function of an isomorphism between two structures is an equivalence relation.

https://doi.org/10.31219/osf.io/7uwnh

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 233, 2021 [JD]

# The terms of a language with one constant, one binary function, and one 4-ary function have an odd number of symbols

[white paper: pedagogical]

We show using induction on complexity that all terms of a language with one constant, one binary function, and one 4-ary function have an odd number of symbols.

https://doi.org/10.31219/osf.io/ue32a

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 231, 2021 [JB]

# Goldbach Conjecture, Twin Primes Conjecture, and Bounded Gap Theorem in the language of number theory

[white paper: pedagogical]

We write the formulas of the theorem and the conjectures highlighted in the title of this white paper in the language of number theory for pedagogical purpose in first-order logic.

https://doi.org/10.31219/osf.io/u45y3

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 229, 2021 [IZ]

# Substitutions and Substitutability

[knowledge base]

SUBSTITUTIONS, SUBSTITUTABILITY, and their underlying definitions are presented in this white paper (knowledge base).

https://doi.org/10.31219/osf.io/k6w7u

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 212, 2021 [II]

# Model

[knowledge base]

MODEL (mathematical logic) and its underlying definitions are presented in this white paper (knowledge base).

https://doi.org/10.31219/osf.io/vs6j9

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 209, 2021 [IF]

# A unidimensional windmill in the plane with varying pivot points

We show that we can choose a point in the plane such that the resulting windmill process with varying pivot uses each point of the plane infinitely many times.

https://doi.org/10.31219/osf.io/mwkaf

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 2, Article 148, 2020 [FT]

# Logical Loop

[microresearch]

We discuss the dubbed term “logical loop” and it’s implication regarding provability in undecidable theorems.

https://doi.org/10.31219/osf.io/c5ezm

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 2, Article 123, 2020 [ET]

# Non-classical logic and undecidability

[conjecture]

We elaborate a conjecture by applying the definition of Non-Axiomatic System  in Non-Classical Logic.

https://doi.org/10.31219/osf.io/h4f7p

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 2, Article 112, 2020 [EI]

# The logic of entanglement in gravity

[microresearch]

We explore the relation between gravity and entanglement using mathematical logic (Modus Ponens).

https://doi.org/10.31219/osf.io/m2dca

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 1, Article 58, 2019 [CF]

# Hilbert-style proof Calculus for Propositional Logic in ABC notation

[microreview]

All nine axioms and a single inference rule of logic (Modus Ponens) within the Hilbert axiomatic system are presented using capital letters (ABC) in order to familiarize the beginner student in hers/his first contact with the topic.

https://doi.org/10.31219/osf.io/jd3gp

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 1, Article 57, 2019 [CE]

# The Incorrectness Lemma

[original idea]

We address two questions. Is there an Incorrectness Lemma? Is this system complete?

https://doi.org/10.31219/osf.io/f8p7k

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 1, Article 53, 2019 [CA]

# A pedagogical approach to convert “if and only if” to “if then”

[microreview]

We show, using the equivalence of some formulas, that (A iff B) ~ ((A v B) => (A ^ B)).

https://doi.org/10.31219/osf.io/xhzc9

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 1, Article 52, 2019 [BZ]

# Generalized Pierce law

[microresearch]

An extension of the Pierce law is presented, considering n “if then” connectives.

https://doi.org/10.31219/osf.io/6fbgp

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 1, Article 50, 2019 [BX]

# Gödel’s incompleteness theorem in a nutshell

[microreview]

We present a microversion of Gödel’s theorem.

https://doi.org/10.31219/osf.io/gphsk

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 1, Article 45, 2019 [BS]

[microresearch]