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A Computationally-Discovered Simplification of the Ontological
A Computationally-Discovered Simplification of the Ontological

pdf file
pdf file

Large cardinals and the Continuum Hypothesis
Large cardinals and the Continuum Hypothesis

... decides CH one way or the other? In fact there are many of these, such as MA or PFA,2 but we will require ϕ to be one of a more special kind. In 1946, that is well before the development of forcing, Gödel entertained the idea of so called stronger axioms of infinity deciding CH (and other independ ...
you can this version here
you can this version here

Gödel Without (Too Many) Tears
Gödel Without (Too Many) Tears

... Gödel’s doctoral dissertation, written when he was 23, established the completeness theorem for the first-order predicate calculus (i.e. a standard proof system for first-order logic indeed captures all the semantically valid inferences). Later he would do immensely important work on set theory, as ...
The Mathematics of Harmony: Clarifying the Origins and
The Mathematics of Harmony: Clarifying the Origins and

Constructions of the real numbers
Constructions of the real numbers

On the specification of sequent systems
On the specification of sequent systems

2.3 Deductive Reasoning
2.3 Deductive Reasoning

Symbolic Logic I: The Propositional Calculus
Symbolic Logic I: The Propositional Calculus

notes
notes

General Dynamic Dynamic Logic
General Dynamic Dynamic Logic

The substitutional theory of logical consequence
The substitutional theory of logical consequence

... For formalized languages the substitutional account of logical validity has largely been superseded by the proof-theoretic and the model-theoretic approaches. According to the proof-theoretic or inferentialist conception, roughly, an argument is valid if and only if the conclusion can be derived fro ...
Prolog 1 - Department of Computer Science
Prolog 1 - Department of Computer Science

Internal Inconsistency and the Reform of Naïve Set Comprehension
Internal Inconsistency and the Reform of Naïve Set Comprehension

relevant reasoning as the logical basis of
relevant reasoning as the logical basis of

Propositional Logic
Propositional Logic

Quine`s Conjecture on Many-Sorted Logic
Quine`s Conjecture on Many-Sorted Logic

AGM Postulates in Arbitrary Logics: Initial Results and - FORTH-ICS
AGM Postulates in Arbitrary Logics: Initial Results and - FORTH-ICS

First-Order Intuitionistic Logic with Decidable Propositional
First-Order Intuitionistic Logic with Decidable Propositional

... their subsets”. Propositional logic can be considered a part of the mathematics of finite sets because of availability of finite models using truth tables. Thus, LEM for propositional formulas is not really a target of intuitionistic criticism of classical logic. The classical assumption that every ...
HKT Chapters 1 3
HKT Chapters 1 3

PDF
PDF

THE EQUALITY OF ALL INFINITIES
THE EQUALITY OF ALL INFINITIES

Quine`s Conjecture on Many-Sorted Logic∗ - Philsci
Quine`s Conjecture on Many-Sorted Logic∗ - Philsci

slides
slides

... If there are infinitely many possible values for X the meaning of this expression cannot be represented using a propositional formula. In AG, the meaning of aggregate expressions is captured using an infinitary propositional formula. The definition is based on the semantics for propositional aggrega ...
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Axiom

An axiom or postulate is a premise or starting point of reasoning. As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy.The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.' As used in modern logic, an axiom is simply a premise or starting point for reasoning. What it means for an axiom, or any mathematical statement, to be ""true"" is a central question in the philosophy of mathematics, with modern mathematicians holding a multitude of different opinions.In mathematics, the term axiom is used in two related but distinguishable senses: ""logical axioms"" and ""non-logical axioms"". Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, ""axiom,"" ""postulate"", and ""assumption"" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. As modern mathematics admits multiple, equally ""true"" systems of logic, precisely the same thing must be said for logical axioms - they both define and are specific to the particular system of logic that is being invoked. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Within the system they define, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow otherwise they would be classified as theorems. However, an axiom in one system may be a theorem in another, and vice versa.
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