MATHEMATICAL LOGIC CLASS NOTE 1. Propositional logic A
... (1) α is not satisfiable iff ¬α is tautology. (But a satisfiable sentence need not be tautology.) (2) α, β are tautologically equivalent iff α ↔ β is a tautology. (3) De Morgan’s law: ¬(φ ∧ ψ) ↔ ¬φ ∨ ¬ψ; ¬(φ ∨ ψ) ↔ ¬φ ∧ ¬ψ are both tautology. A literal is a sentence which is a sentence symbol, or the ...
... (1) α is not satisfiable iff ¬α is tautology. (But a satisfiable sentence need not be tautology.) (2) α, β are tautologically equivalent iff α ↔ β is a tautology. (3) De Morgan’s law: ¬(φ ∧ ψ) ↔ ¬φ ∨ ¬ψ; ¬(φ ∨ ψ) ↔ ¬φ ∧ ¬ψ are both tautology. A literal is a sentence which is a sentence symbol, or the ...
Section 1.3 Functions
... Characteristics of a function from Set A to Set B (page 27) 1. Each element in A must be matched with an element in B. 2. Some elements in B may not be matched with any element in A. (leftovers) 3. Two or more elements in A may be matched with the same element in B. 4. An element in A (the domain) ...
... Characteristics of a function from Set A to Set B (page 27) 1. Each element in A must be matched with an element in B. 2. Some elements in B may not be matched with any element in A. (leftovers) 3. Two or more elements in A may be matched with the same element in B. 4. An element in A (the domain) ...
Natural Deduction Proof System
... • Natural Deduction tries to follow the natural style of reasoning. Most of the proof consists of forward reasoning, i.e. deriving conclusions, deriving new conclusions from these conclusions, etc. Occasionally hypotheses are introduced or dropped. • A derivation is a tree where the nodes are the ru ...
... • Natural Deduction tries to follow the natural style of reasoning. Most of the proof consists of forward reasoning, i.e. deriving conclusions, deriving new conclusions from these conclusions, etc. Occasionally hypotheses are introduced or dropped. • A derivation is a tree where the nodes are the ru ...
2015Khan-What is Math-anOverview-IJMCS-2015
... 3. Axioms or postulates; 4. Theorems and their proofs. We now discuss each of them as follow. UNDEFINED TERMS: To build a mathematical system based on logic, the mathematician begins by using some words to express their ideas, such as `number' or a `point'. These words are undefined and are sometime ...
... 3. Axioms or postulates; 4. Theorems and their proofs. We now discuss each of them as follow. UNDEFINED TERMS: To build a mathematical system based on logic, the mathematician begins by using some words to express their ideas, such as `number' or a `point'. These words are undefined and are sometime ...
To What Type of Logic Does the "Tetralemma" Belong?
... distributive law no longer holds. More recently, though, a different type of logical structure has been put forward, in which the rules for combining propositions are the classical ones, but what changes are the rules of inference [8] [9]. It is these “anhomomorphic” logics, I would suggest, that h ...
... distributive law no longer holds. More recently, though, a different type of logical structure has been put forward, in which the rules for combining propositions are the classical ones, but what changes are the rules of inference [8] [9]. It is these “anhomomorphic” logics, I would suggest, that h ...
sets
... List all pairs belonging to the relation. Use set builders to describe the pairs. Example 1: R0 = {(4,3), (9,2), (3,6), (7,5)} is a relation on N. It is also a relation on A x B where A = {4,9,3,7} and B = {3,2,6,5} ...
... List all pairs belonging to the relation. Use set builders to describe the pairs. Example 1: R0 = {(4,3), (9,2), (3,6), (7,5)} is a relation on N. It is also a relation on A x B where A = {4,9,3,7} and B = {3,2,6,5} ...
Principia Mathematica
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1927, it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced ✸9 and an all-new Appendix C.PM, as it is often abbreviated, was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy, being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with the notion of a hierarchy of sets of different 'types', a set of a certain type only allowed to contain sets of strictly lower types. Contemporary mathematics, however, avoids paradoxes such as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory.PM is not to be confused with Russell's 1903 Principles of Mathematics. PM states: ""The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions.""The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century.