# An outline for Logic Systems through time

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* The flowering of logic systems began with the Greeks. *

Western Europe and the Islamic world borrowed from this
treasure chest of logic. In modern times logics based on Greek logic have spread through the entire world.

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### __Conditional logic__ - *Megarian School* - Euclid of Megara *student of Socrates*

#### Euclid of Megara 435-365 BC

Conditional logic is the "if - then " logic.
### __Sylogistic logic__ - *Peripatetics/Lyceum*-Athens Aristotle

#### Aristotle 384–322 BC

Aristotle invented the famous syllogism which is actually a system of logic.
### __Early axiomatic logic__

#### Euclid 300 BC

Euclid of Alexandria did geometry with axiomatic logic.
From a small number of axioms (assumptions) he defined and could prove planar geometry
### __Early Propositional and Conditional logic__ - *Stoics*

#### Chrysippus 279 – 206 BC

Built and refined Megarian logic.
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* Western Europe built upon Greek logical systems *

The Romans borrowed much of Greek Philosophy and later the Western Europeans borrowed Greek Philosophy from Byzantium ( Eastern
Roman) and the Islamic world.

*Western Europe's Mathematical logic*

One way to look at logic is that it can be calculated.
Starting with Boole, Western Europe mathematics becomes involved with logic.
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__Early forms of mathematical logic__

#### Leibniz 1646-1716

Much of Leibniz logical work was not published until the twentieth century.
__ Combinatory logic - Boolean Algebra__

#### Boole 1815-1864

Boolean Algebra was early mathematical system of logic which could do combinatory logic.
Today this system of logic is extensively used in Computer Engineering, Computer Science and Control Engineering.
__ Set Theory__

#### Georg Cantor 1845 – 1918

Logical systems could be interpreted using set theory and formed a mathematical basis for modern
logic.
### __ Axiomatic arithmetic logic __

#### Peano 1858 – 1932

Peano created axioms using set theory to explain arithmetic.
### __ Formal logic __

#### Frege 1848-1925

Invented modern propositional and predicate logic to create logical system for arithmetic.
He also used Peano's axioms and set theory.
### __ Principia Mathematica __

#### Russell Russell 1872 – 1970 and Whitehead 1861 – 1947

Basically takes what Frege did and fixed Russell's paradox. This
starts a few philosophical schools - Logical Atomism, Logical Positism and Analytical Philosophy. Also several famous
philosphers and mathemacians made their start with Principia Mathematica such as Wittgenstein. Was supposed to
be the basis of all mathematics and philosophy until Godel came along.
__ Propositional logic __

#### Emil Post 1897 – 1954

Proved propositional logic of the Principia used was consistent and complete.Consistent means there are no
contradictions. Completeness means that either a proof or disproof exists for every theorm.
__ Predicate logic __

#### David Hilbert 1862 – 1943 William Ackerman 1896 – 1962

Hilbert and Ackerman proved that Predicate logic was consistent.
__ Predicate logic __

#### Kurt Godel 1906 – 1978

Godel proved that Predicate logic was complete.
__ Principia Mathematica is incomplete__

#### Kurt Godel 1906 – 1978

Godel proved that using Peao's axioms, set theory in the Principia as a basis for mathematics was incomplete.
__ Nu recursion__

#### Kurt Godel 1906 – 1978

Godel's Nu recursion system was Turing complete and could potentially solve Hilberts decidability problem.
Decidability means that there's an algorithm for finding a proof or disproof.
Hilbert's decidability question: does there exist a “definite method” that, when given any
possible statement in mathematics, can decide whether that statement is true or false?
__ Lambda calculus__

#### Alonso Church 1903 – 1995

Church gets credit for solving Hilberts decidability problem with lambda calculus. His lambda
calculus proved it was not decideable. Lambda calculus was later a basis for functional programming.
__ Universal Turing Machine__

#### Alan Turing 1912 - 1954

Turing also solves Hilberts decidability problem. The Turing Machine proved it was not decideable with the halting
phenomenon. His system was the most intuitive and useful and is used by Computer Science.
Anything that is equivalent to a Universal Turing Machine is considered a theorectical __ computer __ such
as lambda calculus, nu recursion etc.
Turing Machines are state machines also known as __automata __
Other state machines which have more practical uses are __ Finite State Automota __ and __ Push Down Automota. __
__ Finite State Machines __

#### Warren McCulloch 1898 - 1969 Walter Pitts 1923 - 1969

McCulloch was a Psychologist and Pitts was a self taught logician who met with an interdisciplinary team interested
in the mind and computers. They were first to define finite automata.
__ Artificial Neural Networks __

#### Warren McCulloch 1898 - 1969 Walter Pitts 1923 - 1969

McCulloch was a Psychologist and Pitts was a self taught logician who met with an interdisciplinary team interested
in the mind and computers. They created a computational model for neural networks. These loosely model the neurons in the
human brain. These neural networks allow computers to do pattern recognition and allow the solving of problems in
AI, machine learning and deep learning.
__ Chomsky's Hierarchy__

#### Noam Chomsky 1928 -

Chomsky believes Humans acquire early language very quickly because they have a language acquition system equivalent to
a Universal Turing Machine in their brain. However, apart from linquistic theory, this Hieracrchy is very useful to the study of
logical systems and automata.
Grammar Languages Automaton
Type-0 Recursively enumerable Turing machine
Type-1 Context-sensitive Linear-bounded non-deterministic Turing machine
Type-2 Context-free Non-deterministic pushdown automaton
Type-3 Regular Finite state automaton