# 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
```