An outline for Logic Systems through time


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.


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.

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.

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