11.4.4: History of Sentential Logic

Sentential Logic was created in 225 B.C.E. by the ancient Greek logician Chrysippus. That knowledge of logic was lost in the Dark Ages but was rediscovered by the French philosopher Abelard in the 12th century. The truth table system for Sentential Logic was invented in 1902 by the American logician Charles Peirce to display how the truth of some sentences will affect the truth of others. Truth tables were rediscovered independently by Ludwig Wittgenstein and Emil Post.

We haven't explored a proof system in this book, but a proof system for Sentential Logic was developed in 1879 by the German logician Gottlob Frege to enable us to create proofs analogous to proofs in plane geometry in which rules of inferenc e and axioms are used to infer a sentence from previously established sentences without knowing anything about which sentences are true. A proof is a list of sentences, a sequence of steps. A typical rule of inference is modus ponens:

From any two steps of a proof that have the forms P and also P→Q you may add a new step of the form Q.

Sentences P and Q can be complicated; they don’t need to be simple sentence letters. When applying this or any other rule of inference, there is no need to know or mention the truth-value of the sentences involved. The proof system for Sentential Logic is often called Sentential Calculus. If Sentential Logic goes instead by the name Propositional Logic, then its proof system is called Propositional Calculus.

Any argument which can be shown to be valid by the method of truth tables is also one whose conclusion can be proved from its premises by using axioms and rules of inference in the proof system. This property of Sentential Logic is called completeness. The completeness of Sentential Calculus was first proved by the American logician Emil Post in 1921.

Sentential Logic also has the reverse property, that any argument that is provable is also valid. So, validity and provability come to the same thing in the sense that the set of arguments that are valid is also the set of arguments whose conclusion can be proved from its premises.

When there is a mechanical method that in principle could answer all questions of a certain kind without ever giving a wrong answer and always giving some answer in a finite time and never relying on probability or creativity, the method is said to decide the question. The truth table method can be used to decide the question of which sentences of Sentential Logic are logical truths, that is, tautologies. We also make the point by saying logical truth in Sentential Logic is decidable. The question of whether an arbitrary sequence of symbols is a well-formed sentence of Sentential Logic is also decidable. Can you imagine how to design a computer program which, when given the input of any string of symbols, will always correctly tell you whether it is a well-formed sentence in Sentential Logic?

Computers are logic machines in two senses: their electronic design follows basic principles of symbolic logic, and their programs are also based on principles of symbolic logic. The first programming language evolved from a formal language for symbolic logic.

Three-valued logic was invented by the British logician Hugh MacColl in 1906. Jan Łukasiewicz was another early advocate of three-valued logic. He delivered a “farewell lecture” to the University of Warsaw in 1918, in which he announced dramatically, “I have declared a spiritual war upon all coercion that restricts man's free creative activity.” The logical form of this coercion, in Łukasiewicz's view, was Aristotelian logic, which restricted propositions to true or false. His own weapon in this war was three-valued logic.