Logic and argument: views across world philosophy

What makes an argument good, and can the rules of correct reasoning be written down? Aristotle was the first to treat them systematically. His syllogistic shows how a conclusion follows from two premises by their form alone, whatever the terms mean. The Stoic Chrysippus built a different system on the connections between whole propositions, made with words such as “if”, “or”, “and” and “not”. In China, the Mohists did not develop a formal logic: they treated reasoning as judging whether a new case is of the same kind as an accepted model.

In India, logicians asked what makes a reason a good one. Dharmakīrti required a reason to meet three conditions, linking the property cited as a reason to the thing being proved. In the Islamic world Avicenna made Aristotle’s syllogism the tool for verifying a claim, by finding the middle term that explains it. In the fourteenth century William of Ockham grounded his logic in a language of thought and a theory of what terms stand for in a sentence.

Gottlob Frege remade logic in 1879 with a notation that could analyse statements about “all” and “some”, the basis of the predicate logic still used today. Bertrand Russell found the paradox that undid Frege’s attempt to derive arithmetic from logic, and with Whitehead built a theory of types to avoid it. Logicians today recognise many kinds of logic beyond Aristotle’s, but his remains the first.

Through time

5th century BCE

Mozi

fl. 440–420 BCE · Chinese

The Mohists did not develop a formal logic. They saw all reasoning as distinguishing kinds: cite a model or standard (fa), show whether a case conforms to it, and so judge whether it is “this” or “not”. Later Mohists called this “extending kinds”, treating things alike in some respects as alike in another.

4th century BCE

Aristotle

384–322 BCE · Western

Aristotle gave the first systematic account of correct reasoning. His syllogistic shows how a conclusion follows from premises by their form alone, whatever the terms mean, and he extended it to necessity and possibility. Logic belongs to no one science: it sets out principles of good argument for every inquiry.

3rd century BCE

Chrysippus

c. 280 – c. 207 BCE · Western

As an alternative to Aristotle’s logic of classes, Chrysippus built a logic of propositions joined by “if”, “or”, “and” and “not”. He listed five basic forms of argument whose validity is beyond question, modus ponens among them, and hoped that every valid argument could be reduced to them.

Drew on and argued against Aristotle

About 900 years later

7th century CE

Dharmakīrti

c. 600 – c. 660 CE · Indian

A good reason must meet three conditions: the property cited as a reason must belong to the subject; it must be found only where the property to be proved is present; and it must be wholly absent where that property is absent. Inference, unlike perception, is conceptual thinking that proceeds by such reasons.

About 400 years later

11th century

Avicenna

c. 970 – 1037 CE · Arabic and Islamic

Logic is the instrument of philosophy. To verify a claim that A is C, one must find a middle term B that explains the connection and build the syllogism “A is B, B is C, so A is C”, following Aristotle’s account of demonstration in the Posterior Analytics.

Drew on Aristotle

13th century

Afdal al-Din al-Khunaji

1194–1248 · Arabic and Islamic

Avicenna’s logic needed revision, not reverence. Khunaji objected to its account of what a true statement must be about, and reworked its treatment of propositions, modality and conditionals, shaping logic in Arabic for centuries.

Argued against Avicenna

14th century

William of Ockham

c. 1287 – 1347 · Western

Written language depends on spoken language, and spoken language on mental language, the language of thought. His Summa of Logic moves from terms to propositions to arguments, using a theory of supposition, what a term stands for in a sentence, to give the truth conditions of propositions.

Drew on Aristotle

About 600 years later

19th century

E. E. Constance Jones

1848–1922 · Western

An informative statement joins terms that differ in meaning but pick out the same thing; seeing this resolves puzzles about what statements assert.

Gottlob Frege

1848–1925 · Western

Frege reconceived logic by building the first predicate calculus, in his Begriffsschrift of 1879. It analysed statements about “all” and “some”, and defined proof in terms still accepted today. He believed he had reduced arithmetic to logic, until Russell’s letter showed that his system was inconsistent.

20th century

Suzuki Daisetsu

1870–1966 · Japanese

Affirmation can come through negation: a thing is truly itself by not being itself. This ‘is and is not’ logic expresses what ordinary logic cannot.

Bertrand Russell

1872–1970 · Western

He refined Frege’s predicate calculus, still the basis of most systems of logic, to argue that mathematics is in a significant sense reducible to logic. In 1901 he discovered the paradox of the set of all sets that are not members of themselves, and between 1903 and 1908 answered it with his theory of types.

Drew on Gottlob Frege; argued against E. E. Constance Jones

Susan Stebbing

1885–1943 · Western

Logic is not only for specialists: it can be turned on the language of politicians, advertisers, journalists and popular scientists, to expose bad reasoning that affects public life.

Drew on Bertrand Russell

Ruth Barcan Marcus

1921–2012 · Western

Modal logic, the logic of necessity and possibility, can be formalised rigorously with quantifiers, against Quine’s doubts, and the result underpins a legitimate metaphysics of modality.