Why modern logic took so long A potted history in five periods to arrive: Three lectures 1. The classical Greeks, 4th century BC Aristotle introduces the idea of checking arguments by Wilfrid Hodges showing that they conform to valid argument patterns. He describes a systematic collection of valid argument Herons Brook, Sticklepath, patterns (syllogisms) and some sporadic ones (topics). Okehampton EX20 2PY Chrysippus describes some propositional argument March 2009 patterns. http://wilfridhodges.co.uk/history11.pdf 3 1 2. The Roman Empire, 2nd to 6th centuries AD Aristotle’s works, edited by Andronicus of Rhodes, become the basis of liberal education. Late 3rd century, Porphyry protects Aristotle’s logic from Lecture One: What traditional logic covered ideological disputes by separating it from metaphysics and and in particular the place of relational reasoning proposing a basis in terms of meanings. Details worked out by his followers and the Alexandrian school of Ammonius. 6th century, Boethius paraphrases in Latin Porphyry’s views on logic. 4 2 3. The Arabs, 8th to 13th centuries Excellent translations made of Aristotle. Translations of 5. Renaissance and Enlightenment, 15th to 19th centuries Roman Empire commentators become available in libraries Logic emerges from the universities and becomes education of connoisseurs as far east as Afghanistan. for barristers and young ladies. 10th century, Al-Far¯ ab¯ ¯ı writes major commentary (now 17th century, attempts to adapt logic to the spirit of the age mostly lost) on Aristotle’s logic. of Descartes and Newton. Main figures Arnauld and Nicole 11th century, Ibn S¯ına¯ an independent thinker in the (Port-Royal Logic) and above all Leibniz. aristotelian tradition (compare Leibniz and Frege — as we 19th century, old idea of logic as protection against error will). One of his textbooks of logic is over 2000 pages. revived in more sophisticated form, in particular by Frege. Later writings on logic (e.g. Ibn Rushd 12th century, Tusi With Peano (1890s) the connection to Aristotle is finally lost. 13th century) show less independence. 7 5 4. The Scholastics, 12th to 15th centuries Aristotle’s syllogisms 12th century, Abelard virtually reinvents logic on basis of A typical syllogism, as Aristotle wrote it (Prior Analytics i.6): Boethius and an incomplete set of texts of Aristotle. If R belongs to every S,andP to no S, there will be a 12th to 13th centuries, Terminists develop theory of deduction that P will necessarily not belong to some supposition. (Unclear whether it’s about the notion of R. reference or about infinitary proof rules.) Here ‘R’, ‘P ’, ‘S’ stand for nouns. (Proper names are treated Early 14th century, Jean Buridan probably the most as common nouns.) Thus for example: technically proficient of the Scholastics. If every animal is mobile, and no animal is eternal, Throughout this period, theory of logic was confined to a then necessarily some mobile thing is not eternal. handful of universities. 8 6 Later logicians saw that to make this example work, we Hilbert used this topic as an argument for adopting need to assume something about existence. first-order logic: S The usual assumption was that if there are no sthen “If there is a son, then there is a father,” is certainly a • ‘Some S is an R’isfalse; logically self-evident assertion, and we may demand of any satisfactory logical calculus that it make • ‘Every S is an R’isfalse; obvious this self-evidence, in the sense that the • ‘Some S is not an R’ (negation of ‘Every S is an R’) is asserted connection will be seen, by means of the true; symbolic representation, to be a consequence of simple logical principles. • ‘No S is an R’istrue. 11 9 Topics Also known as places. The ones for use in general situations This is from Hilbert’s Gottingen¨ lectures of 1917–1922, are called common places; Renaissance scholars suggested published in 1928 in his textbook with Ackermann. keeping a commonplace book for them. The most general In these lectures Hilbert created the syntax and semantics of topics are called maximal places, abbreviated to maxims. first-order logic. Example: What’s so hot about first-order logic, if this topic was already known to Boethius in the 6th century? If one of the correlated things is posited, the other is posited. (Peter of Spain, Tractatus V para. 28.) Answer (in Hilbert’s text): First-order logic is a logical calculus that analyses both syllogisms and relational This is the pattern behind the inference arguments like this one down to ‘simple logical principles’. If there is a father then there is a child. 12 10 For the rest of this lecture, we ask why nobody before the end of the 19th century seriously tried to integrate syllogisms and relational topics into a calculus covering Some Basics both. How are syllogisms supposed to be used, in practice, One meets three views: for validating arguments? • (Joachim Jungius) Relational topics are sui generis. There are some scattered remarks about this at the end of • (Leibniz) Relational topics can be reduced to syllogisms Aristotle’s Prior Analytics i. plus paraphrasing. They show that Aristotle’s practice was probably almost identical to modern elementary logic courses. • (Buridan, De Morgan) Syllogisms and some relational topics are justified by a higher-level principle called dictum de omni et nullo. 15 13 Jungius is chiefly memorable for having stimulated Leibniz. Given an argument in Greek, Aristotle shows that it fits The other views, plus relevant opinions of Ibn S¯ına,¯ some valid syllogism by showing what terms in the all revolve around the question whether one can really argument correspond to the term symbols in the syllogism. reason with anything beyond syllogistic sentences E.g. (Prior Analytics i.35): (Some/Every) A (is/isn’t) a B, A : having internal angles that sum to two right angles. where the terms in place of A and B are taken as B impenetrable. :triangle. C : isosceles triangle. The view that one can’t is what I call Top-Level Processing, TLP for short. (Some refinements will follow.) He calls this setting out the terms. 16 14 Setting out terms (in this sense) disappears after Aristotle Natural language paraphrase: and reappears only in Boole 1847. What takes its place? No (time needing to be set aside for action) is a To check an argument, we have to verify that (for example) (thing that God has). Some (right moment for action) thefirstpremisemeansthesameas‘EveryB is an A’, where is a (thing that God has). Therefore some (right ‘A’, ‘B’ are interpreted as in the setting out of terms. moment for action) is not a (time needing to be set Instead, the traditionals rewrote ‘Every B is an A’usingthe aside for action). expressions in the setting out of terms, and checked that the The subject and predicate terms are marked with brackets. result means the same as the premise. I refer to the corresponding parts of the original argument as So they made a natural language paraphrase of the premise. the eigenterms (adapting Gentzen). 19 17 Example (Aristotle, Prior Analytics i.36) God doesn’t have times that need to be set aside for Why does the paraphrase of each sentence have to be a action. God does have right moments for action. syllogistic sentence? Therefore some right moment for action is not a time that needs to be set aside for action. From a modern perspective, a no-brainer. We are validating by syllogisms, and this just is the form of the sentences in Setting out: syllogisms. In another logic, other forms would be used. A : thing that God has. B : time needing to be set aside for action. But traditional logicians often seem to want to show that C : right moment for action. reasoning can only use syllogistic (or very similar) sentences. Syllogism: This blocks any attempt to find a more powerful logic. No B is an A.SomeC is an A. Therefore some C is not a B. 20 18 Leibniz believes that the key to reasoning is that we can paraphrase the premises so as to bring the eigenterms to Ibn S¯ın¯a has a complex and highly integrated theory that nominative case. uses logic to explain the workings of the rational mind, Opuscules p. 287: including the kinds of error that one can make. When we realise for the first time that something is true, A reading of poets is an act by which a poet is read. on the basis of a rational argument, this is because our ...ParisisaloverofHelen,i.e.Parisisalover,and minds analyse the data, notice common features between thereby Helen is loved. So there are two propositions the propositions expressing the data, and by identifying packed into one. If you don’t resolve oblique these features, produce a new combination of ideas. cases into several propositions, you will never avoid being forced (like Jungius) to devise new-fangled ways of reasoning. 23 21 Remark ‘Recombinant’ syllogisms (Aristotle’s syllogisms and their Leibniz’s basic strategy here is that of Peirce ‘The reader is adaptations to propositional logic) express this kind of introduced to relatives’ (1892) and Davidson ‘The logical argument. form of action sentences’ (1967): Our minds are a kind of PROLOG inference engine. Quantify over actions or events, and treat other parts In short, the basic ingredient of reasoning is to bring of the situation as attached to the action or event in together two ideas. standard ways (e.g. as AGENT or OBJECT). The syllogistic sentences express such a union of ideas. Hence Top-Level Processing. This doesn’t get rid of relations, but it limits them to standard ones built into the language.