2. ARISTOTELIAN-MEDIAEVAL LOGIC

1. 2. ARISTOTELIAN-MEDIAEVAL LOGIC Classical logic deals with statements of the form S is P where S is the subject and P the predicate, that is, a property of the subject. For example: “Socrates is a bearded man”“Some Athenians are fat”“All Spartans are Greek” The verb to be is the copulative conjunction, linking the subject S to what we wish to predicate about it, P.

2. Aristotle also considers 3 principles (or laws), which are universally valid: PRINCIPLE OF IDENTITY: A is A, that is, in a correct reasoning, the meaning of terms must be constant PRINCIPLE OF NON-CONTRADICTION: in a given statement, we cannot both state and deny a predicate of the subject, at the same time and within the same range of meaning PRINCIPLE OF EXCLUDED MIDDLE: in a two-value logical system, like ours, a statement is either T or F (used in reductio ad absurdum)

3. There are 4 categorical statements in Aristotelian logic: 1. universal affirmative: “all S are P” [Affirmo]all that is S has the property P:“all ravens are black” 2. universal negative: “no S is P” [nEgo]nothing that is S has the property P:“no elephant is pink” 3. particular affirmative: “some S are P” [affIrmo]only some S have the property P:“some Italians are vegetarians” 4. particular negative: “some S are not P” [negO]some S do not have the property P:“some swans are not white” Each categorical statement is characterized both by its quality (affirmative/negative) and by its quantity (universal/particular)

4. There are 2 kinds of inferences, immediate and mediate ones. As to immediate inferences, they can be easily obtained from the so-called square of opposition: all S are P A “all ravens are black” no S is P E “no elephant is pink” contradictory statements contrary statements subalternate statements subcontrary statements some S are not P O “some swans are not white” some S are P I “some Italians are vegetarians”

5. contradictory statements cannot be both true or both false contrary statements cannot be both true, but they can be both false subcontrary statements cannot be both false, but they can be both true subalternate statements are statements that are either both true or both false, and one describes a situation that can be deduced from the one described by the other Some immediate inferences: a true A-statement entails a false O-statement a true I-statement entails a false E-statement a true A-statement entails a true I-statement a true E-statement entails a true O-statement …

6. ? = the truth-value of the derivative statement is indeterminate:it can be either true or false, depending on the primary statement.

7. ? = the truth-value of the derivative statement is indeterminate:it can be either true or false, depending on the primary statement.

8. There are other kinds of immediate inferences, allowing to validly move from a categorical statement to another, with a different grammatical and rhetorical form: conversion: it is the inference in which the subject and predicate are interchanged. It is only valid for the E and I propositions.There are two versions: conversio simplex (valid for E and I propositions) and conversio per accidens (valid for A propositions) obversion: it is the inference in which the quality of the proposition is changed and the predicate is interchanged with its complement. contraposition: it is the inference in which the subject is interchanged with the complement of the predicate and the predicate is interchanged with the complement of the subject.

9. CONVERSION RULE: subject and predicate are interchanged

10. OBVERSION RULE: the quality is changed and the predicate is interchanged with its complement

11. CONTRAPOSITION RULE: the subject is interchanged with the complement of the predicate and the predicate is interchanged with the complement of the subject

12. A A-converseA-obverse A-contrapositive E E-converseE-obverse E-contrapositive contradictory statements contrary statements subalternate statements I I-converseI-obverse I-contrapositive O O-converseO-obverse O-contrapositive subcontrary statements

13. Having dealt with immediate inferences, we now turn to mediate ones. Given two categorical statements – a “major” premise and a “minor” premise – we can infer another categorical statement – the conclusion. Such a structured argument is a syllogism, that is, “a discourse in which, certain things having been supposed, something different from the things supposed results of necessity because these things are so” (Aristotle, Prior Analytics, 24b18-20). A syllogism (συλλογισμός) is composed by several elements: – major premise, linking one term (major term, i.e. the predicate) to another term (middle term) – minor premise, linking one term (minor term, i.e. the subject) to another term (middle term) – conclusion, linking the major and the minor term.

14. major premise (M-P) minor premise (S-M) M = middle term P = major term (predicate) S = minor term (subject) conclusion (S-P) EXAMPLE “All men (M) are mortal (P)” “All Athenians (S) are men (M) “All Athenians (S)are mortal (P) • Two chief characteristics: • the conclusion results of necessity from the premises: it follows, that is, for the structure of the argument, not for its content • the middle term M does not appear in the conclusion, but it is what makes the inference possible

15. Although there are infinitely many possible syllogisms, there is only a finite number of logically distinct types. The middle term M can be either the subject or the predicate of each premise that it appears in. This gives rise to a classification of syllogisms into 4figures (σχήματα): M-P S-M P-M S-M M-P M-S P-M M-S S-P S-P S-P S-P I figure II figure III figure IV figure

16. EXAMPLES All heavy bodies fallAll books are heavy M-PS-M I figure All books fall S-P II figure No dog is a felineAll cats are feline P-MS-M No cat is a dog S-P III figure Some animals are wildAll animals are living beings M-PM-S Some living beings are wild S-P IV figure Some Italians are ChristiansAll Christians believe in God P-MM-S S-P Some Italians believe in God

17. As we have seen, a categorical statement can be of 4 different kinds: A, E, I and O. Each syllogism is composed by 3 statements. Therefore, each figure has 43 = 64 different types. All in all, then, the four figures have 64∙4 = 256possible syllogisms (or 512 if the order of the major and minor premises is changed, although this makes no difference, from the logical point of view). However, the vast majority of the 256 possible syllogisms are invalid, since the conclusion does not follow from the premises. In order to establish which syllogisms are valid, we lay down a few rules.

18. Rules of syllogisms: • There can be only three terms (major, minor and middle). • The middle term cannot be present in the conclusion. • From two negative premises no conclusion follows. • From two affirmative premises an affirmative conclusion follows. • From two particular premises no conclusion can follow. • The “strength” of the conclusion equals that of the weakest premise.

19. With these rules (and a few others), the 256 possible syllogisms reduce to 24 valid ones: 19 + 5 “weak”. In the Middle Ages logicians gave different names to each of them. Such names were chosen in order to immediately identify the structure and properties of each syllogism, as well as the way it could be proved: - the syllogisms of the first figure were taken to be the most perfect ones (to which all others must be reduced), and began with the first four consonants of the alphabet: B, C, D and F; - every name had 3 vowels, each of which referred to the quantity (universal/particular) and quality (affirmative/negative) of the premises and of the conclusion: A, E, I and O;

20. - the initial consonant of the various names indicates to which perfect syllogism it has to be reduced; - the internal consonants of the various names indicate the transformation path of syllogisms of the second, third and fourth figure into a syllogism of the first figure: s = conversio simplex p = conversio per accidens m = mutatio premissarum (switch of premises) c = contradictio Out of the 24 valid syllogisms, 5 are “weak”: their conclusions, that is, are weaker than those their premises could entail. They obtain by subalternating the conclusions of the corresponding normal types.

22. So far, in the syllogisms we have considered, both premises (and the conclusion) are categorical statements. But we can construct syllogisms whose premises are not necessarily categorical. - disjunctive syllogism: Steve is either from Milan or from Pisa Steve is from Milan Steve is not from Pisa [modus tollendo ponens] - hypothetical syllogism: one or both premises are hypothetical (it can be seen as a sorites in hypothetical terms); we may have: a) purely hypothetical syllogism, or b) mixed hypothetical syllogism

23. a) purely hypothetical syllogism: If Henry is bald, then Henry does not use combs If Henry does not use combs, then Henry does not buy combs If Henry is bald, then Henry does not buy combs b) mixed hypothetical syllogism: If Henry is bald, then Henry does not use combs Henry is bald Henry does not use combs [modus ponens] If Henry is bald, then Henry does not use combs Henry uses combs Henry is not bald [modus tollens]

24. Another class of syllogisms is that of imperfect categorical syllogism (enthymeme, or truncated syllogism): either the major premise, the minor premise or the conclusion is missing. [All good men are to be loved] Mike is a good man Mike is to be loved All Greeks are free men [Athenians are Greeks] All Athenians are free men No good politicians can be corrupted Some politicians are corrupted [Some politicians are not good politicians]

25. Another class of syllogisms is that of imperfect categorical syllogism (enthymeme, or truncated syllogism): either the major premise, the minor premise or the conclusion is missing. [All good men are to be loved] Mike is a good man Mike is to be loved All Greeks are free men [Athenians are Greeks] All Athenians are free men No good politicians can be corrupted Some politicians are corrupted [Some politicians are not good politicians]

26. Still other syllogisms are composite, that is, they can be seen as (and therefore reduced to) simple syllogisms. 1. conjunctive syllogism, in which one of the premises contains a conjunction: All Athenians are Greek Socrates and Plato are Athenians Socrates and Plato are Greek

27. 2. polysyllogism (multi-premise syllogism, climax or gradatio) is a string of any number of syllogisms such that each major premise is the conclusion of a syllogism: All Athenians are Greek Some Athenians are bald Some Greeks are bald All bald do not use combs Some Greek do not use combs

28. 3. sorites (“heap” of propositions) is a series of statements so arranged that the predicate of each one that precedes forms the subject of each one that follows, and the conclusion unites the subject of the first statement with the predicate of the last statement. It is, in other words, a polysyllogism lacking some premises, or a chain of enthymemes: The soul is a thinking agent A thinking agent can not be severed into parts That which can not be severed can not be destroyed Therefore the soul can not be destroyed

29. 4. epicherema (“attempt to prove”) is a syllogism in which the justification of the major or minor premise, or both, is introduced with the premises themselves, and the conclusion is derived in the ordinary manner. Each justification can be seen as an enthymeme: so, if we add the lacking statements, the epicherema turns into a polysyllogism. All bodies fall because of gravity The pen is a body The pen falls

30. Reductio ad absurdum As we have seen, in order to prove something, we must move from the premises (hypotheses) to the conclusion (thesis). So far, we have considered only direct proofs, that is, we have concluded directly from the premises, in either an immediate or a mediate way. We shall now consider an indirect way of proof.

31. At the beginning of the Elements, after a few definitions (point, line, surface…), Euclid introduces five postulates: they cannot be proved, and are taken for granted. The 13 books that follow present theorems. Theorems, by contrast, are not taken for granted, but are proved: their premises constitute the initial hypotheses, and the conclusion is the thesis that is to be proved. A theorem is a statement of this kind: from some hypotheses H we deduce the thesis T T, whose truth is to be proved, follows – by means of deduction – from the hypotheses H, whose truth is taken for granted (either because they were previously proved, or because they are axioms)

32. Let us consider the following theorem: “Straight lines parallel to a third lineare parallel to each other” • In order to prove it, let us assume two hypotheses: • the working hypothesis according to which there are two straight lines a and b parallel to a third line; • the hypothesis provided by Euclid’s fifth postulate, according to which given a line, and a point not on that line (P), there is one and only one line that passes through that point and is parallel to the other line. The thesis states that a and b, both parallel to c, are parallel to each other. Let us now deny the thesis and state that a and b, both parallel to c, are not parallel to each other.

33. According to this counter-thesis, through P we can draw two straight lines a and b, each of which will be – according to our working hypothesis – parallel to c. That is to say that through the point P, which is not on the line c, pass two lines parallel to c. This contradicts Euclid’s fifth postulate, which we assumed as our starting hypothesis. In other words, stating the counter-thesis leads to a contradiction with the initial hypotheses. Therefore, the counter-thesis is to be deemed false. But if the counter-thesis, which is the negation of the original thesis, is false, the thesis is true. As a consequence, a and b are parallel to each other.

34. Let us illustrate the various step of our reductio ad absurdum by way of a flow-chart: 1. Statement of thesis 2. Statement of hypotheses 3. Negation of the thesis • Logical consequences of the counter-thesis • Contradiction between one of the consequences and one of the initial hypotheses principle of non-contradiction • The counter-thesis is false principle of excluded middle • The thesis is true – QED