Arabic Logic and Syllogisms

Arabic logic begins in the middle of the eighth century. As with logic in the Latin tradition, it has its foundation in Ancient Greek logic, primarily in Aristotelian logic and syllogistics. The Syriac Christians adopted a teaching tradition in logic that included Porphyry’s Isagoge in addition to Aristotle's Categories, De interpretatione and the first seven chapters of the Prior Analytics. This teaching tradition was adopted and spread through the Arab conquest. During the Abbasid Caliphate (750-1258), there was a continuous and growing interest in philosophy and logic. It is this time period that is often referred to as the ‘Golden Age’ of Arabic philosophy and logic.
Gradually, the whole Organon was made available in Arabic translation (the Arabic tradition was unique in treating Aristotle’s Rhetoric and Poetics as part of logic, unlike later traditions in medieval logic. The first more important Arabic logician was Ishâq al-Kindî (d. 870), who wrote a short overview of the whole of the Organon. After him, more and more substantial works were produced. Abû Nasr Alfarabi (d. 950) made the first original contributions, writing a series of commentaries on Aristotle, although only his commentary on De interpretatione has survived. Avicenna seems to have held his work in very high esteem. By far the most important logician in the Arabic tradition, however, was Ibn Sînâ (d. 1037) or Avicenna, as he was known in the Latin West.
Avicenna had a different attitude to Aristotle’s logic than logicians before him. He did not think that Aristotle was necessarily right. Aristotle had a lot of intuitions about logic that do not all fit together into a coherent whole. They had to be worked out and Avicenna believed that when that happened, it would become clear that Aristotle’s logic was only a fragment of a much larger system. After Avicenna, the general character of Arabic logic was no longer Aristotelian but Avicennan, which is to say that the texts drawn upon by most logicians were no longer Aristotle's but Avicenna’s (with the notable exception of Averroes, known to the Latin tradition as 'the Commentator', i.e., on Aristotle). One work of Avicenna in particular became important for subsequent logicians: what is known as Al-Ishârât wa’l Tanbîhat in Arabic and Pointers and Reminders in English -- or Remarks and Admonitions in S. Inati’s translation (Avicenna 1984).
Tony Street (2002, 2004) has identified three things that make logic Avicennan as opposed to Aristotelian: (1) truth-conditions of absolute (or assertoric, i.e., non-modal) sentences are expressed in modal terms; (2) logical properties of so-called descriptional (wasfi) sentences, such as ‘Every B is A while B’, are studied; and (3) syllogisms are divided into connective and repetitive. If a logician adopts (1)-(3), he is following Avicenna, according to Street.
In Pointer Two of Path Four, Avicenna introduces distinctions between different kinds of sentences. The first distinction is between absolute and modal sentences, although absolute sentences turn out to be modal as well. The basic division is one between absolute sentences that are taken to be definite or indefinite with respect to time.
Avicenna talks about three kinds of absolute sentences, all of which are explicated with reference to time. First are absolute sentences that refer to a definite time, but these play no role in his discussion. The other two are the general and the special absolute sentences. General absolute sentences are sentences taken without limitation with respect to time, which means that they have to take in all individuals - past, present and future. Furthermore, the copula is taken to mean that the Bs are As at least sometime, as in ‘Every human is sometimes moving’. A special absolute sentence is a sentence with limitations with respect to time, its subject term referring to individuals at a specific moment in time - although it is not explicated what moment in time that is. The copula is also understood as a conjunction meaning ‘sometimes B and sometimes not B’, as in ‘Everything running is sometimes walking and sometimes not walking’.
Avicenna is quick to point out that neither general nor special absolute sentences behave as expected. For example, they do not fit the traditional square of opposition. ‘Every B is A’ on the general reading does not contradict ‘Some B is not A’. Thus, he introduces one other kind of absolute sentence, namely a perpetual absolute sentence. In a perpetual sentence the copula is simply read as ‘is always’. The contradictory of a general absolute is the perpetual absolute, and similarly with the special absolute, although it will contradict a disjunction of two perpetual sentences (see Street 2002 and Lagerlund 2009).
The second distinctive Avicennan thesis is the introduction of descriptional (wasfi) sentences. This is again done in the context of modal syllogistics, although, such sentences do not need to be modal at all and Avicenna can be seen to have introduced a logic for descriptional sentences (Street 2002). The example he gives in Pointers and Reminders is:
(3:1) Everything walking is necessarily moving while walking
The addition of ‘while walking’ restricts all moving things to those actually walking, which makes the sentence true. Avicenna distinguishes descriptional sentences from substantial sentences. The example he gives of a substantial sentence is:
(3:2) Every human is necessarily an animal
The logic for substantial sentences is different from the logic for descriptional sentences. A sentences like (3:2) converts according to the standard Aristotelian conversion rules, so:
(3:3) Every human being is necessarily an animal
converts into:
(3:4) Some animal is necessarily a human.
Such sentences are characterized by as kath’ hauto (per se) by Aristotle in Posterior Analytics I.4 (see Lagerlund 2000, 30-1). Part of what Artistotle said about modal syllogistic is valid for such sentences.
But another group of sentences, such as:
(3:6) Every literate being is necessarily a human being
are not substantial and hence do not convert, since this converted sentence is false:
(3:7) Some human being is necessarily literate.
However, if these are read as descriptional sentences, then they do convert:
(3:8) Every literate being is necessarily a human being while literate
converts into
(3:9) Some human being is necessarily literate while literate.
Descriptional sentences have a syllogistic logic like substantial sentences and Avicenna thinks part of Aristotle’s modal syllogistic can be worked out using descriptional sentences (for a comparison with similar logics found in the thirteenth-century Latin tradition, see Lagerlund 2009). But even though Avicenna sketches a syllogistic for descriptional sentences in Pointers and Reminders, he is mostly concerned with substantial sentences and their logic.
The third distinctive mark of Avicenna’s logic is the distinction between so-called connective and repetitive syllogisms, which corresponds roughly to Aristotle’s distinction between categorical and hypothetical syllogisms.
In his history of Arabic logic, Khaled El-Rouayheb divides Arabic logic after 1200 into several distinctive periods (El-Rouyaheb 2010). According to him the next period begins with Fakhr al-Din al-Razi. After Razi, the later Arabic logical tradition became disassociated from Aristotle and more narrowly focused on the predicables, definitions, propositions, and syllogisms.
Most thirteenth-century logic can also be described as post-Avicennan in the sense that logicians in this period all took their departure from Avicenna rather than from Aristotle. In the fourteenth century another transformation took place and the lengthy summaries found in the earlier traditions became very rare. Instead of writing commentaries on the works of Aristotle, Arabic logicians were content with writing glosses. Their interest also shifted from formal logic (syllogisms) to semantic concerns.
Arabic logic began to fragment in the fifteenth and sixteenth centuries and several centers developed. El-Rouyaheb identifies distinct Ottoman Turkish, Iranian, Indo-Muslim, North African, and Christian Arabic traditions. These developed independently of one other, and, according to El-Rouyaheb, it is the Ottoman Turkish tradition that is the most important up to the twentieth century. The basic themes outlined by Avicenna remained dominant in this tradition, however.

4. Peter Abelard

Peter Abelard (1079–1142) was one of the first original medieval logicians in the Latin West. His most thorough treatment of the theory of the syllogism can be found in the Dialectica, though he occasionally discusses it in other works as well, such as the Logica ingredientibus(Minio-Paluello 1958). It is only in the Dialectica, however, that the theory is outlined in full.
Since the logic of the Dialectica is based on Boethius' commentaries and monographs, we find in it a treatise on categorical sentences and categorical syllogisms (Tractatus II), and another on hypothetical sentences and hypothetical syllogisms (Tractatus IV). But neither of these discussions is very extensive. Taken together, they are shorter than the discussion of topical inferences, which indicates that Abelard was most interested in developing a logic for sentences (Green-Pedersen 1984 and Martin 1987). His presentation of syllogistic is condensed but highly original. It reveals that he was not able to study the text of Aristotle's Prior Analytics in any detail. He must have seen it, but he cannot have had access to a copy himself.
Abelard gives the four standard figures and shows how the second, third, and fourth (he treats the fourth figure as part of the first figure with the terms in the conclusion converted) can be reduced to the first in the standard ways using conversion rules and proofs through impossibility, but to clarify and simply the theory he also presents rules showing the validity of the different moods. In the first figure he gives these rules (I have included the terms A, B, C to clarify the rules, though they are not in Abelard's text):

اقتباس :

(4:1)	If something A is predicated of something else B universally and a third thing C places the subject B under it universally, then the same thing C also places the predicate A under it with the same mode, namely universally.
(4:2)	If something A is removed from something else B universally and a third thing C places the subject B under it universally, then the first predicate A is removed from the second subject C universally.
(4:3)	If something A is predicated of something else B universally and some third thing C places the subject B under it particularly, then that thing C also places the predicate A under it particularly.
(4:4)	If something A is removed from something else B and a third thing C places the subject B under it particularly, then the first predicate A is removed from the second subject C particularly.

To these he adds two more rules for the second figure:

اقتباس :

(4:5)	If something B is removed from some other thing A and a third thing C places that predicate B under it, then the first subject A is removed from the second subject C universally.
(4:6)	If something B is predicated of some other thing A universally and that predicate is removed from a third thing C universally, then the [first] subject is removed from the same [subject] C universally.

There are three more rules for the third figure:

اقتباس :

(4:7)	If two different things A and C are predicated of the same B universally, then the first A predicated of the second comes together particularly.
(4.8)	If something B is removed from something A universally and something third C is predicated of the same subject B universally, then the first predicate A is removed from the second C particularly.
(4:9)	If something A is predicated of something B particularly and the same B with another predicate C supposits universally, then the first A is predicated of the second C particularly.

If we allow that the conjuncts in the antecedent of these conditional statements can switch places, and that a universal implies a particular, these rules exhaust the 24 valid syllogisms.
Abelard's rules 1 and 2 are equivalent to the rules of class inclusion that later became the subject of much discussion, i.e., the so-called dici de omni et nullo rules. These rules are based on the transitivity of class inclusion and were the standard way in which later medieval logicians explained how the first figure moods are perfect or evident.
It was elegant of Abelard to lay out these rules that entail the valid moods, but then again, the theory of the syllogism is an elegant and simple system. The simplicity of his nine rules reflects the simplicity of Aristotelian syllogistic, since on Aristotle's view only the first two syllogisms and the rules of conversion plus the method of proof by impossibility and a couple of other consequences are needed to demonstrate all 24 valid moods.
Abelard's hypothetical syllogistic does not repeat Boethius' mistake of mixing a term logic like the theory of the syllogism with a sentential logic. Rather, Abelard's work should be seen as a very sophisticated development of a sentential logic. I will therefore not treat it in this overview, since it belongs to the history of sentential logic rather than syllogistic. It seems that the medievals also rather quickly stopped associating the word ‘syllogism’ with this theory.
Abelard is also associated with the history of modal logic. He is famous as the philosopher who introduced the distinction between de dicto and de re modal sentences. The basic notions of Abelard's modal theory are to be found in the introduction to Chapters XII and XIII of his longer commentary on Aristotle's De interpretatione (ed. Minio-Paluello 1958). Abelard concentrates his analysis on the logical structure of modal sentences, introducing some new distinctions and concepts that were later commonly used by medieval logicians.
According to Abelard, modal terms are strictly speaking adverbs expressing how something said of the subject is actualized, e.g., ‘well’ or ‘quickly’ or ‘necessarily’. Adverbs that do not modify an actual inherence, e.g., ‘possibly’, are called secondary modal terms due to their position in a sentence. Abelard also noticed that in De interpretatione 12–13, Aristotle operates with nominal rather than adverbial modes, e.g. ‘it is necessary that’ or ‘it is possible that’. He seems to have assumed that Aristotle did this because the nominal modes lead to many more problems than simple adverbial modes. This is more clearly seen from the fact that sentences including nominal modes, such as ‘Necesse est Socratem currere’, can be understood either adverbially, ‘Socrates runs necessarily’, or, as suggested by the grammar, ‘That Socrates runs is necessary’. He calls these two alternatives de re necessity sentences and de sensu (or de dicto) necessity sentences, respectively. Abelard seems to be the first to employ this terminology. A de re modal sentence expresses the mode through which the predicate belongs to the subject. The mode is, therefore, associated with a thing, whereas the mode in the de dicto case (as he also calls it) is said of what is expressed by a non-modal sentence.
Abelard also referred to this distinction as the distinction between personal and impersonal readings of a modal sentence, the de re sense corresponding to the personal reading and the de dicto sense to the impersonal reading because when the expression ‘necesse est’ or ‘possibile est’ is used at the beginning of a sentence, it lacks a personal subject. Abelard states that this distinction is related to Aristotle's distinction between per divisionem and per compositionem in the Sophistici Elenchi (4, 166a23–31). What is new is Abelard's contention that modal discussions should proceed by distinguishing the different possible readings of modal sentences, moving on to consider their quantity, quality, and conversion as well as their equipollence and any other relations holding between them on these different readings. Abelard's program thus became the standard operating procedure in medieval treatises on logic.
After Abelard, equipollence and other relations between modal sentences were commonly presented with the help of the square of opposition, which Abelard mentions though it does not appear as such in his works. The square can be taken to refer to de dicto modal sentences or to singular de re modal sentences. Although the distinction between de dicto and de re modal sentences was common in logical treatises on the properties of the terms, syncategorematic terms, and the solution of sophisms, twelfth- and thirteenth-century logicians were mainly interested in the logical properties of singular de re modal sentences. There is no detailed theory of quantified de re modal sentences from this period, and the first movements in this direction by Abelard and his followers were rather confused. A satisfactory theory of de re modal sentences did not appear until the fourteenth century, when the various relations between such sentences was presented by John Buridan in his octagon of opposition.
Medieval logicians generally assumed that Aristotle dealt with de dicto modal sentences in theDe Interpretatione and de re modal sentences in the Prior Analytics. In early commentaries on the Prior Analytics, there is usually no mention of Abelard's distinction between them. One reason may be that the only theory available concentrated on singular de re modal sentences, which are not part of modal syllogistic as developed by Aristotle.
While the de dicto/de re terminology was used, it was not all that common. Medieval logicians preferred to use what they took to be Aristotle's terminology, talking about modal sentences in the composite sense (in sensu composito) and divided sense (in sensu diviso). The structure of a composite modal sentence can be represented as follows:

اقتباس :: (quantity/subject/copula, [quality]/predicate)mode

A composite modal sentence corresponds to a de dicto modal sentence. The word ‘composite’ is used because the mode is said to qualify the composition of the subject and the predicate. The structure of a divided modal sentence can be represented as follows:

اقتباس :: quantity/subject/copula, mode, [quality]/predicate

Here, the mode is thought to qualify the copula and thus to divide the sentence into two parts (hence the name, ‘divided modal sentence’). This type of modal sentence was characterized as de re because what is modified is how things (res) are related to each other, rather than the truth of what is said by the sentence (dictum) (see Lagerlund 2000: 35–39, and the entry on medieval theories of modality for further details).
Like virtually all medievals, Abelard thought that Aristotle's modal syllogistic was a theory for de re modal sentences. He says very little about it in his logical works, however. In less than five pages in the Dialectica (245–249) he treats modal, oblique, and temporal syllogistic logic. Earlier in the same work, he says a little about conversion rules. He argues in both the Dialectica(195–196) and the Logica (15–16) that the conversion rules can be defended even on a de rereading, but the conversions he discusses are not modal conversions since the mode must be attached to the predicate and follow the term in the conversion, making the conversion into the conversion of an assertoric sentence. The conversions of de re modal sentences, as Abelard has defined them, do not hold, as Paul Thom has convincingly shown. (Thom 2003: 57–58.)
There is no modal syllogistic explicitly outlined in any of Abelard's logical works, though in theDialectica, he exemplifies some of the valid mixed moods: M–M in the first figure, MM– in the second, and M–M in the third (M represents a possibility sentence and ‘–’ an assertoric). He also shows that uniform modal syllogisms are not generally valid, so that MMM is not valid unless the middle term in the major premise is read with the mode attached to it, as in:

اقتباس :

Everything which is possibly B is possibly A

Every C is possibly B

Every C is possibly A

A consequence of this, of course, is that the middle term in the minor premise is ‘possibly B’ and hence no longer a modal sentence. MMM is consequently reduced to M–M.
Anything more systematic than this has to be drawn out from Abelard's definition of modal sentences and their semantic interpretation. Thom has done this in his book (Thom 2003), where he claims that there is a very specific system developed that is not at all similar to Aristotle's modal system. Abelard was therefore not attempting an interpretation of Aristotle, but must be seen as developing a new system based on his reading of de re sentences. But this project must overcome several problems, particularly since Abelard cannot use the conversion rule.

» Wholly Hypothetical Syllogisms
» Ancient Logic
» Modal Logic
» Infinitary Logic
» Magna Carta Messed Up the World, Here’s How to Fix It: The “logic” of capitalist development has left a nightmare of environmental destruction in its wake. Noam Chomsky

Arabic Logic and Syllogisms

4. Peter Abelard

Arabic Logic and Syllogisms :: تعاليق

Arabic Logic and Syllogisms

مواضيع مماثلة

مواضيع مماثلة