Boethius

The medieval tradition of logic is generally thought to have begun with Boethius (c. 475–526), who ambitiously tried to preserve what was left of philosophical learning from the declining culture of late antiquity. In fact, however, he was only able to save parts of ancient logic, primarily Aristotelian logic (see Lee 1984 for a discussion of Aristotle's syllogistic in late ancient thought). He wrote extensively on the theory of the syllogism, producing a Latin translation of the Prior Analytics, though it was not used very much before the twelfth century (see Aristotle, Analytica Priora, and the introduction by Minio-Paluello). He also wrote two textbooks on the categorical syllogism: On the Categorical Syllogism (De syllogismo categorico) and Introduction to Categorical Syllogisms (Introductio ad syllogismos categoricos) (for the texts, see Migne 1847 and Thomsen Thörnqvist 2008). In addition, he produced an interesting book called On Hypothetical Syllogisms (De hypotheticis syllogismis), which will be touched on in the discussion below (see Obertello 1969 for the text).
Boethius made no substantial contribution to the theory of the syllogism, though he was an important transmitter of the theory to later logicians and his works offer a clear presentation of the Aristotelian account. But that presentation differs from Aristotle's in one important respect. In Boethius, the categorical sentences are constructed using ‘is’ (’est’) and not ‘belongs’, as in Aristotle. The four sentences thus become:

اقتباس :: A – Every B is A
I – Some B is A
E – No B is A
O – Some B is not A

Put in this way, it is more obvious that they are subject predicate sentences, and moreover, that the syllogisms are deductions rather than conditional sentences. As a result, the four figures look different:

اقتباس :

I.

II.

III.

IV.

B – A

A – B

B – A

A – B

C – B

C – A

C – B

C – A

B – C

C – A

B – C

C – A

In systematic terms, Boethius' change makes no difference and all medieval logicians writing after him adopted it, even though it makes the first figure syllogisms less evident. According to Aristotle, the first syllogism of the first figure (Barbara) should read: ‘A belongs to all B, B belongs to all C; therefore A belongs to all C’. This is obviously valid by the transitivity of inclusion. But if we line up the same syllogism using Boethian formulation we get: ‘Every B is A, Every C is B; therefore Every C is A’. This is not at all as obvious and we have to switch the places of the premises to get the same transitivity characteristic: ‘Every C is B, Every B is A; therefore Every C is A’. Beyond this small but significant change, Boethius does not contribute much to the theory, though he is a little more interested than Aristotle in the different kinds of conversion. His hypothetical syllogistic is, on the other hand, rather novel.
Like most things in the history of logic, hypothetical syllogistic also begins with Aristotle. In thePrior Analytics, he says that every syllogism is either direct or from a hypothesis. Traditional syllogistic is direct and hence all syllogisms that do not fall into the patterns of inference defined by the three Aristotelian figures, but which are nevertheless valid syllogisms, must be hypothetical. Aristotle's principal example is a syllogism through impossibility. If we reason from a hypothesis P via a syllogism to a conclusion Q that is impossible, then we can conclude that not-P is true and P false (An. Pr. 41a23–30).
In the second century C.E., Alexander of Aphrodisias tried to develop this into a theory of the hypothetical syllogism. What emerged from his attempt is something quite strange and even confused, though it has been studied at great length (see esp. Speca 2001 and the list of further references there). Boethius' On the Hypothetical Syllogisms is the only remaining early work on this topic.
A hypothetical syllogism is a syllogism in which one or more premises are hypothetical sentences. Boethius draws the distinction between categorical sentences and hypothetical sentences formally by saying that a categorical sentence involves a predication whereas a hypothetical sentence involves a condition, i.e., it says that something is, if something else is. Typically such sentences are conditional sentences such as ‘if P then Q’, though Boethius also treats ‘P or Q’ as hypothetical, apparently because he thinks that disjunction can be translated in terms of a conditional sentence. Another characteristic of hypothetical sentences is that they are made up of categorical sentences.
The basic hypothetical sentences he gives are:

اقتباس :: (2:1) If it is A, then it is B
(2:2) If it is not-A, then it is not-B
(2:3) If it is A, then it is not-B
(2:4) If it is not-A, then it is B

He also considers sentences involving three terms:

اقتباس :: (2:5) If, if it is A, then it is B, then it is C
(2:6) If it is A, then if it is B, then it is C

though a hypothetical sentence can be even more complicated:

اقتباس :: (2:7) If, if it is A, then it is B, then if it is C, then it is D.

Boethius also thinks that hypothetical sentences can be qualified by modalities such as necessity or possibility, but he never develops this idea.
In trying to establish what combination of premises form valid inferences he proceeds like Aristotle and develops lists or tables in which he can group the valid patterns. The basic sentences (2:1)-(2:4) combined with a simple categorical sentence as the second premise boil down to what we today know as modus ponens and modus tollens. This led some modern interpreters to think that Boethius was developing a sentential logic as the Stoics had done (Dürr 1951), but this idea has been rejected by more recent scholars (Obertello 1969, Martin 1991 and Speca 2001). Whatever Boethius thought he was doing, he was not trying to develop a sentential logic. This becomes obvious if one considers a more complex hypothetical syllogism, such as the following, which he accepts as valid:

اقتباس :

If it is A, then if it is B, then it is C

If it is B, then it is not-C

It is not-A

If Boethius' logic is a sentential logic his syllogism would be translatable into the following:

اقتباس :

A ⊃ (B ⊃ C)

B ⊃ ∼C

∼A

But this is not a valid deduction, which means either that Boethius was simply wrong to accept it or that he had something else in mind. If, on the other hand, we use a term logic such as Aristotle's syllogistic, the above inference schema seems valid, since given the two premises, if something that is A and B is also C and something that is only B is not C, then that very same thing has to be not A as well.
Boethius is conscious of a Stoic logical tradition in which the logical forms of sentences were distinguished according to their linguistic form, such that ‘if … then’ structures indicate conditionals and ‘or’ structures indicate disjunctions, making these terms rather like operators on sentences. He seems to be using these ideas to demarcate his hypothetical sentences, though he is still writing in an Aristotelian fashion and developing an Aristotelian term logic (see Speca 2001 and Marenbon 2003: 50–56). This mix makes his logic quite confused, and the confusion was not sorted out until Abelard was able to develop a proper sentential logic out of Boethius' suggestions. (See Martin 2009 as well.)

Boethius

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Boethius