notas.itmens
Aristotelian Logic
We won't get into the details of figures of syllogisms, but we'll be concentrating on the architectonics.
Motivation and Origins#
Logos: logos, as a method of enquiry and demonstration, shared its name with a perceived rational purpose thought to underlie the entire universe. Logos represented the laws and regularities governing all of nature, and also the process of reasoning by which these laws and regularities were to be discovered. Aristotelian logic is a constraint upon logos, faced with the so-called pathological metaphysics of Heraclitus and Parmenides. Since these pathologies turn on the mismanagement of ambiguity, Aristotle struggled in his first two books of the Organon, the Categories and On Interpretation, to produce a theory of ambiguity and a set of protocols for its avoidance.
The logical theory of the Prior Analytics presupposes Aristotle's theory of syllogisms (and a theory of validity). These theories appear implicitly in the Topics and On Sophisitical Refutations. Logic was born when he began to study the kinds of shapes, or structures, that good, persuasive arguments have.
Architectonic#
Subject-Predicate Structure#
In Plato's Sophist:
The signs we use in speech to signify being are surely of two kinds . . . One kind called ‘names’, the other ‘verbs’ . . . By ‘verb’ we mean an expression which is applied to actions . . . And by ‘name’ the spoken sign applied to what performs these actions . . . Now a statement never consists solely of names spoken in succession, nor yet verbs apart from names.
Namely, a simple statement, namely that which can be true or false, is itself built out of two basic components: name, verb. It seems that Plato takes it as obvious that names and verbs are different kinds of things: one picks out the agent, the other picks out the agent's actions.
A problem arises:
- In Categories, Aristotle classifies things that are said to two classes: one involve combination, one without. In particular, a class of sentences are called affirmatives since they affirm something of a subject, and can be true or false depending on whether the world is as the sentence says.
- In De Interpretationes, this distinction takes on what appears to be a syntactic importance, according to which the name and the verb fulfil specific grammatical roles within a simple statement.
- A name is a spoken sound significant by convention, without time, none of whose parts is significant in separation.
- A verb is what additionally signifies time, no part of it being significant separately; and it is a sign of things said of something else.
Hence in Categories it seems that Aristotle's syllogistic logic is about things in the world, while in De Interpretationes it seems to be about the language we use to describe those things. This difficulty arises as a failure to distinguish between using a word and mentioning a word; questions about the precise nature of the roles of the clearly linguistic items of name and of verb become a foundational issue within the history of the development of logic.
Affirmation and denial#
In Aristotle affirmation and denial are put in equal footing. It is obvious that even the most basic human communication requires that affirmation must be tractable and meaningful, but in De Interpretationes Aristotle also makes denial tractable and meaningful in just the way that affirmation is.
By this Aristotle avoids problems which worried his predecessors. Plato and Parmenidese sought to ground philosophy in what is but then struggled to provide an analysis of what is not. If what is ‘exists’, does what is not ‘not exist’? How can we talk meaningfully about what is not? If truth reflects what is, then is falsity about what is not?
Affirmation and denial are central to Aristotle's logic. For him a premise is a sentence affirming or denying something of something. A conclusion, too, will always be either an affirmation or a denial.
Quantifiers#
Contraries and contradictories involve more than attaching subject and predicate, or name and verb. We must be able to signify both affirmation and denial and we must be able to indicate quantity. When a premise says that “all \(A\) are \(B\)” or that “every \(A\) is \(B\)” or that “none of the \(A\)s are \(B\)”, Aristotle calls the premise a universal. With some this becomes particular.
Considering affirmations and denials which are themselves either universal or particular, Arristotle's notion of what counts as a premise is restricted to a small class of simple propositions called categorical propositions.
- Universal affirmative: B belongs to every A.
- Universal privative: B belongs to no A.
- Particular affirmative: B belongs to some A.
- Particular privative: B does not belong to some A.
Tension between subject-predicate and term Logic#
It is often claimed that Aristotle's logic is term logic and subject-predicate logic, but there's a certain tension in these two forms of logic.
- In De Interpretationes, Aristotle regards the combination of subject and predicate as providing the underlying structure of any meaningful proposition: In “B belongs to some A” (some A are B), A serves as the subject and B as the predicate.
- In Prior Analytics, syntactic roles played by name and verb turn out not to be preserved in the syllogistic figures. Example: in the first figure (A is B, B is C, therefore A is C) the B term is the subject of the minor premise, while it is the predicate of the major premise.
The figures themselves do not respect the earlier syntactic roles, and so the name/verb distinction which Aristotle inherited from Plato is therefore no longer a good fit. Eventually Aristotle steps away from Plato's name/verb distinction, preferring instead to describe name/verb using the more neutral label ‘term’.
This becomes particularly important in conversion, which involves the transposition of subject and predicate. For example, generally,
- if it is true that ‘some A is B’, then it is equally true that ‘some B is A’.
- Similarly, given ‘no A are B’, also ‘no B are A’.
- But, given ‘all A are B’, then we only have ‘some B are A’. - Notice that a problem arises when there are not any As; the problem of empty terms; thus Aristotle's logic is most straightforwardly approached as a system which is about things in the world.