A rchive Date
[ 25-05-2000 ]
Category
[ Philosophy ]
sub-Categoy
[ Greek ]
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[Aristotle: Logical Methods
The greatest and most influential of Plato's students was Aristotle, who established his own school at Athens. Although his writing career probably began with the production of quasi-Platonic dialogues, none of them have survived. Instead, our knowledge of Aristotle's doctrines must be derived from highly-condensed, elliptical works that may have been lecture notes from his teaching at the Lyceum. Although not intended for publication, these texts reveal a brilliant mind at work on many diverse topics.
Philosophically, the works of Aristotle reflect his gradual departure from the teachings of Plato and his adoption of a new approach. Unlike Plato, who delighted in abstract thought about a supra-sensible realm of forms, Aristotle was intensely concrete and practical, relying heavily upon sensory observation as a starting-point for philosophical reflection. Interested in every area of human knowledge about the world, Aristotle aimed to unify all of them in a coherent system of thought by developing a common methodology that would serve equally well as the procedure for learning about any discipline.
For Aristotle, then, logic is the instrument (the "organon") by means of which we come to know anything. He proposed as formal rules for correct reasoning the basic principles of the categorical logic that was universally accepted by Western philosophers until the nineteenth century. This system of thought regards assertions of the subject-predicate form as the primary expressions of truth, in which features or properties are shown to inhere in individual substances. In every discipline of human knowledge,then, we seek to establish the things of some sort have features of a certain kind.
Aristotle further supposed that this logical scheme accurately represents the true nature of reality. Thought, language, and reality are all isomorphic, so careful consideration of what we say can help us to understand the way things really are. Beginning with simple descriptions of particular things, we can eventually assemble our information in order to achieve a comprehensive view of the world.
Applying the Categories
The initial book in Aristotle's collected logical works is the Categories, an analysis of predication generally. It begins with a distinction among three ways in which the meaning of different uses of a predicate may be related to each other: homonymy, synonymy, and paronymy (in some translations, "equivocal," "univocal," and "derivative"). Homonymous uses of a predicate have entirely different explanations, as in "With all that money, she's really loaded," and "After all she had to drink, she's really loaded." Synonymous uses have exactly the same account, as in "Cows are mammals," and "Dolphins are mammals."
Paronymous attributions have distinct but related senses, as in "He is healthy," and "His complexion is healthy." (Categories 1) It is important in every case to understand how this use of a predicate compares with its other uses.
So long as we are clear about the sort of use we are making in each instance, Aristotle proposed that we develop descriptions of individual things that attribute to each predicates (or categories) of ten different sorts. Substance is the most crucial among these ten, since it describes the thing in terms of what it most truly is. For Aristotle, primary substance is just the individual thing itself, which cannot be predicated of anything else. But secondary substances are predicable, since they include the species and genera to which the individual thing belongs. Thus, the attribution of substance in this secondary sense establishes the essence of each particular thing.
The other nine categories—quantity, quality, relative, where, when, being in a position, having, acting on, and being affected by—describe the features which distinguish this individual substance from others of the same kind; they admit of degrees and their contraries may belong to the same thing. (Categories 4) Used in combination, the ten kinds of predicate can provide a comprehensive account of what any individual thing is. Thus, for example: Chloë is a dog who weighs forty pounds, is reddish-brown, and was one of a litter of seven. She is in my apartment at 7:44 a.m. on June 3, 1997, lying on the sofa, wearing her blue collar, barking at a squirrel, and being petted. Aristotle supposed that anything that is true of any individual substance could, in principle, be said about it in one of these ten ways.
The Nature of Truth
Another of Aristotle's logical works, On Interpretation, considers the use of predicates in combination with subjects to form propositions or assertions, each of which is either true or false. We usually determine the truth of a proposition by reference to our experience of the reality it conveys, but Aristotle recognized that special difficulties arise in certain circumstances.
Although we grant (and can often even discover) the truth or falsity of propositions about past and present events, propositions about the future seem problematic. If a proposition about tomorrow is true (or false) today, then the future event it describes will happen (or not happen) necessarily; but if such a proposition is neither true nor false, then there is no future at all. Aristotle's solution was to maintain that the disjunction is necessarily true today even though neither of its disjuncts is. Thus, it is necessary that either tomorrow's event will occur or it will not, but it is neither necessary that it will occur nor necessary that it will not occur. (On Interpretation 9)
Aristotle's treatment of this specific problem, like his more general attempt to sort out the nature of the relationship between necessity and contingency in On Interpretation 12-13, is complicated by the assumption that the structure of logic models the nature of reality. He must try to explain not just the way we speak, but the way the world therefore must be.
Demonstrative Science
Finally, in the Prior Analytics and Posterior Analytics, Aristotle offered a detailed account of the demonstrative reasoning required to substantiate theoretical knowledge. Using mathematics as a model, Aristotle presumed that all such knowledge must be derived from what is already known. Thus, the process of reasoning by syllogism employs a formal definition of validity that permits the deduction of new truths from established principles. The goal is to provide an account of why things happen the way they do, based solely upon what we already know.
In order to achieve genuine necessity, this demonstrative science must be focussed on the essences rather than the accidents of things, on what is "true of any case as such," rather than on what happens to be "true of each case in fact." It's not enough to know that it rained today; we must be able to figure out the general meteorological conditions under which rain is inevitable.
When we reason from necessary universal and affirmative propositions about the essential features of things while assuming as little as possible, the resulting body of knowledge will truly deserve the name of science.
The Four Causes
Applying the principles developed in his logical treatises, Aristotle offered a general account of the operation of individual substances in the natural world. He drew a significant distinction between things of two sorts: those that move only when moved by something else and those that are capable of moving themselves. In separate treatises, Aristotle not only proposed a proper description of things of each sort but also attempted to explain why they function as they do.
Aristotle considered bodies and their externally-produced movement in the Physics. Three crucial distinctions determine the shape of this discussion of physical science. First, he granted from the outset that, because of the difference in their origins, we may need to offer different accounts for the functions of natural things and those of artifacts. Second, he insisted that we clearly distinguish between the basic material and the form which jointly constitute the nature of any individual thing. Finally, Aristotle emphasized the difference between things as they are and things considered in light of their ends or purposes.
Armed with these distinctions, Aristotle proposed in Physics II, 3 that we employ four very different kinds of explanatory principle {Gk. aition [aition]} to the question of why a thing is, the four causes:
The material cause is the basic stuff out of which the thing is made. The material cause of a house, for example, would include the wood, metal, glass, and other building materials used in its construction. All of these things belong in an explanation of the house because it could not exist unless they were present in its composition.
The formal cause {Gk. eidos [eidos]} is the pattern or essence in conformity with which these materials are assembled. Thus, the formal cause of our exemplary house would be the sort of thing that is represented on a blueprint of its design. This, too, is part of the explanation of the house, since its materials would be only a pile of rubble (or a different house) if they were not put together in this way.
The efficient cause is the agent or force immediately responsible for bringing this matter and that form together in the production of the thing. Thus, the efficient cause of the house would include the carpenters, masons, plumbers, and other workers who used these materials to build the house in accordance with the blueprint for its construction. Clearly the house would not be what it is without their contribution.
Lastly, the final cause {Gk. teloV [télos]} is the end or purpose for which a thing exists, so the final cause of our house would be to provide shelter for human beings. This is part of the explanation of the house's existence because it would never have been built unless someone needed it as a place to live.
Causes of all four sorts are necessary elements in any adequate account of the existence and nature of the thing, Aristotle believed, since the absence or modification of any one of them would result it the existence of a thing of some different sort. Moreover, an explanation that includes all four causes completely captures the significance and reality of the thing itself.
The Appearance of Chance
Notice that the four causes apply more appropriately to artifacts than to natural objects. The rise of modern science resulted directly from a rejection of the Aristotelean notion of final causes in particular. Still, the scheme works so well for artifacts that we often find ourselves attributing some purpose even to the apparently pointless events of the natural world.
In many applications the formal, efficient, and final causes tend to be combined in a single being that designs and builds the thing for some specific purpose. Thus, the fundamental differentiation in the Aristotelean world turns out to be between inert matter on the one hand and intelligent agency on the other. As we shall soon see, this provides a natural explanation for the functions of animate natural organisms.
As for things that appear to arise by pure chance, Aristotle argued that since the purposeful origination described by the four causes is the normal order of the world, these instances must either be things that should have had some cause but happen to lack it or (more likely) things that actually do have causes of which we are simply unaware. The craft evident in the manufacture of artifacts, he believed, is evidence for the purposive character of nature, and it shares the same necessity, even though we are sometimes ignorant of its internal operations. (Physics II, 8)
Although I would be hard-pressed to come up with a final cause for the existence of the mosquito that is now biting me, for example, Aristotle supposed that there must ultimately be some explanation for its present existence and activity. Many generations of Western philosophers, especially those concerned with reconciling Christian doctrine with philosophy, would explicitly defend a similar view.
Categorical Propositions
Now that we've taken notice of many of the difficulties that can be caused by sloppy use of ordinary language in argumentation, we're ready to begin the more precise study of deductive reasoning. Here we'll achieve the greater precision by eliminating ambiguous words and phrases from ordinary language and carefully defining those that remain. The basic strategy is to create a narrowly restricted formal system—an artificial, rigidly structured logical language within which the validity of deductive arguments can be discerned with ease. Only after we've become familiar with this limited range of cases will we consider to what extent our ordinary-language argumentation can be made to conform to its structure.
Our initial effort to pursue this strategy is the ancient but worthy method of categorical logic. This approach was originally developed by Aristotle, codified in greater detail by medieval logicians, and then interpreted mathematically by George Boole and John Venn in the nineteenth century. Respected by many generations of philosophers as the the chief embodiment of deductive reasoning, this logical system continues to be useful in a broad range of ordinary circumstances.
Terms and Propositions
We'll start very simply, then work our way toward a higher level. The basic unit of meaning or content in our new deductive system is the categorical term. Usually expressed grammatically as a noun or noun phrase, each categorical term designates a class of things. Notice that these are (deliberately) very broad notions: a categorical term may designate any class—whether it's a natural species or merely an arbitrary collection—of things of any variety, real or imaginary. Thus, "cows," "unicorns," "square circles," "philosophical concepts," "things weighing more than fifty kilograms," and "times when the earth is nearer than 75 million miles from the sun," are all categorical terms.
Notice also that each categorical term cleaves the world into exactly two mutually exclusive and jointly exhaustive parts: those things to which the term applies and those things to which it does not apply. For every class designated by a categorical term, there is another class, its complement, that includes everything excluded from the original class, and this complementary class can of course be designated by its own categorical term. Thus, "cows" and "non-cows" are complementary classes, as are "things weighing more than fifty kilograms" and "things weighing fifty kilograms or less." Everything in the world (in fact, everything we can talk or think about) belongs either to the class designated by a categorical term or to its complement; nothing is omitted.
Now let's use these simple building blocks to assemble something more interesting. A categorical proposition joins together exactly two categorical terms and asserts that some relationship holds between the classes they designate. (For our own convenience, we'll call the term that occurs first in each categorical proposition its subject term and other its predicate term.) Thus, for example, "All cows are mammals" and "Some philosophy teachers are young mothers" are categorical propositions whose subject terms are "cows" and "philosophy teachers" and whose predicate terms are "mammals" and "young mothers" respectively.
Each categorical proposition states that there is some logical relationship that holds between its two terms. In this context, a categorical term is said to be distributed if that proposition provides some information about every member of the class designated by that term. Thus, in our first example above, "cows" is distributed because the proposition in which it occurs affirms that each and every cow is also a mammal, but "mammals" is undistributed because the proposition does not state anything about each and every member of that class. In the second example, neither of the terms is distributed, since this proposition tells us only that the two classes overlap to some (unstated) extent.
Quality and Quantity
Since we can always invent new categorical terms and consider the possible relationship of the classes they designate, there are indefinitely many different individual categorical propositions. But if we disregard the content of these propositions, what classes of things they're about, and concentrate on their form, the general manner in which they conjoin their subject and predicate terms, then we need only four distinct kinds of categorical proposition, distinguished from each other only by their quality and quantity, in order to assert anything we like about the relationship between two classes.
The quality of a categorical proposition indicates the nature of the relationship it affirms between its subject and predicate terms: it is an affirmative proposition if it states that the class designated by its subject term is included, either as a whole or only in part, within the class designated by its predicate term, and it is a negative proposition if it wholly or partially excludes members of the subject class from the predicate class. Notice that the predicate term is distributed in every negative proposition but undistributed in all affirmative propositions.
The quantity of a categorical proposition, on the other hand, is a measure of the degree to which the relationship between its subject and predicate terms holds: it is a universal proposition if the asserted inclusion or exclusion holds for every member of the class designated by its subject term, and it is a particular proposition if it merely asserts that the relationship holds for one or more members of the subject class. Thus, you'll see that the subject term is distributed in all universal propositions but undistributed in every particular proposition.
Combining these two distinctions and representing the subject and predicate terms respectively by the letters "S" and "P," we can uniquely identify the four possible forms of categorical proposition:
- A universal affirmative proposition (to which, following the practice of medieval logicians, we will refer by the letter "A") is of the form All S are P.
- Such a proposition asserts that every member of the class designated by the subject term is also included in the class designated by the predicate term. Thus, it distributes its subject term but not its predicate term.
- A universal negative proposition (or "E") is of the form
- No S are P.
- This proposition asserts that nothing is a member both of the class designated by the subject term and of the class designated by the predicate terms. Since it reports that every member of each class is excluded from the other, this proposition distributes both its subject term and its predicate term.
- A particular affirmative proposition ("I") is of the form
- Some S are P.
- A proposition of this form asserts that there is at least one thing which is a member both of the class designated by the subject term and of the class designated by the predicate term. Both terms are undistributed in propositions of this form.
- Finally, a particular negative proposition ("O") is of the form
- Some S are not P.
Such a proposition asserts that there is at least one thing which is a member of the class designated by the subject term but not a member of the class designated by the predicate term. Since it affirms that the one or more crucial things that they are distinct from each and every member of the predicate class, a proposition of this form distributes its predicate term but not its subject term.
Although the specific content of any actual categorical proposition depends upon the categorical terms which occur as its subject and predicate, the logical form of the categorical proposition must always be one of these four types.
The Square of Opposition
When two categorical propositions are of different forms but share exactly the same subject and predicate terms, their truth is logically interdependent in a variety of interesting ways, all of which are conveniently represented in the traditional "square of opposition."
"All S are P." (A)- - - - - - -(E) "No S are P."
| * * |
* *
| * * |
*
| * * |
* *
| * * |
"Some S are P." (I)--- --- ---(O) "Some S are not P."
Propositions that appear diagonally across from each other in this diagram (A and O on the one hand and E and I on the other) are contradictories. No matter what their subject and predicate terms happen to be (so long as they are the same in both) and no matter how the classes they designate happen to be related to each other in fact, one of the propositions in each contradictory pair must be true and the other false. Thus, for example, "No squirrels are predators" and "Some squirrels are predators" are contradictories because either the classes designated by the terms "squirrel" and "predator" have at least one common member (in which case the I proposition is true and the E proposition is false) or they do not (in which case the E is true and the I is false). In exactly the same sense, the A and O propositions, "All senators are politicians" and "Some senators are not politicians" are also contradictories.
The universal propositions that appear across from each other at the top of the square (A and E) are contraries. Assuming that there is at least one member of the class designated by their shared subject term, it is impossible for both of these propositions to be true, although both could be false. Thus, for example, "All flowers are colorful objects" and "No flowers are colorful objects" are contraries: if there are any flowers, then either all of them are colorful (making the A true and the E false) or none of them are (making the E true and the A false) or some of them are colorful and some are not (making both the A and the E false).
Particular propositions across from each other at the bottom of the square (I and O), on the other hand, are the subcontraries. Again assuming that the class designated by their subject term has at least one member, it is impossible for both of these propositions to be false, but possible for both to be true. "Some logicians are professors" and "Some logicians are not professors" are subcontraries, for example, since if there any logicians, then either at least one of them is a professor (making the I proposition true) or at least one is not a professor (making the O true) or some are and some are not professors (making both the I and the O true).
Finally, the universal and particular propositions on either side of the square of opposition (A and I on the one left and E and O on the right) exhibit a relationship known as subalternation. Provided that there is at least one member of the class designated by the subject term they have in common, it is impossible for the universal proposition of either quality to be true while the particular proposition of the same quality is false. Thus, for example, if it is universally true that "All sheep are ruminants", then it must also hold for each particular case, so that "Some sheep are ruminants" is true, and if "Some sheep are ruminants" is false, then "All sheep are ruminants" must also be false, always on the assumption that there is at least one sheep. The same relationships hold for corresponding E and O propositions.] |