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© 1984, by Ralph E. Kenyon, Jr. *
DEC 11, 1984
In his Theaetetus, Plato examines three possible definitions of 'knowledge': "perception", "true judgment", and "true judgment with an account". While Plato rejects each in turn, certain kinds of knowledge, as we use the term today, seem in close agreement with Plato's account of "true judgment with an account".
Considerable claim can be made for this definition, as Plato explains it. Theaetetus says:
(I). He said that true judgment with an account is knowledge, and the kind without an account falls outside the sphere of knowledge. Things of which there's no account are not knowable, he said -- he actually called them that -- whereas things which have an account are knowable.1
This account is further amplified by Plato in the immediately following passage where he distinguishes primary elements or things from complexes.
(II). In my dream, I seemed to hear some people saying that the primary elements, as it were, of which we and everything else are composed, have no account. Each of them itself, by itself, can only be named, and one can't go on to say anything else, neither that it is nor that it isn't; because in that case, one would be attaching being or not being to it, whereas one oughtn't to add anything if one is going to express in an account that thing, itself, alone. In fact one shouldn't even add itself, or that, or each, or alone, or this, or any of several other things of that kind; because those things run about and get added to everything, being different from the things they're attached to, whereas if the thing itself could be expressed in an account and had an account proper to itself, it would have to be expressed apart from everything else. As things are, it's impossible that any of the primary things should be expressed in an account; because the only thing that's possible for it is to be named, because a name is the only thing it has. But as for the things composed of them, just as the things themselves are woven together, so their names, woven together, come to be an account because a weaving together of names is the being of an account. In that way, the elements have no account and are unknowable, but they're perceivable; and the complexes are knowable and expressible in an account and judgeable in a true judgment. Now when someone gets hold of the true judgment of something without an account, his mind is in a state of truth about it but doesn't know it; because someone who can't give and receive an account of something isn't knowledgeable about that thing. But if he gets hold of an account as well, then it's possible not only for all that to happen, but also for him to be in a perfect condition in respect of knowledge.2
Several premises can be extracted from this passage, some of which seem to be in conflict. I cite the passage from which I draw the premise, and explicate the premise. In some cases I show what seem appropriate modifications or interpretations.
the primary elements . . . of which we and everything else are composed
- P1. Everything is composed of primary elements.
But as for the things composed of them, just as the things themselves are woven together, so their names, woven together, come to be an account because a weaving together of names is the being of an account.
- P2. An account is a weaving together of names.
Each [primary] . . . can only be named. . . . because the only thing that's possible for it is to be named, because a name is the only thing it has.
- P3. A primary element can only be named.
one can't go on to say anything else, neither that it is nor that it isn't; because in that case, one would be attaching being or not being to it, . . . In fact one shouldn't even add itself, or that, or each, or alone, or this, or any of several other things of that kind;
- P4. Attaching anything else to a primary name makes an account of something which is more than just that primary itself.
whereas one oughtn't to add anything if one is going to express in an account that thing, itself, alone.
- P5. If there were to be an account of a primary thing, it could only express that primary thing.
- P6. (P3 & P5) If there were to be an account of a primary thing, it could only name that primary thing.
whereas if the thing itself could be expressed in an account and had an account proper to itself, it would have to be expressed apart from everything else.
- P7. If there were to be an account of a primary thing, it would have to be expressed apart from everything else.
- P7'. If there were to be an account of a primary thing, it could not express anything else.
- P7''. If there were to be an account of a primary thing, it could not express any other primaries.
- P8. (P7'' & P3) If there were to be an account of a primary thing, it could not name any other primaries.
- P9 (P6 & P8) If there were to be an account of a primary thing, it could only name that primary thing and it could not name any other primary things.
because those things run about and get added to everything, being different from the things they're attached to,
- PX. There are some things which get added to everything.
This statement leads to premises which seem problematic or even contradictory.
- PY. These things get added to primaries.
- PZ. These things get added to themselves.
As things are, it's impossible that any of the primary things should be expressed in an account; . . . In that way, the elements have no account and are unknowable,
- P10. No primary thing may be expressed in an account.
- P11. (P10 & P3) No primary thing may be named in an account.
Premises PX, PY, PZ, P10, and P11 seem contrary to the general structure. It is possible to interpret premises P10 and P11 as intending to mean that primaries are not mentioned in definiendums. Plato argues for this interpretation, but it seems to me that he does little more than assert it. This and other inconsistencies allow one to pick materiel to support alternative interpretations.
but they're perceivable;
- P12. Primaries are perceivable.
and the complexes are knowable and expressible in an account and judgeable in a true judgment.
- P13. A complex thing has an account.
- P14. (P13 & Q ==> Q) If a complex thing has an account, then it has an account.
- P15. (P14 & P2) If a complex thing has an account, then that account is the weaving together of names.
I think that Plato has all the element for a modern recursive definition for the term "account". P9 and P15, taken together can define an account which allows for both complex things and primary things.
- P15. If there were an account of a thing then:
- i. If that thing is a primary, then the account is only its name (a weaving together of one name). (P2 & P3 & P9)
- ii. If that thing is a complex, then the account is the weaving together of names. (P2 & P15)
Modern recursion is achieved by the use of "names" in part ii., since these names could be of complexes or primaries.
What is striking about Plato's definition of knowledge as true judgment with an account is that the characterization closely parallels three notions: the form of a definition, the axiomatic method, and the notion of undefined terms.
i. In a definition, we name the definiendum, that is, the thing being defined. Then, in saying how that thing is to be defined, we say what things go together, and in what way, that is, we weave together names of other things in what is called the definiendum. In this form of a definition, the definiens is composed of primaries and the definiendum names the complex being defined.
ii. In the axiomatic method or approach, axioms are taken as given, while theorems are derived from the axioms and some definitions. Axioms cannot be proven. They are accepted as true in the system. From these axioms, other statements are proven. Axioms, then, correspond to primaries, while theorems correspond to complexes. Proving a theorem (complex) refers to, or depends upon, axioms (primaries).
iii. In an approach paradigmatically illustrated by the model of geometry, certain terms remain undefined (point, line, plane, etc.) while others are related by definition and theorem to the undefined terms. Primaries would correspond to these undefined terms, while complexes would correspond to higher level structures.
1. The Form of a Definition. Plato distinguished between "primary elements" and "complexes", and states that the primaries can only be named. He even eschews attaching indexicals ('this', 'that') and reflexive terms ('itself', 'alone') to them. The following paragraphs illustrate how Plato's structure for defining "knowledge" corresponds very closely to good practices in forming definitions. As was first made explicit by Fregé, the use-mention distinction plays an important role in definitions. The term to be defined must be mentioned in the definiendum. This distinction had not been made explicit in Plato's time and the lack of an explicit characterization of the use-mention distinction hampered him in his efforts to characterize knowledge.
When one forms a good definition, one should mention the term to be defined only once in the definiendum and not mention it at all in the definiens. If the use-mention distinction is not available, or is confused, then expressing this limitation becomes problematic.
Plato had such "mention" indicators as 'name' and 'word', however, he seldom used these devices. If we were to define something, we might start out with "The term 'X' is defined as . . .". If we could not use the use-mention distinction, we might start out with "X is defined as . . .". In the later case, it is unclear whether we are speaking of "X" or 'X'. In today's interpretation, the later form causes us to focus on that which the term 'X' refers to, namely X. Since the term being defined should be mentioned in the definiendum, this focus must be shifted back to the name, that is, away from the referent. Plato must choose his language in such a way that he focuses on this signification, while also indicating that the term is not used. The act of naming is distinct from using a name in just such a manner. When we name something, the focus is on the name. In short, in the act of naming something, the name is mentioned. The act of naming also occurs only once. Expressing part of this first rule for forming good definitions with Plato's language limitations yields, "When one forms a good definition, the thing to be defined is named . . .", which closely agrees with the language used by Plato. This seems to take care of the first two transformation problems. Dealing with the lack of that term in the definiens poses a greater problem.
When one forms a good definition, the terms used in the definiens should all be lower order terms. That is, they should all be terms which are already known, or, in the case of recursive definitions, the same term used at lower orders of structure (known levels). A recursive definitions is a modern invention; its structure was not explicitly available to Plato.
As an example, consider the mathematical function known as the factorial function. The factorial of a number, N, written "N!", is the product of that number and all positive numbers less than it. For example, 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720. The factorial function can be defined recursively as follows:
"N!" (N factorial) is defined as:
i. If N=1 then N! = 1.
ii. Otherwise, N! = N times (N-1)!
We notice that "N!" is mentioned in the first line of
the definition, but it is used in the third line (line ii.). Since it is used
for a smaller number (each time) it will eventually get to 1 and be done.
Suppose we go thru an example starting with N=3
"3!" is defined as: -- computing 3! yields...
i. If 3=1 then 3! = 1. -- false so go on to ii.
ii. 3! = 3 times (3-1)! -- save 3 and compute 2!
"2!" is defined as -- computing 2! yields...
i. If 2=1 then 2! = 1. -- false so go on to ii.
ii. 2! = 2 times (2-1)! -- computing 1! yields...
ii. 3! = 3 times (2 times 1!) -- Where we are so far
"1!" is defined as: -- computing 1! yields...
i. If 1=1 then 1! = 1. -- true so 1 is the answer.
ii. --- -- we never get to here...
ii. 3! = 3 times (2 times (1)) -- we got a 1 from 1!
ii. 3! = 3 times (2) -- we got a 2 from 2!
ii. 3! = 6 -- we get a 6 from 3!
The first time we try to see what 3! is, we see that we must save the 3 for multiplication and then see what (3-1)! or 2! is. To see what 2! is we must save the 2 for multiplication and see what (2-1)! or 1! is. We know what 1! is, so we can go back and compute 2! as 2 times 1!. Then we continue on with the result of 2! to multiply by 3 and get our answer. To prevent infinite regress, it is very important that the use of the term being defined recursively is at a lower level in the definiens.
A naive or simple view of definitions would require that there be only two levels, the level of the definiendum and the level of definiens. (Any 'middle' level definition could be converted by substituting their definitional structures for any terms in its definiens, repeatedly, until no more 'middle' level terms remain.) In such a two-level cases, each definiens would contain only terms which are not defined in any definition, and which could be called 'primary' terms, while each definiendum would contain one term which appears in no definiens. Because each such defined term is equated by the 'is of identity' to a complex of primary terms, such a term could be called a complex. Without the use-mention distinction, this entire description would appear to be talking about the referents of the terms. Plato would have to write of 'primary things' and 'complex things' or 'complexes' instead of 'primary terms' and 'complex terms'.
Whenever two or more primary terms are "attached" to each other, a definiens is created; if the definiendum is not explicitly named, what is created is still a complex, though unnamed. In particular, attaching "being" or "not being" to any named primary produces such an unnamed complex. Consequently, no primary can be other than just named, which Plato asserts.
When one forms a good definition, one must see that the definiens contains at least two lower order terms, standing in some relation to each other. If we are to confound the use-mention distinction in an effort to simulate the speaking environment of Plato's day, we are required to use the word "things" instead of the word "terms". Once again, using the linguistic devices of Plato's day to try to indicate that we wish to 'mention' what is being defined, requires emphasizing the name of the things (which is almost un-confusing the distinction); Plato uses the metaphor of weaving a tapestry to indicate the relation among the lower order terms, and explicitly writes of "names" woven together.
An account is the weaving together of names.
Previously, a definition which contained a term in both the definiendum and the definiens was said to be "circular", and was thought to be a bad definition. It took modern computer science and attention to the practical problem of terminating a loop to precisely formulate an acceptable notion of 'circularity' which we call "recursive". When Neil Postman says, "Not all circular definitions are bad.", he is presumably including "recursive" as a species of the genus "circular". If we also call the other (non-recursive) species "circular", the lack of univocal use of "circular" should not be too much of a problem. The meaning should be clear from the context.
The difference between a 'recursive' definition and a 'circular' definition is that, in the recursive definition, the term mentioned in the definiendum is used in the definiens at a lower level in such a way that there is eventually a first case or bottom level; in a circular definition, it is the absence of just such a guarantee of termination which is problematic.
2. The Axiomatic Method. Relating the axiomatic method to Plato's Dream makes use of an analogy between the statements in a theory and the things which might be known (complexes and primaries). P3 requires that a primary be only a named thing. An axiom is a statement which is taken as fundamental. No attempt is made to prove axioms by using other statements in the theory. On the other hand, the other statements, which, if true, are called "theorems", are "proved" by recourse to the axioms, together with a "proof". The proof of a "theorem" is a sequence of statements, each of which, (1) is an axiom, (2) follows from an axiom by a rule of inference, or (3) follows from an earlier statement in the list by a rule of inference, and the last of which is the statement being proved. If each statement corresponds to a "name" in Plato's scheme, then a proof could be characterized as "a weaving together of names". A valid proof yields a true conclusion. In this sense, Plato's "true judgment" would correspond to "valid proof". The word 'theorem' is usually reserved for those "complexes" which have a valid proof, so "theorem" does not directly correspond to Plato's "complex thing". Statements in the theory may be capable of being proved true, capable of being proved false, or undecidable in the system. Any decidable statement would either have a proof, or its negation would have a proof. In this sense, then, "decidable statement" would correspond to Plato's "complex thing". According to this definition of proof (above), an axiom is also a theorem, since it consists of sequence of one statement which is an axiom; if we correspond "proof" with Plato's "account", then premises P5 and P6 are consistent with such a structure.
3. Undefined Terms. If one attempts to define one's terms in a non-circular manner, one soon finds that there are a few undefined terms, and no terms left to define them with. In geometry, such statements as "Two points determine a line", "Two distinct lines intersect in at most one point", etc., define relationships which set limits on how we may use the terms 'line' and 'point', but do not tell us what a point or line may be. Any non-circular system must have undefined terms. In Plato's dream, "primaries" correspond to "undefined terms". Such terms do not have "an account" in terms of other defined terms. In applying theories employing such terms, "things" are selected for these terms to represent. In formal systems, the selected objects are said to "model" the theory. If there is a good match between the structure of the theory containing the undefined terms, and the structure represented by that theory, we might say that this good match is a "true judgment". Each defined term, then, has a definition (an account) and is a good match (expresses true judgment). This fits well with Plato's definitions.
Each of these three perspectives has been fruitful in advancing our understanding of various things, and could be said to have advanced our understanding of "knowledge".
If Plato had known of recursive structure he might have used it to solve his problem of characterizing knowledge. He could have moved as follows. Firstly, he would keep the form of the third definition.
Knowledge is true judgment with an account.
Secondly, he would define "accounted for" recursively.
Something is accounted for if it is:
- a primary.
- a complex of things which are accounted for.
Such a move allows primaries to be known. If any complex is knowable, then it is accounted for by things which are either complexes, or primaries. If those things are complexes, then they are knowable. On the other hand, if they are primaries, then they are also knowable. There is no other case. Consequently, sooner or later, everything which is knowable, is knowable ultimately in terms of primaries. Since primaries are knowable, knowable things do not depend upon unknowable things. That knowable things depend ultimately on unknowable things Plato considered an obvious joke -- an unchallengably false premise.
So if we may argue from the elements and complexes that we're familiar with ourselves to the rest, we'll say that the class of element admits of knowledge that is far clearer, and more important for the perfect grasp of every branch of learning, than the complex; and if anyone says that it's in the nature of a complex to be knowable and an element to be unknowable, we'll take him to be making a joke, whether on purpose or not.3
Modern physics demonstrates empirically that Plato was wrong; knowables are based on unknowables. The 'primaries' are the quantum numbers which are used to characterize the basic particles. Each basic particle is characterized by a unique permutation of the basic quantum numbers. If my move for Plato is used, then we would say that these basic quantum numbers are 'knowable' since they appear to be primary; therefore, Plato was right to assert than knowables cannot depend upon unknowables.
Plato had the basic "stuff" available to him for devising a recursive definition of "account". His ability to use this "stuff" was hampered by the lack of an explicit use-mention distinction. If his definition of knowledge were shored up with these two concepts, he is in very close agreement with three modern ways of looking at knowledge.