Ton Deuteron Ploun

PA Chapter 3

Some people think that because you must know the primatives in order to have a demonstration (and thereby have understanding) that
1) there is no understanding, or
2) there are demonstrations of everything.
Neither of these is correct.

The first group claims that we’re led back ad infinitum. It is impossible to survey infinitely many things. But if there is a stop in the chain then the principles are unknowable because there is no demonstration (and the only kind of understanding comes from demonstration). SO, because you cannot know primatives, then you cannot know what proceeds from them simpliciter, and you can thereby only posit as supposition that they are the case.

The second group agrees about understanding but says that all things can be demonstrated because demonstrations can proceed in a circle or reciprocally.

Aristotle asserts not all knowledge is demonstrative. Indemonstrable understanding is possible for immediate items. This must be so because
1) if you must understand the items which are prior;
2) and these items are the things from which the demonstration proceeds, then
3) things must come to a stop
4) and these immediates must be undemonstrable

There is not only understanding but some principle (arche^n) of understanding by which we know the definitions.

At 100b15 Aristotle says that the principle of understanding is the nous or comprehension by which we have knowledge of the principles.

There are three arguments that claim that it is impossible to demonstrate simpliciter in a circle if demonstration must proceed from what is prior and more familiar.
1) because it is impossible for something to be prior and posterior at the same time…except in different ways, ways that induction makes familiar. In this case (where it is prior in one sense (in relation to us) and posterior in another (simpliciter)), knowing simpliciter will not be properly defined; it will be ambiguous or it is not demonstration simpliciter.

2) those who advocate circularity say nothing more than if A is the case then A is the case, and it is easy to prove anything this way.

3) Circular reasoning is only possible for items which follow one another. So even if it is possible, it is only possible for objects counterpredicted (convertible propositions). There are very few counterpredicted items in demonstrations, and so it is impossible to say demonstrations may be reciprocal and thus there can be demonstrations of everything.

AP Book 2

We think we understand something simpliciter when we think we know of the explanation (because of which the object holds) that it is the explanation and that it is not possible to be otherwise.

Therefore, if there is understanding simpliciter of something, it is impossible for it to be otherwise.

The definiendum is understanding simpliciter
The definiens contains two conjuncts
1) to do with the explanation (aitia, to dioti, to dia ti)
2) to do with necessity

To give the aitia for something is to say why it is the case.

The first conjunct of the definiens of understanding:
1) a understands x only if a knows that y is the explanation of x
The second conjunct of the definiens is ambiguous. it is either:
2a) a understands x only if x cannot be otherwise
2b) a understands x only if a knows that x cannot be otherwise.
(Barnes goes for 2b)

These two (necessary) conditions for understanding are jointly sufficient. So we get a definition for understanding simpliciter:
a understands x = df a knows that y is the explanation for x and a knows that x cannot be otherwise

Note how restrictive 2b is – it only allows knowledge of necessary matters. Even Aristotle seems to want (at times) to broaden the notion of episteme to matters which hold only for the most part.

There may be another type of understanding (see Barnes’ discussion of this). For now we’ll assert that we know things through demonstration.
Demonstration = scientific deduction
Scientific deduction = a deduction by possessing which we understand something

Demonstrative understanding must proceed from items which are
1) true
2) primitive
3) immediate
4) more familiar than
5) prior to
6) explanatory

True because you can’t understand what is not the case.
Primitive and indemonstrable because otherwise you would not understand unless you possess a demonstration of these things

Primitive: there is no Q prior to P…there is no Q from which knowledge of P must be derived. Indemonstrable: needing no demonstration. This does not imply that it is immediate because there may be a vaild but non-demonstrative syllogism concluding to P

Explanatory because we only understand something when we know its explanation

If the only knowledge necessary for having a demonstration of P is knowledge of the principles from which P is deducible, then the principles must contain the explanation of P.

Prior (and more familiar) because they are explanatory and we already know them (not only in the sense of grasping them but of knowing they are the case)

Priority is knowledge; knowledge that Q requires knowledge that P but not vice versa. A. treats familiarity the same as priority.

Immediate: to lack a middle term. A syllogistic proposition AxC is immediate iff there is no term B distinct from A and C such that AxB, BxC |– AxC is a syllogism.

These characteristics concentrate on the implication of the explanatory conjunct of the definiens (see above); the implication of the necessity definiens will be taken up in A4.

The six characteristics are divisible into two groups:
1) absolute features of demonstration
2) relative features of demonstration

The six features characterize principles or axioms of a demonstrative science.

Things are prior and more familiar in two ways:
1) in relation to us (items nearer to perception)
2) simpliciter (items further away; most universal)

A principle of a demonstration is an immediate proposition.
* a proposition is immediate if there is no other proposition prior to it.

A proposition is one part of a contradictory pair.
* A bit further on, A says that a statement is also one part of a contradictory pair. A contradictory pair is a pair of opposites between which there is nothing.

If there is nothing ‘between’ P & Q then there is no third possibility apart from P or Q.

A proposition is:
* dialectical if it assumes either part (of the contradictory pair) indifferently
* demonstrative if it determinately assumes one part because it is true

Types of immediate deductive principles:
posit: it can’t be proved but need not be grasped by anyone who is to learn anything
axiom: it must be grasped by anyone who is gong to learn anything

A posit which
* assumes either of the parts of a contradictory pair (that something is or that something is not), is a SUPPOSITION
* does not assume this is a DEFINITION

Aristotle use the phrases ‘common axiom’ as well as ‘axiom’ to mean the same thing. It is called a common axiom because it is shared with more than one science.

We should take suppositions to be exclusively existential propositions.

A. should have said that there are two kinds of posits:
1) that something is (supposition)
2) what something is (definition)

I need to read up on posits. It looks like a vexed subject.

Because
1) you must be convinced about some object and know it insofar as you have a demonstration
AND 2) given that there is a deduction that these items are the case; then you must
3) already know the primatives (all or some)
4) know them better

“Hence if we know and are convinced of something because of the primatives, then we know and are convinced of them better, since it is because of them that we know and are convinced of the posterior items.”

Anyone who is going to have understanding through a demonstration must
1) not only get to know the principles better than what is being proved but
2) there must be no item more convincing or familiar among the opposites of the principles from which a deduction of contrary error may proceed.
Why? Because anyone who understands anything simpliciter must be incapable of changing is mind (and if 2, then he might well change his mind).

look back at Barnes’ analysis of this argument.

AP, book 1

(Note: I wrote up a summary of this before…but because I put down the AP a while ago and have just now picked it back up, I want to work through the text from the very beginning. So this will be a bit repetitive, although certainly has some new/differently thought about stuff.)

All teaching and learning proceed from preexisting knowledge.

Deductive and inductive arguments effect their teaching in similar ways.
* Deductive: assuming items that we are presumed to grasp.
* Inductive: proving something universal by way of the fact that the particular cases are plain.

There are two ways that we must already have knowledge:
1) we must already believe that they are
2) we must grasp what the items spoken about are.

Knowledge is of two sorts.
1) knowledge of propositions (that x is)
2) knowledge of terms (what x means)

(Barnes says: the knowledge presupposed by a teacher is of two sorts…” I don’t get this reference to a teacher – perhaps it is looking back to 71a5(6?) where A. says that both deductive and inductive arguments “effect their teaching through what we already know”?)

It is possible to acquire knowledge when you get knowledge of some things earlier and you get knowledge of the others at the same time.

So if I learn that Sophie is a mammal by inference from the following premises:
1) all cats are mammals and
2) Sophie is a cat
Then though I must already know that all cats are mammals, I may learn that Sophie is a mammal and that Sophie is a cat at the same time.

Note: A. doesn’t say that we learn the premise as we learn the conclusion but rather as we are being led to the conclusion.

Before you are led to a conclusion (before being given a deduction), you should be said to understand it in one way (universally) but not understand it in another way (simpliciter; haplos). (Because if you don’t know there is such a thing (simpliciter) how would you know it has two angles (simpliciter)?

Read more about knowledge simpliciter/episteme haplos. It’s hard to get my mind around right now.)

Barnes: Universal knowledge is knowledge of things like ‘all triangles have three angles’ but perhaps not particular knowledge (about this triangle).

We should not be convinced by folks who make assumptions about every number or triangle simpliciter, not about everything of which they know it is a triangle or a number.

(For example, you might say, all triangles have three angles. Someone would then present you with a triangle that you had never seen before. They would then say that because you have not seen all triangles, you can’t possibly know that all triangles have three angles…your knowledge of that claim is limited to the triangles that you’ve seen. Aristotle says this is baloney – you have universal knowledge even if you don’t have particular knowledge.)

Nothing prevents us from understanding in one sense and being ignorant in another.

OK…new semester and renewed need to study for this exam. I’m working my way through the Posterior Analytics. As noted in a previous entry, the translation and commentary I’m using is by Jonathan Barnes. Because Barnes’ commentary is so extensive (and excellent), I decided that I need a way to distinguish his commentary from my questions and notes…so I’ve created two different styles of blockquotes. The box with a white background will have notes from Barnes’ commentary and the box with the blue background will have my own notes and comments. Hopefully this will make everything a bit clearer.

I’m not sure, going into this, how I’m going to best go through it. My guess is that I’ll go through it slowly, outlining (as I’ve done thus far) and asking questions. I’ll also be looking back at Barnes’ commentary (the PA is 74 pages, the commentary is almost 200 pages. That should be an indication of something, I think.) throughout. And, if the time it took me to get through chapter 1 is an indication, this is going to be very slow going.

(This is a close paraphrase to direct quotation at times. I only wish I knew it well enough to be able to explain it on my own.)

Chapter 1

All learning and teaching come from pre-existing knowledge. This is also the case with deduction and induction, they effect their teaching through things already known.
Deduction: assuming items which we are presumed to grasp
Induction: proving something universal because the particular instances are clear

There are two ways in which one must already have knowledge:
1) we must already believe that they are (that x is)
2) we must grasp what the items spoken about are (what the term ‘x’ means)
(For some things, we must know both things. For example, when we say something about a triangle we must know what it means and ‘that it is’)

(COMMENTARY: knowledge presupposed by a teacher is of two sorts: 1) knowledge of propositions; 2) knowledge of terms)

“It is possible to acquire knowledge when you have acquired knowledge of some items earlier and get knowledge of others at the very same time” (71a17)
(COMMENTARY: “If at t I learn that a is F by inference from the two premisses that a is G and that everything G is F, then I must have known before t that everything G is F, but I may learn that a is G “at the very same time” as I learn that a is F.”)

For example, consider the following argument:
1) Every triangle has two R
2) This figure in the semi-circle is a triangle
3) This figure in the semi-circle has two R
Thus I have to have known (1) before time t, but I might learn both that the figure in the semi-circle is a triangle and that the figure in the semi-circle has two R.

In some cases, learning occurs in this way (what way? in the way (2) is learned? In the way (3) is learned?) and the last term doesn’t become known through the middle term. This happens when the items are particulars and aren’t said of any underlying subject.

Once you deduce the conclusion, you can be said to know the conclusion in one way and not in another…namely, you know the conclusion universally (all triangles have 2R but you don’t know the conclusion simpliciter (haplos)

(COMMENTARY : Barnes reconstructs this sort of scenario:
(1) b knows that everything G is F
(2) a is G
(3) b does not know that there is such a thing as a
(4) (we can thus conclude that) b does not know that a is F
(5) b knows that a is F (from 1 and 2)

The knowledge in (4) is knowledge simpliciter. This is an ordinary knowledge claim.
The knowledge in (5) is knowledge universally. (Although maybe we want to claim that it is (1) that is universal knowledge and not (5).))

Thus, if someone will go up to you and say that do you know that a pair is even, and you say that it is, and then they show you a pair that you’ve never considered before, you can say that you knew that the pair was even (ie university) even if you didn’t know that that pair was even simpliciter (because you’d never considered the pair before). We can claim this sort of knowledge because we assume it holds for every case.

It is only absurd to say that you are learning something in the same sense in which you already know it. (You can learn it in one sense even if you know it in another.)

Before moving on to practical knowledge, I’m going to spend a bit of time coming to grips with theoretical knowledge. Thus, I’m going to be spending the next several entries working through the Posterior Analytics. I’ll be relying on Barnes’ translation (2nd ed.) and notes and thus any quotes that I make will come from that translation and edition.

To start with, though, I want to work through Barnes’ own introduction to the work and make notes of things that seem interesting and noteworthy:

I

Book A

Book A does not lay out a scientific methodology…or indeed any sort of methodology that one is directed to follow. Instead it concerns itself with how we are to organize and arrange the results of our research.

“it’s aim is to say how we may collect into an intelligible whole the scientist’s various discoveries – how we may so arrange the facts that their interrelations, and in particular their explanations, may best be revealed and grasped.” (xii)

Demonstration is meant to organize what is already discovered, not discover things previously unknown.

I can see how this works with the first principles, given how Aristotle says that they’re discovered. How does this work for the principles/conclusions derived from demonstration? Will these already be known prior to putting them into such form? It seems at least possible that one could discover new things via the method of demonstration. Although perhaps this isn’t really the point…A. might not deny that you can learn things via demonstration, but still insist that it’s primary purpose isn’t discovery but the organization of things you’ve already discovered.

The essential thesis of book A: the sciences are properly expounded in axiomatic form. (Aristotle wanted to do this with every branch of knowledge.) The axiomatization must be formalized in a well-defined language.

The influence of Plato and his dialectic?

Was mathematics already axiomatized at this point, or was Euclid the one that axiomatized mathematics? (This ties back, too, with the idea that Plato wanted knowledge to be modeled after mathematics…)

Book B.

“The axioms of any science…must be (or at least include) definitions or statements of essence; and its main burden is to ask how these definitions are to be elicited and exhibited.” (xiii)

“the essence of a kind K is that characteristic, or set of characteristics, of members of K upon which any other properties they have as members of K depend.” (Me: ??!) It’s sort of like the underlying structure upon which we explain other, more superficial, characteristics or properties.
– this just assumes that in all things there are some properties that are explanatorily basic and others that are explanatorily derivative.

Like having a shape of a certain kind, colors in a certain configuration, etc as primary properties and it being beautiful as a derivative property?

Other notes

The Posterior Analytics was written when Aristotle was still teaching at the Academy and thus much of the material stems from/is tailored to concerns in the Academy at the time (which also means that what is just baffling to us might be absolutely apparent to that audience).

II

The Analytics shouldn’t be seen as a chronological succession (one before the other) but rather as twin contributions to the same lecture course. Although the Posterior Analytics does presuppose the syllogistic reasoning put forward in the Prior Analytics.

These were very much a work in progress, a series of lecture notes that were treated as such by Aristotle, and so it doesn’t really make sense to say one is ‘written’ before the other.

The Posterior Analytics depends on syllogistic reasoning…which is both a serious accomplishment and a problem.
1) Aristotle created a logic that is rigorous and elegant
2) Syllogism is only a fragment of logic (and this shows in the Posterior Analytics itself…Some of Aristotle’s scientific and mathematic examples don’t work all that well on the schema as he’s presented it.

Fundamentals of syllogistic:

“Syllogistic is a series of relations that hold between syllogistic propositions.”

A syllogistic proposition is of the form AxB where A and B are terms (A=predicate and B=subject) and x is one of four syllogistic relations (a, e, i, o)

SO AaB is a universal affirmation proposition (Every B is an A). Aristotle expresses such propositions in one of the following ways:
* A holds of every B
* A is said of every B
* A is predicated of every B
* A follows every B

AeB is a universal negative (No B is an A)

AiB is a particular affirmative (Some B is A)

AoB is a particular negative (Some B is not A)

The most important relations that hold between these various propositions are:

the Laws of Conversion:
(AeB -| |- BeA ; AiB -| |- BiA; AaB |- BiA)

The Laws of Subalternation:
(AaB |- AiB; AeB |- AoB)

A mood is an ordered sequence of three syllogistic propositions:
The first two propositions have one term in common; the third proposition conjoins the other two terms (the ‘extremes’).
There are three arrangements of premise pairs that define the syllogistic figures:
(I: AxB, BxC), (II: MxN, MxX), (III: PxR, SxR).

There are 192 moods of which Aristotle accepts 14 as valid argument forms.

An example of this? All men are mortals, Socrates is a man (would this be a case of AiB?), Socrates is a mortal. (?) Perhaps Every cat is a mammal, every mammal bears live young, therefore every cat bears live young?

The fourteen valid arguments have traditional names and the three vowels of each name give in order the syllogistic relations of the mood:
(I)
bArbArA
cElArEnt
dArII
fErIO

(II)
cEsArE
cAmEstrEs
fEstInO
bArOcO

(III)
dArAptI
fElAptOn
dIsAmIs
dAtIsI
bOcArdO
fErIsOn

SO Barbara = AaB, BaC |- AaC
and Celarent would be AeB, BaC |- AeC

Aristotle also uses modal notions of ‘necessarily’ and ‘possibly’. A demonstration is a species of modal syllogism…all of the propositions are necessary. Barbara is the paradigm demonstrative mood. So the paradigm demonstrative mood will be: ☐AaB, ☐BaC |- ☐AaC

A demonstrative science is one that can be displayed through demonstrations.

Supplementary Notes

“A Pst is primarily concerned to investigate how the various facts and theories which practising scientists discover or construct should be systematically organized and intelligibly presented. The connection with teaching is this: in so far as a teacher is concerned to transmit a body of knowledge, he will best do so by preseinting it in a form in which its organization and explanatory coherence are intelligibly revealed.” (xix)