Sorglose Nacht : Prior Analytics

Overview

In Book One of his Analytics, Aristotle discusses the structure of the syllogism, his logical procedure.

Outline: Book One

Structure of the syllogism

Preliminary Discussions
1. Subject and scope of the Analytics; certain definitions and divisions
  1. The subject of the Prior Analytics is demonstration and the faculty that carries it out.
  2. Premise: a sentence affirming or denying something.
    1. Universal: Something belongs to all or none of something else.
    2. Particular: That something belongs to some or not to some or not to all of something else.
    3. Indefinite: A premise that doesn’t indicate its universal/particular status: “Pleasure is not good.”
    4. Premises are demonstrative insofar as they proceed by statement rather than questioning. Premises are offered by the arguer, rather than culled from his interlocutor.
  3. Term: That into which the premise is resolved (In “Socrates is a man” - both “Socrates” and “man”)
  4. Syllogism: A discourse in which one thing (a conclusion, consequence) necssarily follows from some other statements.
  5. Perfect and imperfect Syllogisms: Perfect syllogisms need nothing but what’s in the premises to get the conclusion. Imperfect syllogisms rely on external propositions.
  6. Inclusion and non-inclusion of terms in others: We say that one term is included in another insofar as it is predicated (e.g.) of all of another whenever no instance of the latter can be found of which the former cannot be asserted.
2. Conversion of pure propositions
  1. Every premise is either affirmative or negative.
  2. Universal Premises
    1. Negation: Should be always universally convertible: If no pleasure is good, then no good will be pleasure.
    2. Affirmation: Convertible, but not universally: If every pleasure is good, then some good must be pleasure.
  3. Particular Premises
    1. Negative: Non-convertible: If some animal is not a man, it does not follow that some man is not an animal.
    2. Affirmation: Convertible in part: If some pleasure is good, then some good will be pleasure.
3. Conversion of necessary and contingent propositions
  1. The same logical statuses in (A.I.2) will hold good for necessary premises.
  2. For possible premises, the same affirmative structures will hold, but the negative ones won’t. In fact, their conversion potential inverts:
  1. Negative universal possible becomes non-convertible: If it is impossible that every pleasure is good, that does not necessarily imply that it is also impossible that every good will be a pleasure.
  2. Negative particular possible becomes convertible: It is possible that no garment is white, then it is possible for nothing white to be a garment.
4. Generally, the three conversions (inverting of subject & predicate) that are sound are:
  1. Eab -> Eba
  2. Iab -> Iba
  3. Aab -> Iba
Preliminaries for the Exposition of the Three Figures
1. II-a. The three figures of Syllogisms
  
  Figure First Figure Second Figure Third Figure
  
  &nbsp Pred Subj Pred Subj Pred Subj
  
  Premise A B A B A C
  
  Premise B C A C B C
  
  Conclusion A C B C A B
2. II-b. Terminology
  1. “Aab” = a belongs to all b (Every b is a)
  2. “Eab” = a belongs to no b (No b is a)
  3. “Iab” = a belongs to some b (Some b is a)
  4. “Oab” = a does not belong to all b (Some b is not a)

Exposition of the Three Figures

Proper syllogisms in the first figure

Syllogisms Overview
1. Demonstration is a form of syllogism, and not every syllogism is a demonstration.
2. Whenever three terms are so related that the last is wholly contained in the middle, which is wholly contained in the first (positively or negatively), we have a perfect syllogism.
3. Syllogisms are just like a formal structure for the transitive relation of propositions.

Chart of first-figure syllogisms

All pure syllogisms in the first figure are perfect

Form	Mnemonic	Proof
Aab, Abc ¦ Aac	Barbara	Perfect
Eab, Abc \| Eac	Celarent	Perfect
Aab, Ibc \| Iac	Darii	Perfect; also by impossibility, from Camestres
Eab, Ibc \| Oac	Ferio	Perfect; also by impossibility, from Cesare

Spelling out the first-figure syllogisms
1. All A is B, All B is C: All A is C
2. No A is B, All B is C: No A is C
3. All B is A, some C is B: Some C is A
4. No B is A, some C is B: Some C is not A

Proper syllogisms in the second figure

Chart of Second-figure syllogisms

There are no perfect syllogisms in the second figure.

Form	Mnemonic	Proof
Eab, Aac \| Ebc	Cesare	(Eab, Aac)>(Eba, Aac) \| Cel^Ebc
Aab, Eac \| Ebc	Camestres	(Aab, Eac)>(Aab, Eca)=(Eca, Aab) \| Cel^Ecb>Ebc
Eab, Iac \| Obc	Festino	(Eab, Iac)>(Eba, Iac) \| Fer^Obc
Aab, Oac \| Obc	Baroco	(Aab, Oac +Abc)\|Bar(Aac, Oac) \| Imp^Obc

Spelling out the second-figure syllogisms
1. No B is A, All C is A: No C is B.
2. All B is A, No C is A: No C is B
3. No B is A, Some C is A: Some C is not B
4. All B is A, Some C is not A: Some C is not B

Proper syllogisms in the third figure

Chart of third-figure syllogisms

There are no perfect syllogisms in the third figure.

Form	Mnemonic	Proof
Aac, Abc \| Iab	Darapti	(Aac, Abc)>(Aac, Icb) \| Dar^Iab
Eac, Abc \| Oab	Felapton	(Eac, Abc)>(Eac, Icb) \| Fer^Oab
Iac, Abc \| Iab	Disamis	(Iac, Abc)>(Ica, Abc) = (Abc, Ica) \| Dar^Iba>Iab
Aac, Ibc \| Iab	Datisi	(Aac, Ibc)>(Aac, Icb) \| Dar^Iab
Oac, Abc \| Oab	Bocardo	(Oac, +Aab, Abc) \| Bar^(Aac, Oac) \| Imp^Oab
Eac, Ibc \| Oab	Ferison	(Eac, Ibc)>(Eac, Icb) \| Fer^Oab

Spelling out the third-figure syllogisms
1. All C is A, All C is B: Some B is A
2. No C is A, All C is B: Some B is not A
3. Some C is A, All C is B: Some B is A
4. All C is A, Some C is B: Some B is A
5. Some C is not A, All C is B: Some B is not A
6. No C is A, Some C is B: Some B is not A

Common properties of the three figures
1. Syllogisms always result from conversions (3.c), and changing universals to particulars affects the results.
2. All syllogisms in the second and third figures are provable with those in the first figure. Further, it is possible to reduce all proper syllogisms to the universal syllogisms in the first figure (Barbara and Celarent).
  1. Also, the particular syllogisms in the first figure (Darii, Ferio) can be proven by those in the second figure (Camestres and Cesare respectively).
3. In what follows, Aristotle will be doing something like this:
  1. Two necessary premises (8)
  2. One necessary and one assertoric premise (9-11)
  3. Two possible premises (14,17,20)
  4. One assertoric and one possible premise (15,18,21)
  5. One necessary and one possible premise (16,19,22)
4. More terminology:
  1. Since this is his procedure, it is convenient to describe modal syllogisms in terms of the corresponding non-modal syllogism plus a triplet of letters indicating the modalities of premises and conclusion:
  2. N = “necessary”, P = “possible”, A = “assertoric”.
  3. Thus, “Barbara NAN” would mean “The form Barbara with necessary major premise, assertoric minor premise, and necessary conclusion”.
  4. I use the letters “N” and “P” as prefixes for premises as well; a premise with no prefix is assertoric. Thus, Barbara NAN would be NAab, Abc : NAac.
Syllogisms with two necessary premises
1. There are three classes of premise possible for a syllogism, (a) a necessary one, (b) a contingent/possible one, and (c) a simple/assertoric/pure one.
2. With the exceptions of Baroco (5.a.4) and Bocardo (6.a.5), conclusions will be proved to be necessary by conversion (3.c).
Syllogisms with one assertoric and one necessary premise in the first figure
1. Universals: When the major premise of a first-figure syllogism is necessary, the conclusion is necessary.
2. Particulars: When the universal premise is necessary, the conclusion is necessary.
Syllogisms with one assertoric and one necessary premise in the second figure
1. Universals: When the negative premise of a second-figure syllogism is necessary, the conclusion is necessary.
2. Particulars: When the negative premise is both universal and necessary, the conclusion is necessary.
Syllogisms with one assertoric and one necessary premise in the third figure
1. Universals: When one of the two premises of a third-figure syllogism is necessary and both are affirmative, the conclusion will be necessary.
2. Particulars: When the universal premise is necessary, and both are affirmative, the conclusion is necessary.
Comparison of assertoric and necessary conclusions. In overview:
1. You need at least one necessary premise to get a necessary conclusion.
2. Assertoric conclusions are reached by two simple premises.
Prelimiary discussion of the contingent/possible
1. For Aristotle, “Possibly P” is equivalent to “not necessarily P” and “not necessarily not P”.
  1. Hence the conversion looks like Pp -> [!Np, !N(!p))]. That said, this difference has weird logical consequences.
  2. Entailments:
    1. PAab -> PEab
    2. PEab -> PAab
    3. PIab -> POab
    4. POab -> PIab
  3. Modern modal logic, contrawise, treats necessity and possibility as interdefinable:
    1. “Necessarily P” is equivalent to “not possibly not P”,
    2. “Possibly P” is equivalent to “not necessarily not P”.
    3. Like this: (i) Np -> !P(!p), and (ii) Pp -> !N(!p)
  4. Aristotle acknowledges that there is a certain sense of “possible” that is more like the modern equivalece:
Syllogisms in the first figure with two possible premises
1. PAab, PAbc -> PAac
2. Universals: When the major premise is a universal, and the minor premise is particular, there will be a perfect syllogism.
3. Particulars: When the major premise is particular, no syllogism is possible.
Syllogisms in the first figure with one possible and one assertoric premise
1. PAab, Abc -> PAac
2. Aab, PEbc -> PEac
Syllogisms in the first figure with one possible and one necessary premise
1. PAab, NAbc -> PAac
2. NEab, PAbc -> NEac
3. PEab, NAbc -> PAac
Syllogisms in the second figure with two possible premises
1. No syllogism is possible in this combination.
Syllogisms in the second figure with one possible and one assertoric premise
1. Eab, PAac -> Eba, PAac -> PEbc
2. Aab, PEac -> Aba, PEac -> PEbc
Syllogisms in the second figure with one possible and one necessary premise
1. NEab, PAac -> NEba, PAac -> PEbc, Ebc (otherwise it would be impossible that Aac)
2. NEab, PEac -> NEba, (PEac -> PAac) -> PEbc, Ebc (cf. 19.a, 13.a.ii)
3. Rule: If there is a universal, negative and necessary premise, a syllogism is possible.
Syllogisms in the third figure with two possible premises
1. PAac, (PAbc -> PIcb) -> PIab
2. PEac, PAbc -> POab
3. A syllogism with two negative possible premises lead nowhere.
Syllogisms in the third figure with one possible and one assertoric premise
1. Aac, (PAbc -> PEbc) -> PIab (cf. 13.a.i, 15.b)
2. Abc, POac -> POab
3. Whenever both premises are indefinite or particular, syllogism is impossible.
Syllogisms in the third figure with one possible and one necessary premise
1. NAac, PAbc -> NAac, PIcb -> PAab, Aab
2. PEac, NAbc -> PAac, NAbc -> PEab (cf. 19.b)
3. NEac, (PAbc -> PEbc) -> POac -> Oac -> Oab (* I don’t get this one. It might be wrong, but it seems like this is what he’s saying.)

Supplementary Discussions
1. Every sylllogism is in one of the three figures, is completed through the first figure, and reducible to a universal mood of the first figure.
  1. All of the above syllogisms can be reduced to the univeral syllogisms in the first figure (Barbara, Celarent).
    1. To prove A has some relationship to B, you need some C that unites them.
    2. If this is the case, in order to predicate A of B, you need to predicate either (1) A of C and C of B, (2) C of both A and B, or (3) both A and B of C.
      1. (1) Possible syllogism: [(Eac | Aac),(Ecb | Acb)]
      2. (2) Possible syllogism: [(Aca | Eca), (Acb | Aca)]
      3. (3) Possible syllogism: [(Aac | Eac), (Acb | Acb)]
    3. Which are the three figures (4-6), which we just proved reduce to Barbara and Celarent in 7-22 above.
2. Quality and quantity of the premises of a syllogism
  1. Every syllogism requires at least one affirmitive and one universal premise.
  2. Further, one of the premises must be like the conclusion in both its affirmitive/negative quality and in terms of its necessary/possible/assertoric status.
3. Number of the terms, propositions, and conclusions
  1. Every demonstration requires three terms and no more. (The fact that multiple minor premises can be used to assess a single conclusion does not create extra premises, but extra syllogisms.)
  2. It follows from this that every conclusion follows from two premises and no more.
  3. In the case of prosyllogisms or continuous middle terms, we can generally state that:
    1. Terms = premises +1
    2. Premises = relations of predication (e.g. A,E,I,O)
    3. When you add terms, conclusions grow proportionally where: newTerms = oldTerms++; conclusions+=(oldTerms-1);
4. The kinds of proposition to be established or disproved in each figure.
  1. The universal affirmative is only proved through Barbara.
  2. The universal negative is proved through Celarent in the first figure, Cesare and Camestres in the second.
  3. The particular affirmative is proved through Darii in the first figure, and Darapti, Disamis and Datisi in the third.
  4. The particular negative is proved through Ferio in the first, Festino and Baroco in the second, and Felapton, Bocardo, and Ferison in the third figure.

MODE OF DISCOVERY OF ARGUMENTS
1. General
  1. Rules for categorical syllogisms, applicable to all problems
    1. Individuals (Socrates) cannot be predicated of universals, but universals can be predicated of them (Socrates is human).
      1. Predicating a sensible particular (Socrates) on something else is always incidental: The white thing is Socrates.
      2. The ‘upward limit’ of predication is yet to come (Posterior 1. 19-22); we assume it now.
    2. The aspiring syllogist should collect a cache of universal premises (by comprehending relations of definition and properties).
      1. The aforementioned aspirant should take care to realize that some things that apply universally to the species are not so applicable to the genus, and while this is not the case vice versa, one should still avoid applying species predicates to a genus for propriety’s sake.
  2. Rules for categorical syllogisms, peculiar to different problems
    1. To build a syllogism, you have to look at subjects and their attributes.
    2. Suppose (1) B entails A, which entails C and D’s cannot be predicated of As and (2) E’s have attribute(s) F, can’t have attributes H, and are entailed by G.
      1. C=A, then Afe,Aac->Aae (first figure)
      2. C=G -> Iae (last figure)
      3. F=D -> Eaf,Afe -> Eae (first figure, second figure)
      4. B=H -> Aba, (Ehe->Ebe) -> Eae
      5. D=G -> (Ead->Eag),Ige -> Oae (last figure)
      6. B=G -> Aba, (Aeg->Aeb) -> Aea & Iae
    3. Hence, we must find out which terms in the inquiry are identical
  3. Rules for reductio ad impossibile, hypothetical syllogisms, and modal syllogisms
    1. What is proved ostensively may also be concluded syllogistically per impossibile and vice versa.
      1. Aba, Iae -> Ibe: But it Ebe was assumed. Hence, it must be the case that Eae.
      2. Eae, Aeg -> Eag: But Aag was assumed. Hence it must be the case that Iae.
    2. Generally, a ostensive syllogism has two true premises, and in the reductio ad impossibilie, one of the premises is assumed falsely.
    3. Hypothetical syllogims:
      1. C=G, Aeg -> Aae
      2. D=G, Aeg -> Eae
    4. The method works the same way whether the relation is necessary or possible.
2. Proper to the several Sciences and Arts
  1. “It is the business of experience to give the principles which belong to each subject.”
3. Division
  1. Division (cf. The Sophist) is a sort of weak syllogism - it begs the question and proves something more general than it ought.
    1. Division takes the universal as a middle term. E.g.:
      1. A = Animal, B = mortal, C = immortal, D = man
      2. Division assumes all A is either B xor C, so if D is A, then D = B xor C, which Aristotle doesn’t believe.
      3. Funny example then where B = footed, C = footless.
Analysis (I) of arguments into figures and moods of syllogism
1. Rules for the choice of premises, terms, middle term, figure
  1. In attempting to select the premises, ensuring at least one universal premise and two total premises.
  2. Further, we need to discern that nothing unnecessary is assumed, and nothing necessary is omitted.
  3. This established, we need to take as the middle term that which is stated in both premises.
2. Quantity of the premises
  1. That one premise be universal - which is to say that one term is premised of all of another term - is absolutely required.
3. Concrete and abstract terms
  1. An easy fallacy to encounter is one in which the terms are “set out wrong”.
    1. E.g. A=Health,B=Disease,C=Man -> Eac, which is obviously wrong.
    2. Subustituting more concrete terms -> A=Healthy,B=Diseased,C=Man, we get better results.
4. Expressions for which there is no one word
  1. Let A=180 degrees, B=Triangle, C=Isosceles triange
    1. It appears that while Aac because of Aab, there is no middle term for AB.
    2. This is because, Aristotle says, the middle must not always be assumed to be an indivdual thing, but sometimes a complex of words.
5. The nominative and the oblique cases
  1. Terms should always be used in the nominative (man, good, contraries) and not the oblique (of man, of good, of contraries).
    1. Eg.: If Wisdom (a) is knowledge (b), and wisdom (a) is of the good (c), then knowledge is of the good. I think the problem here is just an equivocation of “is of the good”. Or maybe the translation suffers from this and the original doesn’t. Who knows.
  2. Premises ought to be understood by case: dative, genitive, accusative, nominative.
6. The various kids of attribution
  1. Any derivation of “belonging” (”This belongs to that”) can be understood in as many ways as there are categories.
7. Repetition of the same term
  1. When you have a term that repeats another term,
    1. A=”knowledge that it is good”, B=”good”, C=justice. * universals here to keep it simple
    2. P1: There is of the good (b) knowledge that it is good (a). (Aba)
    3. P2: Justice (c) is good (b). (Acb)
    4. But if we add “that it is good” to B (good that it is good), we still get P1, but P2 becomes sensless.
  2. Hence, if you’re going to repeat a term, add it to an extreme (A or C) and not a middle term.
    1. He is deeply confusing about this.
8. Substitution of equivalent expressions
  1. We can exchange equivalent expressions in syllogisms.
9. The definite article
  1. The premise “Pleasure is good” is different from “Pleasure is the good”.
  2. The use of either requires consistency in the use of the definite article.
10. Interpretation of certain expressions
  1. The main point seems to be that “A is said of all of which B is said” is equivalent to “A is said of all the things of which B is said”, a point so lukewarm it’s hard to imagine I actually understand the passage.
11. Analysis of composite syllogisms
  1. A composite syllogism can be composed of simple syllogisms from multiple figures; it can be internally heterogenous.
12. Analysis of definitions
  1. Sometimes syllogisms use throwaway terms in definitions. E.g. if a given syllogism proves that water is a drinkable liquid, then we only really proved either drinkablility or liquidity (assuming the other).
13. Analysis of arguments per impossibilie and of other hypothetical syllogisms
  1. Reduction of hypothetical syllogisms (syllogisms with at least one hypothetical premise) is impossible.
  2. Neither can arguments brought to conclusion per impossibile.
    1. These differ from (I.C.44.a) insofar as those latter require an agreement on a hypothetical premise, whereas in the former men accept the reasoning because the falsity of a premise is patent and required for the conclusion.
14. Analysis (II) of syllogisms in one figure into another
  1. The conversion of syllogisms (cf. I.A.I.3) continues to apply, and that we can reduce complex syllogims to their components, which can in turn be reduced to the first-figure syllogisms as before, and that these can be proved reductio ad impossibile.
15. ‘Is not A’ and ‘is not-A’
  1. The clearest example he gives of why these two phrases are different:
    1. They are not identical: “It is not a white log” is not identical to “it is a not-white log”.
    2. Nevertheless, there seems to be some relationship as it is impossible that “It is a white log” and “It is a not-white log”.
  2. Succintly, the appeal to a third term seems to be required, The general four-term logical matrix is:
    1. A=”white”, B=”not white”, C=”not-white”, D=”not not-white”.
    2. Nothing can be (1) [A,B] or (2) [C,D], but everything must be (3) either A or B and (4) either C or D. (Noncontradiction, Excluded Middle)
    3. D “follows” B: Since (4), and C cannot belong to that which B belongs (since B “carries along” A and (1)).
    4. Since C “does not reciprocate” with A, but (4), then it is possible that something could be [A,D].
    5. Since A follows C, then B and C cannot belong to the same thing.
    6. B “does not reciprocate” with D either, since (I.C.46.b.iv).
  3. Out of those five conclusions (ii-vi), only one makes sense to me: ii. The quoted language is just lost on me. Email would be great.

Whew. Next week, Book Two: Properties and defects of syllogism; Arguments akin to syllogism.

Form	Mnemonic	Proof
Eab, Aac \| Ebc	Cesare	(Eab, Aac)>(Eba, Aac) \| Cel^Ebc
Aab, Eac \| Ebc	Camestres	(Aab, Eac)>(Aab, Eca)=(Eca, Aab) \| Cel^Ecb>Ebc
Eab, Iac \| Obc	Festino	(Eab, Iac)>(Eba, Iac) \| Fer^Obc
Aab, Oac \| Obc	Baroco	(Aab, Oac +Abc)\|Bar(Aac, Oac) \| Imp^Obc

Form	Mnemonic	Proof
Aac, Abc \| Iab	Darapti	(Aac, Abc)>(Aac, Icb) \| Dar^Iab
Eac, Abc \| Oab	Felapton	(Eac, Abc)>(Eac, Icb) \| Fer^Oab
Iac, Abc \| Iab	Disamis	(Iac, Abc)>(Ica, Abc) = (Abc, Ica) \| Dar^Iba>Iab
Aac, Ibc \| Iab	Datisi	(Aac, Ibc)>(Aac, Icb) \| Dar^Iab
Oac, Abc \| Oab	Bocardo	(Oac, +Aab, Abc) \| Bar^(Aac, Oac) \| Imp^Oab
Eac, Ibc \| Oab	Ferison	(Eac, Ibc)>(Eac, Icb) \| Fer^Oab

Sorglose Nacht

Prior Analytics - Book 1

The following is an outline of a philosophical text which is provided with no claim with regard to it's accuracy or neutrality. Use freely, but at your own risk.

Overview

Outline: Book One

Figure	First Figure		Second Figure		Third Figure
&nbsp	Pred	Subj	Pred	Subj	Pred	Subj
Premise	A	B	A	B	A	C
Premise	B	C	A	C	B	C
Conclusion	A	C	B	C	A	B