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Prior Analytics - Book II

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

In Book One of his Analytics, Aristotle discusses the structure of the syllogism, his logical procedure. In Book Two, he discusses some of the properties and defects of syllogisms, and some types of reasoning that are related to syllogisms.

Outline: Book Two

Properties and defects of syllogism; Arguments akin to syllogism
1. Properties
  1. The drawing of more than one conclusion from the same premises
    1. Very nice recapitulation of book one: (1) The number of figures, (2) the character and number of premises, (3) when and how a syllogism is formed, (4) what we must look for when refuting and establishing propositions, (5) how we should investigate a given problem, (6) and by what means we will attain principles appropriate to each subject.
    2. Universal and affirmative particular syllogisms yield more than one result (as these are all convertible propositions).
      1. (E.g.) Universal: Aab -> Aba
      2. (E.g.) Particular Affirmative: Iab -> Iba
      3. (E.g.) Particular Negative: Oab -//> Oba
    3. You can also get this conclusion for universal syllogisms insofar as that which is subordinate to the middle term can be inferred.
      1. Adb, Aba -> Ada, Aad
  2. The drawing of true conclusions from false premises; the first figure
    1. Given true premises, false conclusions are impossible.
    2. True conclusions (wrt fact, not reason) may be drawn from false premises.
      1. If both premises are wholly false, the conclusion can be true. (All men are stones, All animals are stones -> All men are animals).
      2. If both premises are partially false. (Some men are stones, Some animals are not stones -> Some animals are not men)
      3. If only one premise is false:
        
        When the first premise (AB) is wholly false the conclusion will be false.
        
        When AB is partially false, the conclusion can be true.
        
        When the second premise (BC) is wholly false, the conclusion can be true.
        
        When BC is partially false, the conclusion (C) can be true.
      4. If one premise is wholly and one is partially false:
        
        When AB is partially false, C can be true.
        
        When BC is partially false, C can be true.
  3. The drawing of true conclusions from false premises; the second figure
    1. In each of the above situations, in the middle figure, it is possible to reach a true conclusion from one or more false premises.
  4. The drawing of true conclusions from false premises; the third figure
    1. In each of the above situations, in the third figure, it is possible to reach a true conclusion from one or more false premises.
  5. Circluar proof; the first figure
    1. Reciprocal demonstration is when any [P1,P2,C] of a syllogism is provable by assuming the others.
    2. Reciprocal demonstration is only possible if propositions and terms [A,B,C] are convertible (cf. Book 1:A.I.2; i.e. Aab->Aba, Eab->Eba).
      1. Positive Universal: AC: Aab, Abc -> Aac. We can also prove AB by Aac, Acb ->Aab. And BC: (Aac->Aca),Aab -> Abc.
      2. Negative Universal: AC: Abc, Eab -> Eac. And then: AB: Eac, (Acb->Abc) -> Eab. Etc.
      3. Positive & Negative Particular: In the particular cases, we can demonstrate the particular premise from the universal and the conclusion, but not the other way around.
        
        E.g. if: (U)AB, (P)BC -> (P)AC –> Aba, Iac -> Ibc. And this only.
  6. Circluar proof; the second figure
    1. Positive Universal & Particular: Reciprocal demonstration is not possible.
    2. Negative Universal: Aab, Eac -> Ebc. (Aab->Aba), Ebc -> Eac, Etc,
    3. Negative Particular: Once again, the particular premise can be proved but not the universal, for the same reasons as in (A.I.5.iii) above.
  7. Circluar proof; the third figure
    1. Positive & Negative Universal: Reciprocal demonstration is not possible. (Third figure conclusions are always particular.)
    2. Particular premises are sometimes possible to prove reciprocally when the other premise is universal.
      1. Both affirmative & the universal concerns the minor extreme.
      1. Eg. Aac, Ibc -> Iac. If (Aac->Aca), Ibc -> Ibc (!Icb). This fails because we can’t prove something universal about the minor extreme, C.
      2. Eg. Abc, Iac -> Iab. Then if (Abc->Acb), Iab -> Iac. This succeeds because we can say something universal about the minor extreme, C.
    3. One premise is universal affirmative and the other negative -> Circular proof can be given.
  8. Conversion; the first figure
    1. Conversion means altering the conclusion of a syllogism to make another syllogism to prove that either:
      1. The last term cannot belong to the middle.
      2. The middle cannot belong to the last term.
      3. Aristotle will refer to this operation as “refuting a premise”.
    2. Conclusions can be converted into their:
      1. Contradictories: “to all”->”not to all”, “to some”->”to none”.
      2. Contraries: “to all”->”to none”, “to some”->”not to some”.
    3. Universal Contrary: Major extreme premise cannot be refuted universally (forces appeal to third figure). AB, BC -> Eac. –> Eac, Aab -> Ebc. –> Eac, Abc -> Oab (Felapton).
    4. Universal Contradictory : Conversion results in conclusions that are negative and particular.
    5. Particular Contrary: Neither premise may be refuted.
    6. Particular Contradictory: Both premises may be refuted.
  9. Conversion; the second figure
    1. Universal Contrary: The major extreme premise (AB) may not be refuted, but AC can. Aab, Eac -> Abc. Abc, Aab -> Aac. Abc, Eac -> Oab.
    2. Universal Contradictory: The major extreme premise (AB) may not be refuted, but AC can. Ibc, Eac -> Oab. Ibc, Aab -> Iac.
    3. Particular Contrary: Neither premise can be refuted.
    4. Particular Contradictory: Both premises can be refuted.
  10. Conversion; the third figure
    1. Universal Contrary: Neither premise can be refuted.
    2. Universal Contradictory: Both premises can be refuted.
    3. Particular Contrary: Neither premise can be refuted.
    4. Particular Contradictory: Both premises can be refuted.
  11. Reductio ad impossibile; the first figure
    1. The syllogism per impossibile is proved when the contradictory of the conclusion is stated and another (incompatible) premise is assumed.
    2. It resembles conversion, except that a conversion leverages an already-agreed to contradictory, whereas in a reduction to the impossible it is simply clear that the contradictory is true.
    3. E.g.: Aab, Abc -> Aac. Now, we pose that Eab, or Oab then Eab, Abc -> Eac. But Eac is impossible.
    4. All the syllogisms in all moods in all figures can be proved per impossibile, except the universal affirmative in the first figure. (Cf. I.A.11.c.)
      1. Example proof: Eab, Abc -> Eac. Aac, Abc -> Aab, which is impossible.
      2. Why it doesn’t work in the universal figure of the first: Aab, Aca -> Abc. Now assume Ebc. Ebc, (Aca->Aac) -> Eab, which is impossible, but the negation of Eab does not necessarily prove Aab. I think.
  12. Reductio ad impossibile; the second figure
    1. Reductio ad impossibile is possible in all syllogisms in this figure; proofs.
  13. Reductio ad impossibile; the third figure
    1. Reductio ad impossibile is possible in all syllogisms in this figure; proofs.
  14. Comparison of reductio ad impossibile and ostensive proof
    1. Reductio ad impossibile
      1. posits what it wishes to refute by reduction to a statement admitted to be false.
      2. takes one premise from which the syllogism starts and the contradictory of the original conclusion.
      3. it is necessary to suppose that the conclusion is not true.
    2. Ostensive proof
      1. starts from admitted positions.
      2. takes the premises from which the syllogism starts.
      3. it is not necessary that the conclusion is known or true.
    3. Both
      1. Both take two admitted premises.
      2. Anything that can be proved with one can be proved with the other.
    4. Figural dependencies for proving syllogisms.
      1. Proving a syllogism in the first figure by RAI and ostensive proof.
        
        If negative: Proof with the middle figure.
        
        If affirmative: Proof with the last figure.
      2. Proving a syllogism in the second figure by RAI and ostensive proof.
        
        Proof will accomplished using the first figure.
      3. Proving a syllogism in the third figure by RAI and ostensive proof.
        
        If negative: Proof with the middle figure.
        
        If affirmative: Proof with the first figure.
  15. Reasoning from opposites
    1. Possible types of oppositions
      1. Universal affirmative to universal negative
      2. Universal affirmative to particular negative
      3. Particular affirmative to universal negative
        
        Note that particular affirmative to universal negative doesn’t qualify, presumaly along the same logic as (A.I.11.d.ii). Aristotle says they are “only verbally opposed.”
      4. All the universals are “contraries”, all the particulars are “contradictories.”
    2. The first figure
      1. No syllogism can be made from opposed premises.
    3. The second figure
      1. Syllogisms can be made any opposed premise
      2. Science (B) is good, No science (C) is good -> Science (B) is science (C).
    4. The third figure
      1. No syllogism can be made from opposed premises.
      2. A negative syllogism is possible whether the terms are universal or not: Some medicine is a science (B), No medicine is a science (C) -> Some science is a not science [Iab,Eac->Oac].
    5. The types of opposites engender six sets of two premises [e.g. (A.I.15.a.i): Aab, Eac; Aac, Eab;]
    6. It is not possible to draw a true conclusion from opposed false premises.
2. Defects
  1. Petitio principii (Begging the question)
    1. Begging the question is trying to prove something that’s not self-evident by means of itself.
    2. Basically, using A -> B -> C -> A to prove A is begging the question.
      1. When it is uncertain whether A belongs to C, and uncertain whether A belongs to B, but one assumes A belongs to B, one might be begging the original (AC) question.
      2. If in the above it turns out that B = C or B < -> C, the question is begged.
    3. Syllogisms are question-begging when either their predicates are identical or their subjcets are identical.
  2. False cause
    1. ‘False cause’ describes a situation in which the conclusion would have been reached with or without the hypothesis on which it was based.
    2. This is most obvious when the premise is completely irrelevant to the conclusion.
    3. It can also happen when the premise is related to the conclusion, but the conclusion does not follow from it.
  3. Falsity of conclusion due to falsity in one or more premises
    1. A false argument depends on the first false statement in it, be this the conclusion or one of the premises.
      1. A false syllogism cannot be drawn from true premises (cf. A.I.2.a).
  4. How to impede opposing arguments and conceal one’s own
    1. Don’t allow the person against whom you are arguing to use the same term twice in his premises. Be watchful: The middle term is necessary!
    2. Start from the outside: Assume we are set out to prove AF from B, C, D, and E. We need to prove AB and EF first, no BC. Our tricky interlocutor may attempt to start at the middle, and confound us!
  5. When refutation is possible
    1. A refutation is a syllogism which establishes the contradictory of the original conclusion.
      1. A refutation is possible only when at least one of the terms is affirmative.
      2. A refutation is possible only when at least one of the terms is universal.
  6. Error
    1. It turns out that when you do these in practice, its easy to logically know one thing and think the opposite.
      1. A set of premises like this could arise: [Aab, Eac, Abd, Acd] which entails a contradiction.
      2. This is the case with particulars too: [180 degrees, triangle, some particular triangle]. While someone can know ABC holds, she is not per-se required to think that C exists.
    2. Criticism of Meno and the theory of learning by recollection: “It never happens that a man starts with foreknowledge of the particular, but along with the process of being led to see the general principle he receives a knowledge of the particulars, by an act (as it were) of recognition.”
      1. In seeing some particular and not recognizing the universal, one can be led to error as well. E.g.: One can think [all mules are sterile, this is a mule, this animal is with foal] by simply not recalling AB in the presence of some compelling circumstantial evidence of C.
    3. These points show the three senses of “to know”, which, we will note, dictate the three kinds of error above.
      1. To have knowledge of the universal
      2. To have knowledge of “proper to the matter at hand” (of the particular)
      3. To exercise such knowledge
3. Arguments akin to Syllogism
  1. Rules for conversion and for comparison of desirable and undesirable objects
    1. Whenever the extremes (A,C) are convertible, the middle (B) must be convertible with both.
    2. Let => equal “more preferable”. Given {x:{A,B},{C,D}} where A,B and C,D are sets of opposites:
      1. If A=>B and D=>C, then if {A,C}=>{B,D} -> A=>D
      2. Since they are opposites, A and B are in an equal relationship of preferability with inverse magnitude e.g.: (1,-1).
      3. If A=>B and D=>C, if A==D -> {A,C}=={B,D}
      4. Also the example he gives here is incredible (this being the Analytics): “To recieve affection is preferable in love to sexual intercourse. Love then is more dependent on friendship than on intercourse…”
  2. Induction
    1. Every belief comes either through syllogism or from induction (not only demonstrative and dialectical syllogisms thus far, but rhetorical syllogisms and other forms of persuasion).
    2. The syllogism that springs out of inducution, which works from a premise and a conclusion rather than two premises:
      1. [Long-lived, Bilelessness, Particular long-lived animals]: We know Aac, and Abc, so we can induce that Aab as long as C is wider in extension than B. His example is terrible. A more contemporarily comprehensible version is if we swap B for some Darwinian thing like “has been selected for in its ecosystem”.
      2. Also, this will cover syllogisms by probability: [Fire a cooked my hot dog, Fire b cooked my hot dog, Fires cook hot dogs.]
  3. Example
    1. Reasoning by example works when (in syllogism ABC) AB is proved by means of AD where D resembles C.
      1. E.g. ABCD:[Evil, making war against one’s neighbors, Athenians against Thebans, Thebans against Phocians]
      2. To prove AC, we appeal to [AB,BC] and we attempt to prove AB by appeal to AD.
    2. So, if deductive reasoning is reasoning from whole to part, and inductive reasoning is from part to whole, then reasoning by example is from part to part.
  4. Reduction
    1. Reduction involves attempting to clarify a term relationship by reducing one of the terms via another syllogism.
      1. E.g. [What can be taught, knowledge, justice]. AB is clear, but it is unclear that virtue is knowledge. But if BC is clearly equally or more true than AC, we have a reduction.
      2. So, assuming virtue is D, we can reduce the uncertain premise AC to a more certain premise set AD,CD: Aab, [Aad, Acd->Aac] -> Abc.
  5. Objection
    1. An objection is a like a premise contrary to a premise, with the exception that there are no universal/particular restrictions with regard to objections, even in, e.g., Barbara or Celarent. In other words you can validly offer a particular objection to a universal syllogism.
    2. In the attempt to raise an objection, one starts from premises which will result in a contrary conclusion.
      1. This will not work in the second figure, which cannot produce an affirmative conclusion.
  6. Enthymeme
    1. Enthymeme is a syllogism which requires an unstated assumption to be true for the premises to lead to the conclusion.
      1. Universal enthymeme (irrefutable, first figure): [To be with child, To have milk, A lactating woman]: AC,BC->AB.
      2. Particular enthymeme (refutable third figure): [To be with child, To be pale, A pale woman]: AC,BC->AB.
      3. Recall that enthymeme is not refutable in the second figure.
    2. The middle term (B) of an enthymeme may be called an index.
      1. Arguments derived from the middle term are those in the first figure, and are most generally accepted to be true.
    3. The extreme terms (A,C) are signs.
      1. It is possible to infer character from features, if we assume that (1) body and soul are changed together by natural affections, and (2) for each change there is a corresponding sign.
      2. E.g. Lions have courage, Lions have large extremities, Large extremities are signs of courage –> Tigers have large extremities…etc.
      3. An enthymeme of this type would require that, e.g. all and only courageous animals have large extremities.

Written by Paul Tulipana, and posted to Sorglose Nacht on October 13, 2008

Tags:

Aristotle, Logic, Outlines, Philosophy, Syllogism

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

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.

Written by Paul Tulipana, and posted to Sorglose Nacht on October 6, 2008

Tags:

Aristotle, Logic, Outlines, Philosophy, Syllogism

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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

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