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

  1. 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:
            1. When the first premise (AB) is wholly false the conclusion will be false.
            2. When AB is partially false, the conclusion can be true.
            3. When the second premise (BC) is wholly false, the conclusion can be true.
            4. When BC is partially false, the conclusion (C) can be true.
          4. If one premise is wholly and one is partially false:
            1. When AB is partially false, C can be true.
            2. 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.
            1. 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.
            1. If negative: Proof with the middle figure.
            2. If affirmative: Proof with the last figure.
          2. Proving a syllogism in the second figure by RAI and ostensive proof.
            1. Proof will accomplished using the first figure.
          3. Proving a syllogism in the third figure by RAI and ostensive proof.
            1. If negative: Proof with the middle figure.
            2. 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
            1. 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.