Sorglose Nacht

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

Metaphysics I-VI

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

The first six books of Aristotle’s metaphysics serve to give the problem of being its historical and theoretical context. Book I discusses the definition and purposes of philosophy, and gives a short recapitulation of its history. Book II seeks to address in advance concerns about Aristotle’s metaphysics, by making the argument against the possibility of an infinite regress. Later in the Metaphysics, this will be developed into the famous argument for God. Book III provides a sketch of the main problems of philosophy. Book IV details a few additional premises of Aristotle’s argument, namely the arguments for the principles of non-contradiction and the excluded middle. Book V is a philosophical lexicon, giving the meanings of 30 key philosophical terms. Book VI, finally, leads into the main argument (given in parts VII-IX), by excluding two of the senses of Being detailed in Book V as the proper object of study for metaphysics.

BOOK I/BIG ALPHA
1. The advance from sensation through memory, experience, and art, to theoretical knowledge.
  1. Since we all desire to know, we rejoice in our senses. Particularly in sight.
  2. Sensation gives some animals memory, and those which have memory and hearing can be taught.
  3. Humans also have art and reasoning.
  4. Art arises when from many notions gained by experience, one universal judgment may be made (not particular, e.g. medicine good for all people with symptom n, not just Socrates).
  5. Nonetheless, it must be remembered that art can only be applied to particulars (aka. one cures Socrates, not disease y).
  6. Wisdom, though, is in knowing why the thing is so, and not simply in knowing that it is.
  7. Hence, Wisdom is knowledge about certain principles and causes.
2. Characteristics of ‘wisdom’ (philosophy).
  1. Generally wise people: can know many things (although not necessarily in detail), can know hard things to know, can teach well, etc.
  2. Things far from the senses are hardest for men to know. Knowing these universals is a good indicator of wisdom.
  3. The most worthy of knowing among these are the first principles and causes. This is philosophy.
  4. There is no doubt straightaway that this is not “a science of production”, but rather a slow, arduous process of uncovering.
  5. There may also be some concern that philosophy’s objectives are either beyond human means or that their achievement would make God jealous.
    1. Not so: “God is thought to be among the causes of all things and to be a first principle…”
    2. Not so: “such a science either God alone can have, or God above all the others.”
3. The successive recognition by early philosophers of the material, efficient, and final causes.
  1. Causes are spoken of in four senses:
    1. The essence/substance
    2. Matter or substratum
    3. The source of the change
    4. The purpose and the good
  2. The first philosophers identified material constituting the essential part of all things (c.i.ii above) as causal: The essence of Socrates constitutes Socrates whether he is being beautiful, or musical, or not.
    1. Ancient philosophers are generally materialists (aka. one or some combination of the four elements): Yet they don’t agree on the number or nature of these principles.
  3. For those among the ancients who abide multiple causes, we have things like fire as having independent essences (c.i.i above).
  4. But /why/ do these materials form things? And how do elements cause things like beauty? Anaxagoras talked of a /reason/ throughout nature (c.i.iii?).
4. Inadequacy of the treatment of these causes.
  1. As an exemplar for someone who had an idea, but didn’t carry it out systematically, Aristotle poses Empedocles as the first to mention the bad and the good as principles (c.i.iv?). He was also the first to pose four material elements (though he treaded them as two: fire and its opposites).
  2. But, generally (and there are some specifics here), their treatment of the causes was quite inadequate.
5. The Pythagorean and Eleatic schools; the former recognizes vaguely the formal cause.
  1. The Pythagoreans were the first to take up mathematics; they thought that all things (justice, soul, reason, etc) were expressible numerically. Numbers were their first principle.
    1. This gives way to a principle where even and odd are two causes, and from one springs all numbers. This in turn gives way to binary cognates: even/odd, male/female, one/many, left/right, good/bad.
    2. From this we can learn that the contraries are the principles of things.
    3. Notably (later, 987:13) unique to Pythagoreans, also, is the thought that finitude and infinity are not attributes of other things, but are themselves the substance of the things of which they are predicated.
  2. In particular, what’s germane to Aristotle is Parmenides’ conception of the One:
    1. Seeing as being is everything that exists and nothing that doesn’t, it is one.
    2. But our senses show us many things.
    3. Parmenides then gives us a two-cause/principle system: hot/cold qua existent/nonexistent.
6. The Platonic philosophy; it uses only the material and formal causes.
  1. Socrates was busying himself about ethical matters, but in seeking the universal in these, fixed his thought for the first time on definitions.
  2. Since sensible things are always changing, any common definition could not be of them, and must rather be of an Idea. Sensible things are “named after these” in virtue of their participation in them.
    1. Note: Objects of mathematics apparently fit somewhere between sensible things and Forms, since they are eternal and also many.
    2. The participation relationship as such, provides the possibility of a unique /separation/ between the one and the many.
    3. Yet, what /happens/ appears to be contrary to this, since (i.e.) a man who makes tables applies the form, and though he is one, makes many tables. (What is the argument here?)
  3. So Plato recognized two types of causes:
    1. Essential: The Forms are the essences of things, and the One is the essence of the Forms
    2. Material
7. The relation of the various systems to the four causes.
  1. Almost everyone gets the matter causes: be it fire and water, the infinite, atoms, the great and the small.
  2. Some others have mentioned the source of movement, e.g. friendship and strife, or reason, or love.
  3. No one has expressed the essence (i.e. substantial reality) distinctly. Plato hints at it with the Forms.
  4. The good as a cause is both said and unsaid in the philosophers of movement’s causes. For those who say that the One or the existent is the good, and that it is the cause of substance, but not that is for the sake of this. It is not then a cause qua good, but only incidentally.
8. Criticism of pre-Platonic philosophers.
  1. One cause (ie. fire or water or air) is not a tenable position, because it gives no account of movement, or essence, etc. Similarly, multiple material elements.
  2. Anaxagoras has an interestingly modern position, but is still part of a camp that deals only with the sensible, and thus, cannot offer us a compellingly complete account of ontology.
  3. The Pythagoreans, who now deal with things visible and invisible (including numbers), still only deal with the physical world, implying that they actually agree with the physical philosophers that the real is constituted by only perceptible, sensible things.
9. Criticism of the doctrine of ideas.
  1. The Forms are nefariously difficult to prove.
    1. There’s no convincing way to prove that they exist in the first place.
    2. Further, of the more accurate arguments, some lead to Ideas of relations, of which we may say there is no independent class, and other introduce the ‘third man’ (infinite regress of forms: F1:[a,b,c]->F2[F1,a,b,c]->…)
    3. How could the substance and that of which it is the substance exist apart?
  2. The Forms are not even that beneficial if they exist
    1. The doctrine of the Forms seems to necessitate as many Forms as there are things in the world. Apparently, this will also require Forms for the negations of things.
    2. Given (j.i) and (j.ii.i) it seems entirely unclear what exactly the Forms are contributing to either ontology or epistemology.
  3. A second thread, a participation relation tells us nothing about causation.
    1. How do sensible things come into existence given the existence of the Forms?
    2. Numbers cannot be Forms because Platonists speak of the One has homogeneous.
    3. How would a theory of Forms account for (e.g.) points on a line? [What’s the argument here?]
  4. The overview
    1. There’s no convincing account of the causes of movement from a theory of Forms.
    2. The proofs of oneness show not the oneness of all things, but the existence of a One in itself, which requires us to grant a lot of assumptions.
    3. It’s unclear how things combine to allow things like points, lines, and planes from numbers: These aren’t Forms, nor intermediates, nor perishable things. They seem to be a fourth class.
    4. How could we /learn/ the Forms of all things (contra Socrates’ recollection model)? How can we know “straightness” outside of straight things?
    5. Finally how can we comprehend sense-primitives with Formal concepts? (He doesn’t say this, but it’s Kant’s left-right intuition from the Prolegomena.)
10. The history of philosophy reveals no causes other than the four.
BOOK II/LITTLE ALPHA
1. General considerations about the study of philosophy.
  1. Philosophy is the attempt to attain knowledge of truth. The (eternal) truth which causes all other truths is the sublime object of philosophy.
2. There cannot be an infinite series, an infinite variety of kinds, of causes.
  1. There is a first principle, and the causes of things are neither (b.i.i) an infinite series nor (b.i.ii) infinitely varied in kind.
    1. Cause itself necessitates that the series of causes be bounded. If (a) every (temporal) events is caused, and (b) there is no beginning of this series, then (c) every event is an intermediate event (requiring a causal agent that precedes it), and therefore (d) nothing causes anything else. Better:
      1. A contingent being exists (a contingent being is such that if it exists, it can not-exist)
      2. This contingent being has a cause or explanation of its existence.
      3. The cause or explanation of its existence is something other than the contingent being itself.
      4. What causes or explains the existence of this contingent being must either be solely other contingent beings or include a non-contingent (necessary) being.
      5. Contingent beings alone cannot cause or explain the existence of a contingent being.
      6. Therefore, what causes or explains the existence of this contingent being must include a non-contingent (necessary) being.
      7. Therefore, a necessary being (a being which, if it exists, cannot not exist) exists.
    2. If the kinds of causes are infinite, then knowledge (this seems to mean, “complete knowledge”) is impossible, because we cannot account for/abstract from infinite types causes in finite time.
3. Different methods are appropriate to different studies.
  1. Getting knowledge (ontology) and getting the way of attaining knowledge (epistemology) are two different things, and require different modes of discourse.
BOOK III/BETA
1. Sketch of the main problems of philosophy.
2. Fuller statement of the problems: -
  1. Can one science treat of all the four causes?
  2. Are the primary axioms treated of by the science of substance, and if not, by what science?
  3. Can one science treat of all substances?
  4. Does the science of substance treat also of its attributes?
  5. Are there any non-sensible substances, and if so, of how many kinds?
  6. Are the genera, or the constituent parts, of things their first principles?
  7. If the genera, is it the highest genera or the lowest?
  8. Is there anything apart from individual things?
  9. Is each of the first principles one in kind, or in number?
  10. Are the principles of perishable and of imperishable things the same?
  11. Are being and unity substances or attributes?
  12. Are the objects of mathematics substances?
  13. Do Ideas exist, as well as sensible things and the objects of mathematics?
  14. Do the first principles exist potentially or actually?
  15. Are the first principles universal or individual?
BOOK IV/LAMBDA
1. Our object is the study of being as such.
  1. In order to grasp first causes/principles, we need to study being as being. We arrive at this conclusion because it must be something about being as such that is /necessary/ to the existence of things.
2. We must therefore study primary being (viz. substance), unity and plurality, and the derivative contraries, and the attributes of being and of substance.
  1. All things that are said to be refer to a single “thing”, namely, substance.
    1. “Substance” is the stuff of being, it seems, because things are said to be insofar as they are related to (are, or are qualities of, or are negations of, etc.) substance.
    2. Hence, substance is the subject of philosophy.
  2. Now, there are as many parts of philosophy as there are kinds of substance. What we are detailing here is “first philosophy” (here, ontology).
    1. Being and unity have a strict causal relationship (”one being” parses out to the same content as “being”). This means that the study of unity is part of the study of being, and hence falls under first philosophy’s domain. (Entails sameness, etc.)
    2. Likewise, as difference is simply the negation or privation of unity, this too must fall under the domain of our first science. Which entails of course unlikeness, otherness, contrariety, etc.
    3. Also, the history of philosophy tells us that all things are either contraries or composed thereof (hot/cold, love/strife, limited/unlimited), and hence in this way too we can see that first philosophy entails a study of being as sameness and otherness.
    4. So our first philosophy will examine being qua being, and also the attributes that belong to it qua being: prior/posterior, genus/species, whole/part, etc.
3. We must study also the primary axioms, and especially the law of contradiction.
  1. Truths that hold good for everything there is (axioms) doubtless also belong to the domain of first philosophy. The reason is that what unites these axioms is being itself, so they are axioms that hold good for all things qua being.
  2. And here Aristotle introduces, as the fundamental axiom, the law of non-contradiction.
    1. Remember, this is a term logic: “the same attribute cannot at the same time belong and not belong to the *same subject* and in the same respect.” (My emphasis: Not subject/predicate-combo [proposition]).
4. Fatal difficulties involved in the denial of this law.
  1. This law is so solid that no educated person should ever demand its demonstration.
  2. We can however demonstrate it negatively:
    1. Reasoning is possible because words at least one meaning. (E.g. Men are ‘two-footed animals’.)
      1. If they had several, we could create new words for each of the meanings.
      2. If they had infinite meanings, reason would be impossible. Likewise if they had none.
    2. So, if a name has one and only one meaning. This entails that “being a man” cannot mean “not being a man”.
    3. Any confusion of the signifier/signified relationship is just that: confusion. It doesn’t point to the possibility that words have multiple meanings.
  3. Given that, we understand that non-contradiction is necessary. (E.g. a man can not both be and not be a two-footed animal.)
  4. Further, this incontradictable “manness” is the very substance/essence of what it is to be a man.
    1. So, attributes are not essential (e.g. against the view that not-manness could be an attribute of a man, I suppose) because this would entail infinite predication:
      1. An accident is of a subject, not another accident: The white is not musical, the man is white and musical.
      2. But, in “Socrates is musical” both terms are “accidental to something else.” So, two senses:
        
        White is accidental to Socrates, and Socrates the white has not yet another accident.
        
        White cannot have musicality.
      3. On overview: Sense (d.iv.i.i) reduces to sense, and (d.iv.i.ii), in this an infinite number of accidents combined together is impossible; there must be substance somewhere.
    2. The end result of this is you’re just talking about infinite indeterminate subjects, all of which must be predicated by the affirmation and negation of every attribute.
      1. We say x is y.
      2. We say z is b.
      3. We say x is not z qua b. This entails x is not b.
      4. Without N/c we say x is b.
      5. This entails in turn that x is in fact z, and every other subject, etc.
  5. Two arguments in conclusion
    1. Nicely summed up: “If it is true that a thing is a man and a not-man, evidently also it will be neither a man nor a not-man.”
    2. Either N/c is true of everything, or else of nothing. If it were true of something, then the possible predicates of that would be subject to it, and so on…
  6. Two more tempered arguments
    1. Practically, we don’t see people walking off cliffs all the time, so they must be capable of some kind of judgements, and hence telling the difference between good and not-good.
    2. Additionally, there seems to be “more truth” in thinking that 4 is 5 than there is in thinking that it’s 1000. In other words, he admits of some “degree” of attributes that’s possible in things (terms). [Again, this would appear to be contra propositional logic.]
5. The connexion of such denial with Protagoras’ doctrine of relativity; the doctrine refuted.
  1. Nonetheless, we see contradictions cropping up everywhere - aka. two men will have contradictory opinions on what is good. This can lead to a certain relativism, that all opinions are right!
  2. This is due first to a confusion about two senses of “be”: namely, something can potentially at the same time two contraries, but not actually.
  3. Secondly this confusion arises as entailed by a confusion about from whence truth comes. Namely, some think that truth arises from the sensual appearances. (Aka. they think that since things become, they are neither being nor non-being exactly, or rather both.)
    1. There’s an appeal here that things are changing only in quantity, not in quality.
      1. Note generally that the required product is to show that there’s something changeless.
    2. Not all appearances are true; people (e.g. doctors) and senses (e.g. sight) have different degrees of authority on various objects of appearance. But the appearance of (e.g.) sweetness as such will never be changed (sweetness will always be sweet).
6. Further refutation of Protagoras.
  1. If not all things are relative, and some are self-existent (e.g. the objects of sensation), not everything that appears will be true.
  2. If a thing is one, this entails that it is in relation to either one or a definite number of things; that “that which thinks” is in relation to infinite things is impossible. (This is an argument against solipsism.)
7. The law of excluded middle defended.
  1. There cannot be an intermediate between contradictories. If there was, saying “it is” or “it isn’t” is bankrupt of its content.
  2. Another regress: If there is a term B which is neither A nor not-A, there will be a new term C which is neither B nor not-B.
  3. So, if (4.e.iii) then everything is true, and if (4.g.i) then everything is false.
8. All judgements are not true, nor are all false; all things are not at rest, nor are all in motion.
  1. Any of this requires us to postulate the notion of “meaning”. Namely, that we know what it is for something to be true or false.
  2. Hence, given (4.c.ii) and (4.h.i), some judgements must be true and some must be false.
  3. This dictates also that there are both motion and rest.
    1. Admitting that there is some truth in (4.e.i), and that some propositions can be true and false at different times, there must be movement.
    2. And, if everything is constantly in motion, nothing can be true. We can be assured that the former clause of the previous sentence is false by appeal to (2.b.i) [among other things].
BOOK V/DELTA: PHILOSOPHICAL LEXICON
1. Beginning
  1. The ways in which it is spoken of.
    1. The first point or the best point at which to start.
    2. The immanent (a house’s foundation) or non-immanent (parents to child) start of something.
    3. The mover/changer of something.
    4. The condition of knowability of something.
  2. What these ways have in common.
    1. They are the first point from which a thing comes to be (known).
2. Cause
  1. The ways in which it is spoken of.
    1. The material stuff (bronze:statue) or the pattern (or essence) of something.
    2. Both the beginning (5.a.ii) and the end of something (one walks for health).
  2. What follows
    1. There are several causes of one thing.
    2. Causes and effects usually play both roles reciprocally (excercise is a cause of good health, which causes exercise).
    3. Contraries are usually causes for contrary effects (the presence and privation of the steersman:safety and shipwreck).
  3. Four senses of causes: Material substrata, essences, sources of change, ends.
  4. Genus and accident:
    1. Causes as either the individual, or the genus, or as the accidental, or as the genus that includes the accidental.
    2. Genus-causes are also inherited from parent objects (the sculpture is caused by Ron, and man, and animal, and living thing, etc.).
    3. Accidental causes are “accidentally” inherited from individuals (the sculpture is caused by Ron, who is musical, so the musical caused the statue).
  5. What these ways have in common.
    1. They may all be taken as acting or having a capacity, although this works in different ways.
3. Element
  1. The ways in which it is spoken of.
    1. The (indivisible) primary component(s) immanent in a thing.
    2. Indivisible primary things that are useful for many purposes (aka. atoms) - cf. /elemental/.
  2. What these ways have in common.
    1. The element of each thing is the first component immanent in each.
4. Nature
  1. The ways in which it is spoken of.
    1. The genesis, the cause of genesis (seed), and the source of the primary movement of growing things (mother:baby).
    2. (a) The primary material out of which an object is made (wood:bed), or (b) the essence of a natural object.
    3. By extension of (5.d.i.ii.b), every essence in general.
  2. What these ways have in common.
    1. The essence of things which have in themselves a source of movement.
5. Necessary
  1. The ways in which it is spoken of.
    1. (a) That without which a thing cannot live (breathing, food) and (b) that without which good cannot come (medicine).
    2. The compulsory and compulsion (doing one’s taxes).
  2. What these ways have in common.
    1. That which cannot be otherwise than it is.
6. One, Many
  1. The ways in which it is spoken of.
    1. One by accident (Coriscus is musical and just; musical:just)
    2. One by its own nature (continuous things: straight lines are more “one” than bent lines).
    3. One by virtue of homogenous substratum.
    4. One by virtue of participation in a genus ([horse, man, dog] qua “animal”).
    5. One by indistinguishability (Leibniz’s identity of indiscernables).
    6. Generally, one by continuity, form, or definition. (qua form: Circle is more “one” than straight line.)
    7. Beginning in number.
  2. What these ways have in common.
    1. Oneness by (a) number by essence, (b) species by definition, (c) genus by shared figures of predication, (d) analogy by relation to a third or fourth thing.
  3. “Many” will be the opposite of these.
7. Being
  1. The ways in which it is spoken of.
    1. Accidentally. By unessential attributes: musical, white.
    1. Synthetic predication: Both belong to the same thing, and this is.
    2. The subject of which the attribute is predicated is.
    3. The attribute which is predicated on a subject is.
  2. Essentially: By their own nature.
    1. By the categories (inc. analytic/tautological predication).
  3. A statment that is true. (”Socrates is musical.”)
  4. That which is potentially and actually (the half line is in the line, we still call the first corn sprouts corn).
8. Substance
  1. The ways in which it is spoken of.
    1. The simple bodies (earth, fire, water, etc); everything else is predicated on them.
    2. That which, being a subject’s unpredicated attribute (e.g. an animal’s soul) and its cause.
    3. The enabling condition of an individual, the loss of which would entail the loss of the individual (e.g. plane to line).
    4. The essence or definition of a thing.
  2. What these ways have in common.
    1. The ultimate substratum.
    2. The separable nature of the shape or form of each thing.
9. The same, Other, Diffferent, Like, Unlike
  1. The ways in which it is spoken of.
    1. The same as accidental (coincidence of attributes in an individual). Aka. the musical man is the same as the musical.
    2. The same by their nature: Sameness as a unity of treating many as one (these Warhols) or one as many (my “self” qua mind/body dualism).
    3. Different: (a) things which though other are the same in some respect, (b) those whose genus is other, to contraries, etc.
    4. “Like” things have the same attributes in every respect, or many same attributes, those whose quality is one, sharing in the most salient attribute(s).
10. Opposite, Contrary, Other in species, The same in species
  1. The ways in which it is spoken of.
    1. Contradictories and contraries, relative terms, qua privation and possession.
    2. Contrary:
      1. Attributes differening in genus that can’t belong at the same time to the same subject
      2. The most different of things in the same genus
      3. The most different of attributes/things in the same subject/faculty
      4. The most different absolutely or in genus or in species.
      5. Things which being in the same genus have a difference (”man and horse” via animals).
11. Prior, Posterior
  1. The ways in which it is spoken of.
    1. Empirically
      1. Prior: Some things because they are nearer to some (absolute, natural or referential) beginning.
      2. Posterior, because they are farther.
      3. Beginnings can be spatial, temporal, qua movement (boy:man) or power, or in arrangement.
    2. Qua Knowledge: Prior in definition (e.g. universals).
    3. The attributes of prior things are prior: Straightness (attribute of line) is prior to smoothness (attribute of plane).
    4. Metaphysically: E.g. for Plato, this Forms would have been prior to concrete particulars.
  2. What these ways have in common.
    1. Generally, prior things can exist without posterior things but not vice versa. (And again, in some ways, the same things may alternately occupy prior and posterior positions in relation to each other.)
12. Potency, Capable, Incapacity, Possible, Impossible
  1. The ways in which it is spoken of.
    1. The (a) intrinsic or extrinsic source of movement or change, (b) the condition of possibility for (a) - its capability, (c) the capability of performing /well/ [white men can’t jump], (d) the states in virtue of which something is unchangeable.
    2. Capacity is the intrinsic ability of something for (a)-(d) above, and incapacity is its opposite.
    3. The possible is that which is not of necessity false, and the impossible is that which is (of necessity false).
13. Quantum
  1. The ways in which it is spoken of.
    1. That which is divisible into two or more constituent parts, each of which is “one” and “this”.
    2. This can be either a plurality if it numerable, or a magnitude if it is measurable.
      1. Plurality: divisible into non-continuous parts.
      2. Magnitude: divisible into continuous parts.
    3. This can, like most of these things, be essential or accidental (that to which musicality and whiteness belong is a quantum).
14. Quality
  1. The ways in which it is spoken of.
    1. The differentia of essence (e.g. man is an animal of a certain quality).
    2. In mathematics: E.g. factors of a number. (”6″ is the quality of six, where “2×3″ and “3+3″ are some of its quantitative attributes.
    3. All the modifications of substances that move (E.g. hot or cold, white or black) which, when changed, alter the substance.
  2. What these ways have in common.
    1. Properly: The differentia of the essence.
    2. Vulgarly: The modifications of things that move.
15. Relative
  1. The ways in which it is spoken of.
    1. Reciprocal relations to a common term. E.g. As 1/2:2 or 1/3:3 qua 1.
      1. This can be definite (qua above) or indefinite (e.g. “many times n”).
      2. All these relations refer to unity/likeness/sameness.
    2. Active to passive. E.g. As that which can heat to that which can be heated.
    3. “…as the measurable to the measure, and the knowable to knowledge, and the perceptible to perception.
  2. What these ways have in common.
    1. (5.p.i) & (5.p.ii): Something’s very essence (thing that heats, half of one) includes a reference to something else.
    2. (5.p.iii): Something else’s essence (an inch) includes a reference to it (measurability, knowability).
  3. Finally, this all works by extension: Medicine is relative because its genus, science, is.
    1. And this extension can be accidental (white is relative if the same thing happents to be double [a relative term] relative term) or essential (equality).
16. Complete
  1. The ways in which it is spoken of.
    1. That outside of which it is impossible to find any of its parts.
    2. That which cannot be improved upon.
    3. Things which have attained their end.
17. Limit
  1. The ways in which it is spoken of.
    1. The last point of each thing; the first point beyond which it is not possible to find any part, and the first point within which every part is.
    2. The form of a special magnitude (e.g. in a thing that has magnitude).
    3. The end, substance, or essence of each thing (the limit of an object is equivalent to the limit of its knowability).
18. That in virtue of which, In virtue of itself
  1. “In virtue of which…”
    1. The form or substance of each thing (a man is good in virtue of the good itself).
    2. The proximate subject in which it is the nature of an attribute to be found (color in a surface).
    3. For what end? (”In virtue of what has he come?”)
    4. What is the cause? (”In virtue of what has he wrongly inferred…”)
    5. In reference to position (e.g. ‘at which he stands’ or ‘along which he walks’)
  2. “In virtue of itself”
    1. The essence of each thing (”Callias is in virtue of himself Callias”).
    2. Whatever is present in the thing (”Callias is in virtue of himself an animal”).
    3. Whatever attribute a thing receives in itself directly or in one of its part (”The surface is white in virtue of itself”).
    4. That which has no cause other than itself (”Man is man in virtue of himself”).
    5. Whatever attributes belong to a thing alone.
19. Disposition
  1. The ways in which it is spoken of.
    1. The arrangement of that which has parts, in respect of either place or potency or kind.
20. Having or habit
  1. The ways in which it is spoken of.
    1. “Having”: The relationship between the haver and the had. Evidently, we cannot have this having, in virtue of an infinite regress.
    2. A disposition of one who is (well or ill) disposed (e.g. “a health habit”).
    3. A portion of such disposition.
21. Affection
  1. The ways in which it is spoken of.
    1. A quality in respect of which a thing can be altered (white, sweet, etc.)
    2. The one of these alterations actually accomplished.
    3. Especially, injurious alterations actually accomplished.
    4. And hence, misfortunes in general.
22. Privation
  1. The ways in which it is spoken of.
    1. Something none of the attributes a thing might naturally have (even if the thing would not naturally have it). “The plant is deprived of eyes.”
    2. A special case of (5.y.i) where the thing would naturally have it (a blind man vs. a mole).
    3. The violent taking away of something.
23. Have or hold, Be in
  1. Have or hold, to be in something
    1. To treat a thing according to one’s own nature (”fever has him”).
    2. When something is present in something receptive of it (”the bronze has the form of a statue”, “he has a disease”).
    3. The container of something (”the casks hold the wine”).
    4. Something that hinders something else from moving or acting (”the pillars hold up the roof”).
24. From
  1. The ways in which it is spoken of.
    1. As from matter: (a) from the higest genus (all meltable things come from water), (b) a statue comes from bronze.
    2. As from the first moving principle (fight from abusive language).
    3. As a part from a whole (a verse from the Illiad), or the whole from a part (words from letters).
    4. Something comes from something insofar as it comes from a part of it (”Plants come from the earth”).
    5. Following in time (night comes from day).
25. Part
  1. The ways in which it is spoken of.
    1. That into which a quantum can in any way be divided.
    2. Of the parts in (5.bb.i.i) only those which measure a whole.
    3. The elements into which a kind might be divided apart besides quantity (species are parts of genus).
    4. The elements into which a whole is divided (as “bronze cube” and “statue” are to “bronze”).
    5. The elements in a definition (the genus now as part of the species).
26. Whole, Total, All
  1. The ways in which it is spoken of.
    1. That from which is absent none of its natural parts.
    2. That which contains things and these things form a unity:
      1. The whole of living things includes [man, horse, god].
      2. Something is whole by nature (a tree, I suppose).
    3. Quanta to which position makes a difference are wholes (say, a person), those to which it does not are totals (water). See (dd) below.
27. Mutilated
  1. Totals (qua [5.cc.i.iii] above) cannot be mutilated. You can’t mutilate water, or six, or fire.
  2. Wholes, on the other hand, can. You can chop off a man’s arm, and he ceases to be “whole”.
28. Race or genus, Other in genus
  1. The ways in which it is spoken of.
    1. A continuous generation of formally similar things (the race of men).
    2. The thing which first brought things into existence (Hellenes come from Hellen, they are her race).
    3. The extensional concept (e.g. ‘plane’ to all planar figures).
    4. The substratum of the qualities; the part of the definition whose differentiae gives the qualities of a participating particular.
29. False
  1. A false thing:
    1. Cannot be put together (is non-existent).
      1. Always: “the diagonal of a square is = its side”
      2. Sometimes: “I am done taking notes on Aristotle”
    2. Representations (a sketch, a dream) that are not the things the appearance of which they produce in us.
  2. A false account:
    1. “A false account is not an account of anything, except in a qualified sense.”
    2. A true account attains to the essence or accidental qualities of a thing. A false account is the opposite of this.
  3. A false man:
    1. A person who is fond of false accounts (5.ff.ii) for their own sake.
30. Accident
  1. The ways in which it is spoken of.
    1. That which can be asserted of something (S) but is neither necessary nor usually part of S.
      1. Accidents thus have indefinite or chance causes.
    2. All that attaches to each thing in virtue of itself but is not its essence.
      1. E.g. that a triangles’ add up to 180 degrees. (Spurious, I know, but that’s what he says.)
      2. Hence this type of accident may be eternal, but no accident of the other sort can ever be.
BOOK VI/EPSILON
1. Distinction of ‘theology’, the science of being as such, from the other theoretical sciences, mathematics and physics.
  1. Most sciences (wissenshaft) “bracket” the question of being in total, and deal with on particular aspect of being.
    1. The natural/physical sciences are “theoretical” sciences, and focused on one particular sort of being. Namely, the concrete stuff of nature/the world, or stuff as it is embedded in concrete particulars.
    2. Mathematics is also theoretical, but whence its objects are to be categorized (qua existence/being) is still unclear.
      1. That said, it is clear that some mathematical theorems consider things eternal/unmoveable/etc., but A wonders if that shouldn’t more properly be the domain of another science.
      2. Some part of maths also deals with things that are eternal but which are nonetheless embedded in concrete particulars (he appeals to the movements of the planets).
    3. The third theoretical philosophy is theology. This will be the most important of the three, assuming that there is some kind of immovable substance. If there is, “theology” will be first philosophy.
2. Four senses for ‘being’. Of these (i) accidental being is the object of no science.
  1. Recapitulating, Being is (a) accidental, (b) true (’non-being’ being the false), (c) figures of predication, (d) potential or actual existence.
  2. Accidental being (6.b.i.a, 5.g.i.i) cannot be treated scientifically.
    1. Accidents are those things which are not always or for the most part so.
    2. Science is either of that which is always or is for the most part. How else would it be learnable/teachable?
3. The nature and origin of accident.
  1. Accidental things exist, for otherwise everything that is will be necessary (aka. it would be necessary that Socrates is musical). I think this is basically done by the law of the excluded middle.
  2. I think that what happens here is that he says that in one sense (contra 6.c.i) everything will happen of necessity, because of the causal deterministic nature of the world. So accidents have first causes too, although determining these causes is a sticky wicket.
4. (ii) Being as truth is not primary being.
  1. Being as truth and falsity (6.b.i.b, 5.g.i.iii) is not the subject of philosophy either. Truth and falsity are determined in thought, and thus being in this sense is a distinctly second-stage type of being.
  2. For the record, though these notes stop here, we are now prepared to consider “being qua being”, obviously either in terms of (6.b.i.c) or (6.b.i.d).

Written by Paul Tulipana, and posted to Sorglose Nacht on September 15, 2008

Tags:

Aristotle, Becoming, Being, Categories, Causality, Lexicon, Logic, Matter, Motion, Outlines, Philosophy, Physics

Propositional Calculus for Computer Programmers: Background and First Part

This post is the inaugural element in what I plan to be an ongoing project into attempting to develop a framework for translating philosophical propositional calculus (PC) into terms that computer programmers can easily understand.

Personally, I find the confluence of factors that motivate this project incredibly interesting; but of course that’s because they’re largely about my life. The next section will talk about some of my motivations, as well as some of the sources I plan to draw on for this project. If you want to skip this section, you can click here.

Background

I told you before that my philosophical background is in the so-called continental tradition. What does this mean? Well, one thing that it means is that in my previous philosophical career, I never had to understand the sentential calculus. It’s just not a form of arugmentation that people in that tradition are prone to use.
At the same time, I told you that I was applying for PhD studies in American philosophy programs. Most of these programs are couched in the analytic tradition, which is a tradition that is deeply enamored of propositional logic as an argumentation tool. American analytic philosophy in the Russelian tradition - at least in some (admittedly much rarer now) cases - actually situates logic as first philosophy. This is quite a change for a person who comes very distinctly from a tradition that has historically situated everything but logic (i.e. metaphysics, epistemology, ethics) in the big chair.
Luckily, in addition to my philosophical interest, I have also been a computer programmer for ten years. This means that I have spent a lot of time negotiating symbolic logic, and as such I imagine or — at least hope — that there’s got to be a strategy to map these things together.

Pedagogy

When I was eighteen, I took a class called Formal Foundations of Computer Science, which was a symbolic logic class. I did rather poorly at this, which should confuse you, given the aforementioned. It confused the heck out of me for just long enough to be disastrous. Only recently, spurred by my partner Nick’s research project in programming pedagogy, did I understand this: the pedagogical model for teaching formal logic to computer programmers should be way more practical than it is (or at least was in my case).

Hence, the following attempt to make logic work. Throughout the following, I will be using the class notes and assignments most kindly provided by the Massachusetts Institute of Technology’s Open Courseware project. The syllabi will be provided courtesy of Prof. Vann McGee’s Logic I and Logic II classes in the MIT Linguistics and Philosophy department. The pedagogical materials will be provided courtesy of Delicious.com and Nicholas Senske of the University of Michigan.

Finally, expect this series to take some time to complete. Let’s get started.

Propositional Calculus for Computer Programmers: Part 1

Today’s course is easy. All we’re going to do is look at the basic syntax of the propositional calculus, and then the truth assignments for its basic operations. Really, nothing to it. I’ll be doing this using C-style syntax.

Syntax

PC symbol	English	C Operator or Fragment
∧	and	`&&`
∨	or (inclusive)	`\|\|`
¬	not	`!`
→	if…then (only if)	`onlyIf(a,b)`
↔	if and only iff	`ifOnlyIf(a,b)`

You’ll notice that the last two rows above are logical operations that don’t have C operators. Programming languages usually handle these operations by selection statements. For brevity’s sake, I’ll be using the ternary operation.

function onlyIf(a,b)   { a ? (b ? return true : return false) : return true; }
function ifOnlyIf(a,b) { a ? (b ? return true : return false) : (b ? return false : return true); }

Now, you may remember from way back in your beginning CS career drawing truth tables. Well, that’s what we’re going to be working on now. A final note on my conventions is that you should just assume, from now on that 1 == true and 0 == false.

Truth Assignments for simple PC operations

a	b	(a ∧ b)	(a ¬ b)	(a → b)	(a ↔ b)	¬a
1	1	1	1	1	1	0
1	0	0	1	0	0	0
0	1	0	1	1	0	1
0	0	0	0	1	1	1

One last thing to note that’s really convenient for programming types is that when you draw truth tables, an easy algorithm for mapping out your variable columns (in the above case, the first two columns: a, b), you just start from the bottom and count in binary (above: 00,01,10,11). This holds true for n variables.

So the plan is that in the next one of these, we’ll look at the axioms for the PC. I’m hoping that we can figure out ways to make this stuff feel more intuitive for the pragmatic programmer.

Written by Paul Tulipana, and posted to Sorglose Nacht on August 30, 2008

Tags:

Computer Programming, Continental Analytic Divide, Logic, Propositional Calculus, Syntax, Truth Tables

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

Posts tagged Logic.

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

Outline: Book Two

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

Metaphysics I-VI

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

BOOK I/BIG ALPHA

BOOK II/LITTLE ALPHA

BOOK III/BETA

BOOK IV/LAMBDA

BOOK V/DELTA: PHILOSOPHICAL LEXICON

BOOK VI/EPSILON

Propositional Calculus for Computer Programmers: Background and First Part

Background

Pedagogy

Propositional Calculus for Computer Programmers: Part 1

Syntax

Truth Assignments for simple PC operations