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Venn diagrams for modal logic 5

2012-04-18T18:15:00.001+03:00

Following are patterns for all universal syllogisms (i.e. syllogisms with universal conclusion). By changing letters for subject, middle, and predicate terms we can generate from these diagrams universal syllogisms in every figure, and by changing modality we can generate universal syllogisms in whatever combination of de dicto modalities, provided modalities apply simultaneously, conclusion being always in the weakest modality present in premises:

[-C+B]+[-B+A]=[-C+A]
[-C+B]+[+A-B]=[-C+A]
[+B-C]+[-B+A]=[-C+A]
[+B-C]+[+A-B]=[-C+A]
[+A-B]+[+B-C]=[+A-C]
[+A-B]+[-C+B]=[+A-C]
[-B+A]+[+B-C]=[+A-C]
[-B+A]+[-C+B]=[+A-C]

[-C+B]+[-B-A]=[-C-A]
[-C+B]+[-A-B]=[-C-A]
[+B-C]+[-B-A]=[-C-A]
[+B-C]+[-A-B]=[-C-A]
[-A-B]+[+B-C]=[-A-C]
[-A-B]+[-C+B]=[-A-C]
[-B-A]+[+B-C]=[-A-C]
[-B-A]+[-C+B]=[-A-C]

[+C+B]+[-B+A]=[+C+A]
[+C+B]+[+A-B]=[+C+A]
[+B+C]+[-B+A]=[+C+A]
[+B+C]+[+A-B]=[+C+A]
[+A-B]+[+B+C]=[+A+C]
[+A-B]+[+C+B]=[+A+C]
[-B+A]+[+B+C]=[+A+C]
[-B+A]+[+C+B]=[+A+C]

[+C+B]+[-B-A]=[+C-A]
[+C+B]+[-A-B]=[+C-A]
[+B+C]+[-B-A]=[+C-A]
[+B+C]+[-A-B]=[+C-A]
[-A-B]+[+B+C]=[-A+C]
[-A-B]+[+C+B]=[-A+C]
[-B-A]+[+B+C]=[-A+C]
[-B-A]+[+C+B]=[-A+C]

Some examples of syllogisms in different modalities:

[[-C+B]+[-B+A]]=[[-C+A]]
[[-C+B]+[+A-B]]=[[-C+A]]
[[+B-C]+[-B+A]]=[[-C+A]]
[[+B-C]+[+A-B]]=[[-C+A]]
[[+A-B]+[+B-C]]=[[+A-C]]
[[+A-B]+[-C+B]]=[[+A-C]]
[[-B+A]+[+B-C]]=[[+A-C]]
[[-B+A]+[-C+B]]=[[+A-C]]

([-C+B]+[-B-A])=([-C-A])
([-C+B]+[-A-B])=([-C-A])
([+B-C]+[-B-A])=([-C-A])
([+B-C]+[-A-B])=([-C-A])
([-A-B]+[+B-C])=([-A-C])
([-A-B]+[-C+B])=([-A-C])
([-B-A]+[+B-C])=([-A-C])
([-B-A]+[-C+B])=([-A-C])

[+C+B]+[[-B+A]]=[+C+A]
[+C+B]+[[+A-B]]=[+C+A]
[+B+C]+[[-B+A]]=[+C+A]
[+B+C]+[[+A-B]]=[+C+A]
[[+A-B]]+[+B+C]=[+A+C]
[[+A-B]]+[+C+B]=[+A+C]
[[-B+A]]+[+B+C]=[+A+C]
[[-B+A]]+[+C+B]=[+A+C]

([+C+B])+[[-B-A]]=([+C-A])
([+C+B])+[[-A-B]]=([+C-A])
([+B+C])+[[-B-A]]=([+C-A])
([+B+C])+[[-A-B]]=([+C-A])
[[-A-B]]+([+B+C])=([-A+C])
[[-A-B]]+([+C+B])=([-A+C])
[[-B-A]]+([+B+C])=([-A+C])
[[-B-A]]+([+C+B])=([-A+C])

Problematic modalities are funny, because, as said, they don't have to be coherent. Thus diagram 6 is valid because both premises are taken to be possibly true simultaneously. Were they taken to be possibly true separately (diagram 9; I use different shades of red to stress separated possibilities), then the conclusion does not follow necessarily, even if it remains a possibility. I mean: if it is possible that all C is B, and it is possible that no B is A, then it does not necessarily follow, that it is possible, that no C is A, but as it is not definitly impossible, we may count it as a possibility. But then the structure of deduction has changed from "what does necessarily follow" to "what is not necessarily excluded". If one possibility is in the range of the other (i.e. given one possibility, the other would be possible also; so called accessibilty-relation), then the deduction is valid necessarily.

Venn diagrams for modal logic 4

2012-04-11T11:07:00.001+03:00

In section IX the Logician states that "It sometimes happens that we get an apodeictic syllogism even when only one of the premisses -- not either of the two indifferently, but the major premiss -- is apodeictic : e.g. if A has been taken as necessarily applying or not applying to B, and B as simply applying to C. If the premisses are taken in this way A will necessarily apply (or not apply) to C. For since A necessarily applies (or does not apply) to all B, and C is some B, obviously A must also apply (or not apply) to C." This assertion is often taken as false. As writes Hugh Tredennick, commenting the text we are looking at, "The argument is fallacious [...] The relation of A to C cannot be apodeictic unless B is necessarily 'some C'."

But why does Aristotle see this as 'obvious'? St. Albertus Magnus offers an explanation: "Dico autem quod minor debet esse de inesse simpliciter et non de inesse ut nunc. Et voco de inesse simpliciter idem quod substantialiter inesse, quod secundum rem quidem est necessarium, quamvis non sit modo necessitatis determinatum. Et tunc quidem majori existente de necessario, et minori de tali inesse, sequitur conclusio de necessario: et si sit de inesse ut nunc, non sequitur conclusio de necessario." (I say that minor must 'apply simpliciter' and not 'as now'. And I call 'apply simpliciter' same as 'apply substantially', which according to thing is necessary, while not being in mode of determined necessity. And thus when major is by necessity and minor applies in this manner, then conclusion follows necessarily; and if it applies as now, then conclusion will not follow necessarily.) So he distinguishes two aspects in assertoric statement, assertion about actuality (inesse ut nunc) and assertion about substantiality (inesse simpliciter), the latter one having de re necessity. That means he sees it as composition of two necessities, de dicto in major and de re in minor. But it seems de re in major would be enough.

Let's take some example in style of Aristotle, e.g. A="rational", B="man" and C="Socrates": All men are necessarily rational, Socrates is a man, therefore Socrates is necessarily rational (see diagram 1; here I experiment with a notation for singulars -- being singular, it should be indivisible; and circle with green line marks that which is necessarily A, i.e. de re modality). Critics claim that the argument is good only if Socrates is necessarily man. But if we take the major to be in de re modality (see Englebretsen), it is not needed. Anothr example, A="animal", B="man" and C="Greek": All men are necessarily animals, all Greeks are men, therefore all Greeks are necessarily animals (diagram 2).

Looking at the diagram 2 it is clear that all C-s that there might be fall under that which is necessarily A.

De re necessity follows even in case of accidental connection between B and C. E.g. A="rational", B="man", C="white" (dia. 3): All men are necessarily rational, some men are white, therefore some white things are necessarily rational. I.e. not that it is necessary that some white things are rational (here is de dicto modality), but some white things are such, that they are necessarily rational (for they have human nature).

Aristotle also states that, were the modalities distributed differently, i.e. major assertoric and minor apodeictic, conclusion would not be apodeictic. This is confirmed by diagram 4. His example is A="motion", B="animal", C="man": All men are necessarily animals, all animals move, but not necessarily, therefore all men move, but not necessarily.

Actually, the major on this diagram is "all that are necessarily animals, move". And if we take the minor to be in de re modality, then it should indeed be so, otherwise there would be four terms. But the conclusion would be assertoric also in case we take minor to be in de dicto modality, with middel term simply "animals" as on diagram 5.

It remains to agree with Englebretsen that Aristotle is mixing here de re and de dico modalities.

[Apr 12] PS. Diagram 4 could be presented with gradual modalities of the middle term:

Venn diagrams for modal logic 3

2012-04-11T10:23:00.001+03:00

Aristotle continues with assertion that for syllogisms with apodeictic (i.e. necessary) premises "the conditions are, roughly speaking, the same as when they are assertoric." He gives two reasons for this: "For the negative premiss converts in the same way, and we will give the same explanation of the expression 'to be wholly contained in' or 'to be predicated of all'." If we consider (as it seems Aristotle does) apodeictic statements to be a subset of actually true statements (i.e. statements that are true when propositional modality is removed), then it is indeed so, for apodeictic statements have to be coherent just as assertoric statements. Only they have to be coherent "globally" (i.e. in every possible state) while assertoric statements have to be coherent "locally" (i.e. in actuality). And it is the assumption of coherence that gives force to logic, or rather, that makes logic. But problematic statements (i.e. possibility-statements) need not to be directly coherent -- different states of affairs are possible which would exclude each other if they would be actual. And different possibilities have their different sets of relative necessities (and generally, their "coherence sets"), which include but need not to be exhausted by what is absolutely necessary. And as such, relative necessities of different potentialities need not to be directly coherent. By "directly coherent" I mean "possibly true all at once".

Next four diagrams are for apodeictic syllogisms in traditional first figure. Let's add exclamation mark to indicate their apodeictic character.

Necessarily, all S is M
Necessarily, all M is P
Necessarily, all S is P

Necessarily, some S is M
Necessarily, all M is P
Necessarily, some S is P

Necessarily, all S is M
Necessarily, no M is P
Necessarily, no S is P

Necessarily, some S is M
Necessarily, no M is P
Necessarily, some S is not P

These might have been presented also with black-and-white patterns but with green frameline.

Venn diagrams for modal logic 2

2012-04-04T09:36:00.000+03:00

Let's look further into representation of modalities in Venn diagrams, while returning to sources -- to treatment of modalities in Prior Analytics. Aristotle first introduces modality while defining syllogism: "A syllogism is a form of words in which, if certain assumptions are made, something other than what has been assumed necessarily follows from the fact that the assumptions are such." This may be called relative necessity, i.e. what is necessary given certain assumptions. Absolute necessity instead is something that is true without any prior assumptions, e.g. tautologies.
In last post I proposed to represent modalities of statements by colored frames, and it is OK if we have single statement or all the statements represented inside the frame share this modality. But if we have statements with different modalities, then we need either to reintroduce concentric generic modalities or find some other way to represent the modalities of connection between terms.

One alternative way to represent modalities of connections between terms is to use colored connection-markers. Connection-markers are crosses, lines, shaded areas etc. which mark presence or absence of instances of terms in relation to some other term. Following diagrams represent contradictory statements "Some A may be not B" (1.) and "Every A must be (/is necessarily) B" (2.).

In this way we emphasize the modality of copula. These diagrams are logically equivalent to the style of representation of modalities introduced in last post, respectively "Possibly, some A is not B" (3.), and "Necessarily, every A is B" (4.), where emphasis is on the modality added to the statement:

If we apply diagram 2 in context of propositional logic, then we get the form of proof. Let A and B represent (possibly multiple) formulas; then the diagram may be read "Given A, B follows necessarily" or simply "A proves B". And abstracting from A, we may represent B that is proved as:

Let this be the representation for "relatively necessary". It is different from both "Necessarily B" (6.) (absolute necessity) and from B with "gradual modalities" (7.):

Having introduced relative necessity into diagrams, we may continue with Prior Analytics. On page 203 Aristotle distinguishes modalities of premisses: "Now every premiss is of the form that some attribute applies, or necessarily applies, or may possibly apply to some subject. These three types are divided into affirmative and negative in accordance with each mode of attribution; and again of affirmative and negative premisses some are universal, others particular and others indefinite." I believe I have covered these distinctions already (except indefinite terms). Follows discussion of convertibility of terms, and here I have to differ from some of his comments, because of some forms of statements missing from his analysis. But I will not delve into it now.

On page 207 he introduces different meanings of the problematic modality (i.e. "possible"): "... the term "possible" is used in several senses (for we call possible both that which is necessary and that which is not necessary and that which is capable of being)..." It seems he lists these meaning as exclusive of each other, and this corresponds perfectly to different concentric strata introduced in last post. I'll repeat it here, stressing different strata:

Green marks "that which is necessary", grey "that which is not necessary" [but is true], and light-red "that which is capable of being" [but is not true]. (That he means them exclusively seems clear from examples that follow in the text. It is evident also from text starting on page 237, where he specifically begins to treat modal syllogisms: "... 'to apply' is not the same as 'necessarily to apply' or 'possibly to apply' (because there are many predicates which apply, but not necessarily; and others neither apply necessarily nor indeed apply at all, but it is possible that they should apply)...".) In the next paragraph he states what he means by "possible" in his analysis of modal syllogisms: "... in such premisses as are said to be possible in the sense that they are generally or naturally true (for we define the possible in this way)..." Here he introduces non-formal meaning of "possible", defining it by being "generally or naturally true", but not necessary. The closest formal meaning seems to be "strong probability short of necessity". As a "diagrammatic joke" different probabilities could be represented on diagram by continuous line that spirals from "necessary" to "impossible" in one 2π turn (9.). More practical would be just to use light-green circle for representing "that which occurs generally or naturally" (10.):

(Area x should be grey on diagram 9, and it shows it grey in composition mode, but for some reason Firefox doesn't show it as such in presentation mode. Or maybe it is problem of Blogspot?)

OK, so far, so good.

POSTSCRIPTUM 22.04.12:
"What is generally or naturally true" is different both from 'possible' in formal meaning and from 'true' and of course from 'contingent', the latter being what is possible but not necessary. To indicate the difference we could present "true" and "general or natural" separately as on the following diagram:

'True' as previously is the grey area, 'natural' the light-green area, bright-green is 'necessary' and light-red is 'possible', bigger circles implied to include smaller ones. 'Contingent' is then the area between bright-green and white.

For Aristotle, who in Prior Analytics identifies 'possible' formally taken with 'contingent', 'generally true or natural' excludes 'necessary'.

Venn diagrams for modal logic

2012-04-01T23:01:00.000+03:00

Having introduced Venn diagrams for opposite terms it is easy to take a step further and apply this technique to modal terms. Here are modalities of term A:

And, of course, we can apply this to modal propositions as well.

If we drop term letters so that only modalities remain, then we get a general schema for modalities. To complete the general case we have to add a line for actuality:

On diagram above we analyzed term A only, so it fills the entire area with its different modalities. From this we can proceed in two ways. First, we can add other terms with their modalities (plus adding actuality-line). Let's also mark different modality-lines with different colors:

But this is impractical. It will make the diagram too cluttered. E.g. to mark "Some possible A is possible B" we would have to draw a line through all the regions of A's and B's overlapping between red lines. It would be much easier and most clear to draw just one colored line to mark the intended modality of the term. This is then the diagram for "Some possible A is possible B":

In this way we can combine different modalities, with additional benefit that diagram in black lines will be interpreted just as it traditionally should be interpreted.

Second, we can separate modalities from specific terms. Let's interpret this so that modalities in this way separated apply to propositions about terms, not to terms themselves (modalities of terms will be indicated as above, i.e. by color of their line). E.g. following diagram is for "It is possible that some A is not B":

But same considerations as in case of term-modalities (and some others) apply here also. We can simplify this diagram by removing concentric circles of generic modalities and applying intended propositional modality to the frame of the diagram:

There are many more interesting things we can do with color-coded modalities but it is enough for now.

Venn diagrams of opposing terms

2012-03-29T15:23:00.000+03:00

Opposite terms A and Ā can be represented on Venn diagrams as above. They are topologically similar to universally exclusive terms with addition of the necessity of their mutual exclusivity.

We can analyse several opposing terms as well:

1: (Ā+Ē)
2: (-A-Ā+Ē)
3: (-A-Ā-E-Ē)
4: (Ā-E-Ē)
5: (A+Ē)
6: (A-E-Ē)
7: (A+E)
8: (-A-Ā+E)
9: (Ā+E)
10=3

Relations in Venn diagrams

2012-03-21T12:46:00.000+02:00

I looked for Venn diagrams for relations but found none on the web. Probably with good reason, because of multiple quantification in relational sentences, which seems to not lend itself for representation in Venn diagrams. However, if we treat relational expressions similarily to complex terms, then it seems possible. For example: "Every A is r to every B" [-A+r-B] = [-A+[r-B]] is represented as:

In general case these diagrams are read from left to right and cannot be simply converted. Eg. the above diagram may not be read as "Every B is r to every A". So, they are directed or asymmetric diagrams (except for symmetric relations). But they can be read "from backwards", eg. the above diagram may be read as "To every B every A is r".

The above sentence is of course contradictory to the sentence "Some A is not r to every B" (A-[r-B]), which is same as "Some A is not-r to some B" (A-r+B):

This marks absence of relation r from some A to some B, not existence of some other, non-r relation between these. That's because we don't quantify over relations yet. Otherwise it would read "Some A is not only R to some B", meaning that there is some other relation besides R between some pair of A and B.

The above sentence is not equivalent to the sentence "Some A is not r to B", for here r is negated of the B as class, not of some B; explicitly "some A is not r to any B" (A-(r+B))=(A+[-r-B]), and it is represented as:

The line inside A represents existential quantifiation "There is some A", and its path through different regions represents disjunction: this A may be "r to some -B" (r-B), "-r to some -B" (-r-B), or "-r to some B" (-r+B). In all these cases it is "not r to any B" -(r+B).

And negation that there is such A is of course "Every A is r to B" [-A+(r+B)], where particular quantification of B is implied, for we have no negations. This is represented as:

The odd thing is that according to Venn diagram the last sentence "Every A is r to B" seems to be stronger than the first one "Every A is r to every B". But of course it is not. It declares absence of such A that is not r to some B (ie. that is non-r to every B), while first declares absence of such A that is not r to any B (ie. that is non-r to even one B). But still the dominant hatching causes some confusion.

{COMMENT 25.03.2012: Interpretation of hatched regions as disjunction of negations doesn't make sense either. If we read it as conjunction, we get much stronger sense than intended: instead of "Every A is r to some B" we get "Every A is only (r to some B)", ie. "No A is non-r to anything and every A is r only to B". Reading it as disjunction we get "Every A is r to every B, or only to B, or is r to every non-B", which doesn't make sense. To avoid reading of hatched areas as conjunction or disjunction of negations, it would be good to use different hatchings, just as we use lines instead of crosses in mixed particular sentences. For example use half-hatching in mixed cases and cross-hatching in simple cases.}

Here follows the catalogue of elementary relational sentences with Venn diagrams:

Universal sentences with contradictories:
1. No A is r to B [-A-r-B]	2. Some A is r to B (A+r+B)
3. Every A is r to every B [-A+r-B]	4. Some A is not r to every B (A-r+B)
5. A is r to only B [-A-r+B]	6. Some A is r to not only B (A+r-B)
7. Every A is non-r to only B [-A+r+B]	8. Some A is non-r to not only B (A-r-B)
9. Only A is r to B [+A-r-B]	10. Not only A is r to B (-A+r+B)
11. Only A is not r to every B [+A+r-B]	12. Not only A is not r to every B (-A-r+B)
13. Only A is not r to only B [+A-r+B]	14. Not only A is r to not only B (-A+r-B)
15. Only A is not non-r to only B [+A+r+B]	16. Not only A is non-r to not only B (-A-r-B)
Mixed universal sentences with contradictories
17. Every A is r to some B [-A+(r+B)]	18. Some A is not r to some B (A-(r+B))
19. No A is r to every B [-A-[r-B]]	20. Some A is r to every B (A+[r-B])
21. No A is non-r to only B [-A-[r+B]]	22. Some A is non-r to only B (A+[r+B])
23. No A is r to only B [-A-[-r+B]]	24. Some A is r to only B (A+[-r+B])
25. To every B some A is r [(A+r)-B]	26 To some B no A is r ([-A-r]+B)
27. To no B is only A r [-[A-r]-B]	28. To some B is only A r ([A-r]+B)
29. To no B is only A not r [-[A+r]-B]	30. To some B is only A not r ([A+r]+B)
31. To every B some A is not r [(A-r)-B]	32. To some B every A is r ([-A+r]+B)
33. To only B is only A r [-[A-r]+B]	34. To not only B is only A r ([A-r]-B)
35. To only B is only A not r [-[A+r]+B]	36. To not only B is only A not r ([A+r]-B)
37. To only B is no A r [-[-A-r]+B]	38. To not only B no A is r ([-A-r]-B)
39. To only B is every A r [-[-A+r]+B]	40. To not only B is every A r ([-A+r]-B)
41. Only A is r to every B [+A-[r-B]]	42. Not only A is r to every B (-A+[r-B])
43. Only A is not r to B [+A+(r+B)]	44. Not only A is not r to B (-A-(r+B))
45. Only A is non-r to only B [+A-[r+B]]	46. Not only A is non-r to only B (-A+[r+B])
47. Only A is r to only B [A-[-r+B]]	48. Not only A is r to only B (-A+[-r+B])

Aequipollentia

2012-03-16T17:42:00.000+02:00

Universal Forms

Model for the form "Every A is r to every B" is "Every _ is _ to every _", algebraically [-_+_-_].
We have 8 sentences with this form:
1. "Every A is r to every B" [-A+r-B]
2. "Every A is r to every -B" [-A+r-(-B)]
3. "Every A is -r to every B" [-A+(-r)-B]
4. "Every A is -r to every -B" [-A+(-r)-(-B)]
5. "Every -A is r to every B" [-(-A)+r-B]
6. "Every -A is r to every -B" [-(-A)+r-(-B)]
7. "Every -A is -r to every B" [-(-A)+(-r)-B]
8. "Every -A is -r to every -B" [-(-A)+(-r)-(-B)]

Model for the form "Every A is r to only B" is "Every _ is _ to only _", algebraically [-_-_+_].
8 senteces again, arranged according to equivalencies to former sentences:
1. "Every A is -r to only -B" [-A-(-r)+(-B)]
2. "Every A is -r to only B" [-A-(-r)+B]
3. "Every A is r to only -B" [-A-r+(-B)]
4. "Every A is r to only B" [-A-r+B]
5. "Every -A is -r to only -B" [-(-A)-(-r)+(-B)]
6. "Every -A is -r to only B" [-(-A)-(-r)+B]
7. "Every -A is r to only -B" [-(-A)-r+(-B)]
8. "Every -A is r to only B" [-(-A)-r+B]

Model for the form "No A is r to any B" is "No _ is _ to any _", algebraically [-_-_-_].
We have 8 sentences with this form:
1. "No A is -r to any B" [-A-(-r)-B]
2. "No A is -r to any -B" [-A-(-r)-(-B)]
3. "No A is r to any B" [-A-r-B]
4. "No A is r to any -B" [-A-r-(-B)]
5. "No -A is -r to any B" [-(-A)-(-r)-B]
6. "No -A is -r to any -B" [-(-A)-(-r)-(-B)]
7. "No -A is r to any B" [-(-A)-r-B]
8. "No -A is r to any -B" [-(-A)-r-(-B)]

Model for the form "Every A is only to B not r" is "Every _ is only to _ not _", algebraically [-_+_+_].
8 senteces again, arranged according to equivalencies to former sentences:
1. "Every A is only to -B not r" [-A+r+(-B)]
2. "Every A is only to B not r" [-A+r+B]
3. "Every A is only to -B not -r" [-A+(-r)+(-B)]
4. "Every A is only to B not -r" [-A+(-r)+B]
5. "Every -A is only to -B not r" [-(-A)+r+(-B)]
6. "Every -A is only to B not r" [-(-A)+r+B]
7. "Every -A is only to -B not -r" [-(-A)+(-r)+(-B)]
8. "Every -A is only to B not -r" [-(-A)+(-r)+B]

Clusive Forms

Model for the form "Only A is not r to every B" is "Only _ is not _ to every _", algebraically [+_+_-_].
8 sentences, same arrangement:
1. "Only -A is not r to every B" [+(-A)+r-B]
2. "Only -A is not r to every -B" [+(-A)+r-(-B)]
3. "Only -A is not -r to every B" [+(-A)+(-r)-B]
4. "Only -A is not -r to every -B" [+(-A)+(-r)-(-B)]
5. "Only A is not r to every B" [+A+r-B]
6. "Only A is not r to every -B" [+A+r-(-B)]
7. "Only A is not -r to every B" [+A+(-r)-B]
8. "Only A is not -r to every -B" [+A+(-r)-(-B)]

Model for the form "Only A is not r to only B" is "Only _ is not _ to only _", algebraically [+_-_+_].
8 sentences, same arrangement:
1. "Only -A is not -r to only -B" [+(-A)-(-r)+(-B)]
2. "Only -A is not -r to only B" [+(-A)-(-r)+B]
3. "Only -A is not r to only -B" [+(-A)-r+(-B)]
4. "Only -A is not r to only B" [+(-A)-r+B]
5. "Only A is not -r to only -B" [+A-(-r)+(-B)]
6. "Only A is not -r to only B" [+A-(-r)+B]
7. "Only A is not r to only -B" [+A-r+(-B)]
8. "Only A is not r to only B" [+A-r+B]

Model for the form "Only A is r to any B" is "Only _ is _ to any _", algebraically [+_-_-_]. ("To every B only A is r" "Only A is not to every B not r")
8 sentences, same arrangement:
1. "Only -A is -r to any B" [+(-A)-(-r)-B]
2. "Only -A is -r to any -B" [+(-A)-(-r)-(-B)]
3. "Only -A is r to any B" [+(-A)-r-B]
4. "Only -A is r to any -B" [+(-A)-r-(-B)]
5. "Only A is -r to any B" [+A-(-r)-B]
6. "Only A is -r to any -B" [+A-(-r)-(-B)]
7. "Only A is r to any B"; "To every B only A is r" [+A-r-B]
8. "Only A is r to any -B" [+A-r-(-B)]

Model for the form "Only A is non-r to not only B" is "Only _ is not only to _ not _", algebraically [+_+_+_]. ("Only to B is not only A not r" "Only A is not only to B not r")
8 sentences, same arrangement:
1. "Only -A is (not r) to not only -B"; "Only to -B is not only -A not r" [+(-A)+r+(-B)]
2. "Only -A is (not r) to not only B"; "Only to B is not only -A not r" [+(-A)+r+B]
3. "Only -A is (not -r) to not only -B"; "Only to -B is not only -A not -r" [+(-A)+(-r)+(-B)]
4. "Only -A is (not -r) to not only B"; "Only to B is not only -A not -r" [+(-A)+(-r)+B]
5. "Only A is (not r) to not only -B"; "Only to -B is not only A not r" [+A+r+(-B)]
6. "Only A is not only to B not r"; "Only to B is not only A not r" [+A+r+B]
7. "Only A is (not -r) to not only -B"; "Only to -B is not only A not -r" [+A+(-r)+(-B)]
8. "Only A is (not -r) to not only B"; "Only to B is not only A not -r" [+A+(-r)+B]

Comparison

1. "Every A is r to every B" [-A+r-B]
"Every A is -r to only -B" [-A-(-r)+(-B)]
"No A is -r to any B" [-A-(-r)-B]
"Every A is only to -B not r"; "" [-A+r+(-B)]
"Only -A is not r to every B" [+(-A)+r-B]
"Only -A is not -r to only -B" [+(-A)-(-r)+(-B)]
"Only -A is -r to any B" [+(-A)-(-r)-B]
"Only -A is (not r) to not only -B"; "Only -A is not to only -B not r" [+(-A)+r+(-B)]

2. "Every A is r to every -B" [-A+r-(-B)]
"Every A is -r to only B" [-A-(-r)+B]
"No A is -r to any -B" [-A-(-r)-(-B)]
"Every A is only to B not r"; "To only B any A is not r" [-A+r+B]
"Only -A is not r to every -B" [+(-A)+r-(-B)]
"Only -A is not -r to only B" [+(-A)-(-r)+B]
"Only -A is -r to any -B" [+(-A)-(-r)-(-B)]
"Only -A is (not r) to not only B"; "Only to B is not only -A not r" [+(-A)+r+B]

3. "Every A is -r to every B" [-A+(-r)-B]
"Every A is r to only -B" [-A-r+(-B)]

"No A is r to any/no B" [-A-r-B]

"Every A is only to -B not -r" [-A+(-r)+(-B)]
"Only -A is not -r to every B" [+(-A)+(-r)-B]
"Only -A is not r to only -B" [+(-A)-r+(-B)]
"Only -A is r to any B" [+(-A)-r-B]
"Only -A is (not -r) to not only -B"; "Only to -B is not only -A not -r" [+(-A)+(-r)+(-B)]

4. "Every A is -r to every -B" [-A+(-r)-(-B)]

"Every A is r to only B" [-A-r+B]

"No A is r to any -B" [-A-r-(-B)]
"Every A is only to B not -r" [-A+(-r)+B]
"Only -A is not -r to every -B" [+(-A)+(-r)-(-B)]
"Only -A is not r to only B" [+(-A)-r+B]
"Only -A is r to any -B" [+(-A)-r-(-B)]
"Only -A is (not -r) to not only B"; "Only to B is not only -A not -r" [+(-A)+(-r)+B]

5. "Every -A is r to every B" [-(-A)+r-B]
"Every -A is -r to only -B" [-(-A)-(-r)+(-B)]
"No -A is -r to any B" [-(-A)-(-r)-B]
"Every -A is only to -B not r" [-(-A)+r+(-B)]

"Only A is not r to every B" [+A+r-B]

"Only A is not -r to only -B" [+A-(-r)+(-B)]
"Only A is -r to any B" [+A-(-r)-B]
"Only A is (not r) to not only -B"; "Only to -B is not only A not r" [+A+r+(-B)]

6. "Every -A is r to every -B" [-(-A)+r-(-B)]
"Every -A is -r to only B" [-(-A)-(-r)+B]
"No -A is -r to any -B" [-(-A)-(-r)-(-B)]
"Every -A is only to B not r" [-(-A)+r+B]
"Only A is not r to every -B" [+A+r-(-B)]
"Only A is not -r to only B" [+A-(-r)+B]
"Only A is -r to any -B" [+A-(-r)-(-B)]

"Only A is not only to B not r"; "Only to B is not only A not r" [+A+r+B]

7. "Every -A is -r to every B" [-(-A)+(-r)-B]
"Every -A is r to only -B" [-(-A)-r+(-B)]
"No -A is r to any B" [-(-A)-r-B]
"Every -A is only to -B not -r" [-(-A)+(-r)+(-B)]
"Only A is not -r to every B" [+A+(-r)-B]
"Only A is not r to only -B" [+A-r+(-B)]

"Only A is r to any B"; "To every B only A is r" [+A-r-B]

"Only A is (not -r) to not only -B"; "Only to -B is not only A not -r" [+A+(-r)+(-B)]

8. "Every -A is -r to every -B" [-(-A)+(-r)-(-B)]
"Every -A is r to only B" [-(-A)-r+B]
"No -A is r to any -B" [-(-A)-r-(-B)]
"Every -A is only to B not -r" [-(-A)+(-r)+B]
"Only A is not -r to every -B" [+A+(-r)-(-B)]
"Only A is not r to only B" [+A-r+B]
"Only A is r to any -B" [+A-r-(-B)]
"Only A is (not -r) to not only B"; "Only to B is not only A not -r" [+A+(-r)+B]

Algebra of relations

2012-03-14T19:12:00.001+02:00

Let a, b, c... be terms for individuals and r, s, t... be terms for specific relations. That means we shall not quantify over individuals nor over specific relations. Let A, B, C... be terms for classes of individuals and R, S, T... terms for classes of relations. We shall quantify over classes.

Simple relation is represented algebraically by
a+r+b , which is equivalent to (a+r+b)
and it may be read as "a is r to b" or "a is in relation r to b" or "a is r in relation to b".

Until it is clear that r is relation between a and b, indexes may be skipped, so
a₁+r₁₂+b₂
is implied.

This relation can be denied in several ways:

1. Relation r can be denied to exist between individuals a and b
a-r+b "a is not-r to b"
I.e. there is a, there is b, but there is not r between these.

2. It can be denied about individual a, that it is related by r to b
a-(r+b) "a is not r to b"
This is algebraically equivalent to a+[-r-b], i.e. "there is a, but either there is no b, or a is not r to it"="a is r to only -b". This is slightly more general than a-r+b

3. It can be denied about individual b, that a is r in relation to it
(-(a+r)+b) "to b, it is not a that is r in relation to it"

4. The whole sentence can be denied. This is algebraically equivalent to
-(a+r+b) "It is not the case that a is r to b" = [-a-r-b]

Relations between classes

Without quantified relations.
Contradictory pairs:

(A+r+B) "Some A is r to some B"            [-A-r-B] "Every A is -r to every B"
(A+r-B) "Some A is r to not only B"        [-A-r+B] "Every A is r to only B"
(A-r+B) "Some A is -r to some B"           [-A+r-B] "Every A is r to every B"
(A-r-B) "Some A is -r to not only B"       [-A+r+B] "Every A is -r to only B"
(-A+r+B) "Not only A is r to some B"       [A-r-B] "Only A is not -r to every B"
(-A+r-B) "Not only A is r to not only B"   [A-r+B] "Only A is not r to only B"
(-A-r+B) "Not only A is -r to some B"      [A+r-B] "Only A is not r to every B"
(-A-r-B) "Not only A is -r to not only B"  [A+r+B] "Only A is not -r to only B"

(A+[r+B]) "Some A is -r to only B"         [-A-[r+B]] "No A is -r to only B"
(A+[r-B]) "Some A is r to every B"         [-A-[r-B]] "No A is r to every B"
(A+[-r+B]) "Some A is r to only B"         [-A-[-r+B]] "No A is r to only B"
(A+[-r-B]) "Some A is -r to every B"       [-A-[-r-B]] "No A is -r to every B"
(-A+[r+B]) "Not only A is -r to only B"    [A-[r+B]] "Only A is -r to only B"
(-A+[r-B]) "Not only A is r to every B"    [A-[r-B]] "Only A is r to every B"
(-A+[-r+B]) "Not only A is r to only B"    [A-[-r+B]] "Only A is r to only B"
(-A+[-r-B]) "Not only A is -r to every B"  [A-[-r-B]] "Only A is -r to every B"

([A+r]+B) "To some B only A is -r"         [-[A+r]-B] "To no B only A is -r"
([A+r]-B) "To not only B only A is -r"     [-[A+r]+B] "To only B only A is -r"
([A-r]+B) "To some B only A is r"          [-[A-r]-B] "To no B only A is r"
([A-r]-B) "To not only B only A is r"      [-[A-r]+B] "To only B only A is r"
([-A+r]+B) "To some B every A is r"        [-[-A+r]-B] "To no B every A is r"
([-A+r]-B) "To not only B every A is r"    [-[-A+r]+B] "To only B every A is r"
([-A-r]+B) "To some B every A is -r"       [-[-A-r]-B] "To no B every A is -r"
([-A-r]-B) "To not only B every A is -r"   [-[-A-r]+B] "To only B every A is -r"

Exclusive modes

2012-03-14T15:04:00.000+02:00

It seems my exclusive modes have found honorable forefather in Petri Hispani. See p. 272ff in his "Parvorum logicalium":

Opposite Terms

2010-10-09T18:01:00.008+03:00

Let opposite of term A be Ā. Ie. Ā is most distant term from A in its kind (species?). White and black, good and bad, friend and enemy... Then, on condition there is distribution of terms in some dimension, there is square of opposition of terms:

 A   opposite   Ā
   com     ment
       ple
   com     ment
-Ā subopposite -A



 A |          -A
---|----------------|---
       -Ā           | Ā

It is not Greimas' semiotic square, which deals with oppositions of signs.

Back

2010-10-01T21:05:00.004+03:00

Hei-hoo! I'm back from "Sabbatical" when I renovated my new (old) appartment and blogged in Estonian at Tabernaakel. Now, back to singulars :).

First, I'll stick to the earlier conventions of + and - and drop some innovations, eg implicative context. Instead I'll introduce singulars using same symbols, i.e. angle brackets. E.g. speaking of simple terms we have [A] "Only A", (A) "Some A" and now we have also 〈A〉 "This A". Of complex terms we have [A+B] "Only A or B", [A-B] "Only A or non-B", [-A+B] "No A that is not B", [-A-B] "No A that is B", (A+B) "A that is B", (A-B) "A that is not B", (-A+B) "Non-A that is B" and (-A-B) "Non-A that is not B". Speaking of simple existence sentences we have [A] "Only A exists" (or "Everything is A"), (A) "Some A exists" (or "Something is A") and 〈A〉 "This A exists" (or "This is A").

In last message I stayed confused about treatment of singulars. Earlier I suggested that there is no problem with treating singulars like other terms, eg "Socrates is wise" as (S+W), even if this would be read as "Some Socrates is wise". This is awkward, even if we can benefit from being able to express "Only Socrates is wise" [S-W]. The awkwardness remains even if we introduce explicitly names by using quotes ('S'+W), reading it "Someone named 'Socrates' is wise". This is improvement, because we don't let name function directly as predicate any more, but it functions now indirectly the same way: "Someone of those named 'Socrates' is wise". And ['S'-W] would make explicit the flaw we might have not noticed in previous version: "Only someone of those named 'Socrates' is wise". It is not what is meant by "Socrates is wise", where we mean certain Socrates, namely the one we are speaking about (even if you don't know for sure which one I am speaking about). But now let's use specific context for this kind of reference "This Socrates is wise" (ie. the unique one we are talking about): 〈S+W〉. This context-marker has property of being auto-dual: -〈S+W〉=〈S-W〉 ie. "It is not the case, that (this) Socrates is wise"="Socrates is not wise".

It seems we are still able to refer to Socrates as the only wise one: [〈S〉-W].

First act of intellect

2009-01-10T23:53:00.003+02:00

Until now I have treated unemphasized contexts explicitly as applying to propositional calculus. Let's use lowercase letters for propositions and uppercase letters for terms. Thus [pq], (pq) and 〈pq〉 are interpreted accordingly as "p or q is true", "both p and q are true" and "if p is true, q is true". But unemphasized contexts appear also inside term logic categorical expressions, e.g. [-S+(P+Q)] -- All S is P and Q (or [‹S›(PQ)] or 〈S(PQ)〉). So far I haven't dealt explicitly with expressions like [SP], (SP) etc., except in case of set algebra, where uppercase letters were interpreted as sets. But can unemphasized contexts be meaningfully interpreted as outer contexts in term logic? Of course they can. As outer contexts they may be used to construct complex terms, not propositions as in case of emphasized contexts. E.g., given bindings M=man and D=dead, (MD) may be read as "man that is dead", or "dead man", without asserting propositionally that "man is dead". Latter is expressed by (MD). For an other example, consider e.g. 〈MW〉, where M=men and W=went to war, reading it as "all men that went to war", contrasted to 〈MW〉, read as "all men went to war".

In this way unemphasized contexts may be used just to select terms without asserting anything about them. As such they are just complex terms and do not have truth-value. Nevertheless, they have characteristics of consistency and inconsistency, e.g. (A‹A›) is inconsistent term while [A‹A›] is trivial term. This enables us to treat the intellect's first act explicitly.

(Hmm. What about single terms?)

Implicative context introduced

2009-01-06T14:24:00.007+02:00

Let's introduce implicative context by 〈 and 〉.

〈pp〉 = ○

Standard equivalencies:

〈pq〉	[‹p›q]	(‹p‹q››)
〈‹p›q〉	[pq]	(‹‹p›‹q››)
〈‹p‹q››〉	[‹‹p›‹q››]	(pq)
〈‹pq›〉	[‹p‹q››]	(p‹q›)
〈‹p‹q››〉	[‹pq›]	(‹p›‹q›)
〈p‹q›〉	[‹p›‹q›]	(‹pq›)
〈‹p›‹q›〉	[p‹q›]	(‹p‹q››)
〈‹‹p›‹q››〉	[‹p‹q››]	(‹p›q)

〈pqr〉 = 〈p〈qr〉〉 = 〈(pq)r〉
〈〈pq〉r〉 = ([pr]〈qr〉)
〈pq〉 = 〈‹q›‹p›〉
〈‹p›q〉 = 〈‹q›p〉
〈‹p›p〉 = p
〈p‹p›〉 = ‹p›

〈‹‹››pq〉 = 〈pq〉 = 〈‹pq›‹›〉 = p → q
〈‹‹››pp〉 = 〈pp〉= 〈‹pp›‹›〉 = ○
〈‹‹››p〉 = 〈p〉 = 〈‹p›‹›〉 ?
〈‹‹››‹›〉 = 〈‹›〉 = 〈‹‹››‹›〉 ?
〈‹‹››〉 = 〈〉 = 〈‹›‹›〉 ?

Peirce and Bricken accommodated

2009-01-03T10:40:00.008+02:00

(Corrected 4.01.09)

Both Peirce's alpha graphs and Bricken's boundary logic may be seamlessly accommodated into my logic algebra. For this I'll adopt following conventions into my notation:

1) Letters g and f signify any propositional formula (or graph).
2) Negation of a formula is expressed by enclosing this formula between ‹ and ›.
3) Appearance of a formula in odd depth of negation is marked by enclosing this formula between ≺ and ≻; in even depth (incl 0) -- between ≼ and ≽; in any depth (incl 0) -- between ⋘ and ⋙.
4) Contents of the negated formula are joined by conjunction or disjunction depending on immediate context of the negated formula, i.e. [‹g f›] = ‹[g f]› and (‹g f›) = ‹(g f)›.
5) When formula is true independently of the context, context markers may be dropped.
6) Default context (Blank sheet) is conjunctive (i.e. Peircean).

Now we can demonstrate equivalences of three systems:

Peirce α	Bricken BL	Tom
Basic formulas
g	g	g
‹g›	‹g›	‹g›
gf	‹‹g›‹f››	(gf) ⇔ [‹‹g›‹f››]
‹‹g›‹f››	gf	(‹‹g›‹f››) ⇔ [gf]
‹g‹f››	‹g›f	(‹g‹f››) ⇔ [‹g›f]
Transformations
⇔ ‹‹›‹›› ⇔ ‹‹›› ⇔ T	‹›‹› ⇔ ‹› ⇔ T	(‹‹›‹››) ⇔ (‹‹››) ⇔() ⇔ ○ ⇔ ⇔ [‹›] ⇔ [‹›‹›]
‹›‹› ⇔ ‹› ⇔ F	⇔ ‹‹›‹›› ⇔ ‹‹›› ⇔ F	[‹‹›‹››] ⇔ [‹‹››] ⇔ [] ⇔ □ ⇔ (‹›) ⇔ (‹›‹›)
‹›g ⇔ ‹›	‹‹›g› ⇔	(‹›g) ⇔ □ ⇔ [‹‹›g›]
‹‹›g› ⇔	‹›g ⇔ ‹›	(‹‹›g›) ⇔ ○ ⇔ [‹›g]
‹‹g›› ⇔ g	‹‹g›› ⇔ g	‹‹g›› ⇔ g
≼gf≽ ⇒ ≼g≽	≺gf≻ ⇒ ≺g≻	(≼gf≽) ⇒ (≼g≽) [≺gf≻] ⇒ [≺g≻]
≺g≻ ⇒ ≺gf≻	≼g≽ ⇒ ≼gf≽	(≺g≻) ⇒ (≺gf≻) [≼g≽] ⇒ [≼gf≽]
g⋘f⋙ ⇔ g⋘gf⋙	g⋘f⋙ ⇔ g⋘gf⋙	g⋘f⋙ ⇔ g⋘gf⋙

Bricken's boundary logic

2009-01-02T12:58:00.004+02:00

William Bricken's boundary logic (BL) (see also) is interesting dual to Peirce's existential graphs. In his system blank sheet represents FALSE (Peirce: TRUE) and composition of structures represents disjunction (Peirce: conjunction). Accordingly basic logical operations are represented as follows:

‹void› -- FALSE
() -- TRUE
(p) -- NOT p
p q -- p OR q
((p) (q)) -- p AND q
(p) q -- IF p THEN q

Basic transformation rules:

()() = () -- calling [T v T = T]
(()) = ‹void› -- crossing [~T = F]
(() p) = ‹void› -- occlusion [~(T v p) = F & ~p = F]
() p = () -- dominion [T v p = T]
((p)) = p -- involution (~(~p) = p)
p {q p} = p {q} -- pervasion (from any depth) [p v ~(... q v p...) = p v ~(...q...)]

As parentheses operate mostly as negation, they may be replaced by ‹›:
‹p› -- NOT p
‹p q› -- NOT (p OR q)
‹‹p›‹q›› -- NOT (NOT p OR NOT q) = p AND q
‹p› q -- NOT p OR q = IF p THEN q

Hence, it is similar to operation in my notation inside brackets, where negation applies to disjunction:
‹p q› -- [‹p q›] = [-[p+q]] = [(-p-q)]
‹‹p›‹q›› -- [‹‹p›‹q››] = [-[-p-q]] = [(+p+q)]
‹p› q -- [‹p› q] = [-p+q]

Compendium of Peirce's Existential Graphs

2009-01-01T18:05:00.021+02:00

Based on Zalamea. See also Zeman, Sowa and Dau.

p, q, f -- propositional letters
P, Q, F -- predicate letters

Alpha

Peirce's modified notation	My (modified) notation
pq -- conjunction	(pq)
‹p› -- negation	‹p›
‹p‹q›› -- implication	[‹p›q]
‹‹p›‹q›› -- disjunction	[pq]

A1 Erasure: Any evenly enclosed graph may be erased.
A2 Insertion: Any graph may be inserted in any oddly enclosed region.
A3 Iteration: Any graph may be iterated (i.e. repeated) in a strict region of that graph.
A4 Deiteration: Any graph whose occurrence could result from iteration may be deiterated (i.e. erased).
A5 Double cut: A double cut may be inserted or erased around any graph in any region.

A1 (ER): pq ⇒ p                                       (pq) ⇒ p
A2 (IN): ‹p› ⇒ ‹pq›                                 ‹p› ⇒ ‹pq›
A3, A4 (IT and DI): p‹q› ⇔ p‹pq›         (p‹q›) ⇔ (p‹pq›)
A5 (DC): p ⇔ ‹‹p››                                 p ⇔ ‹‹p››

Beta

LI -- Line of Identity

B1 Erasure: Any evenly enclosed portion of an LI may be erased.
B2 Insertion: Any portions of LI's may be joined in an oddly enclosed region.
B3 Continuous iteration: Any LI may be extended towards strict regions. Any LI may branch in its region.
B4 Continuous deiteration: Any LI may be retracted towards regions with lesser cuts.

B1: P---Q ⇒ P- -Q                       (PQ) ⇒ (P)(Q)
B2: ‹P- -Q› ⇒ ‹P---Q›            ‹(P)(Q)› ⇒ ‹(PQ)›
B3, B4: P--- ‹q› ⇔ P-‹- q›    (P)‹q› ⇔ (P‹q›)?

Some X is R to every Y ⇒ To every Y some X is R.
-‹-‹-R2-›-› ⇒ ‹‹-R2-›-›            (X1[‹Y2›R12]) ⇒ [‹Y2›(X1R12)]?

Gamma

{f} -- possibly not f                                          (‹f›)
‹{f}› -- not possibly not f = necessarily f     ‹(‹f›)› = [f]
{‹f›} -- possibly not not f = possibly f          (‹‹f››) = (f)

‹{f}› ⇒ ‹‹f›› ⇒ f ⇒ ‹‹f›› ⇒ {‹f›}

G1: In an even area, any Alpha cut may be half-erased to become a Gamma cut.
‹› ⇒ {}                                                                ‹› ⇒ (‹›)
G2: In an odd area, any Gamma cut may be half-completed to become an Alpha cut.
‹{}› ⇒ ‹‹››                                                      ‹(‹›)› ⇒ ‹‹››

Boundary mathematics and algebra of sets

2008-12-22T18:40:00.013+02:00

(Corrected 23.12.08)

Brandon brought to my attention an interesting comparison between boundary mathematics and algebra of logic as developed in this blog. However I didn't quite follow his equivalences between the two systems. Of course this might be caused by the fact that I met with the subject of boundary mathematics first time in his blog. Nevertheless, the subject provoked me to think about similarities and differences between the two.

Jeffrey James, in his thesis "A Calculus of Number Based on Spatial Forms", bases his system on "making distinctions out of the void". The basic element is a single boundary in the void -- unit, represented by o, or generally instance, represented by (X). The other boundaries are "black hole", represented by □, or generally abstract, represented by [X]; and inverse, represented by △ in empty case or generally by <X>. Only instance has actually a spatial interpretation as boundary. Boundaries can be collected and nested to form complex structures.

In following I will interpret [(X)] ≗ X as set X and formulate set algebra, taking lead from Wikipedia article. Translation into propositional formulas should be pretty straightforward by replacing set letters with propositional letters. I believe it may be regarded as extension of Jeffrey's notation, or as combination of his boundary mathematics with my algebra of logic. It's quite new to me, so there may be lot of things that need to be corrected.

Union of sets A and B: [AB]
Intersection of sets A and B: AB
Complement of set A (to universal set): <A>
Complement of set A relative to set B: <A>B
Difference of set A from B: A
Symmetric difference of sets A and B: [(<A>B)(A)]
Cartesian product of sets A and B: ([A][B])
Inclusion of set A in set B: [<A>B]

Commutativity: [AB] ≗ [BA]; AB ≗ BA
Associativity: [A[BC]] ≗ [[AB]C] ≗ [ABC]; A(BC) ≗ (AB)C ≗ ABC
Distributivity: [A(BC)] ≗ [AB][AC]; A[BC] ≗ [(AB)(AC)]

Null (or empty) set: □
Universal set: [o]?

Identity laws: [A□] ≗ A; A[o] ≗ A
Complement laws: [A<A>] ≗ [o]; A<A> ≗ □

Idempotent laws: [AA] ≗ A; AA ≗ A
Domination laws: [A[o]] ≗ [o]; A□ ≗ □
Absorption laws: [A(AB)] ≗ A; A[AB] ≗ A

Proof of idempotent laws:
[AA] ≗ [AA][o] ≗ [AA][A<A>] ≗ [A(A<A>)] ≗ [A□] ≗ A
AA ≗ [(AA)□] ≗ [(AA)(A<A>)] ≗ A[A<A>] ≗ A[o] ≗ A

DeMorgan's laws: <[AB]> ≗ <A>; <AB> ≗ [<A>]
Double complement or involution: <<A>> ≗ A
Complement laws of universal and empty sets: <[o]> ≗ □; <□> ≗ [o]

Uniqueness of complements: [<([AB]=[o] AB=□)> B=<A>] ≗ [<[AB]<(AB)>> B=<A>]

Reflexivity: [<A>A]
Antisymmetry: [<A>B][A] ≗ A=B
Transitivity: [<[<A>B][C]>[<A>C]]

If A, B and C are subsets of S:
Least element and greatest element: [<□>A]? and [<A>S]
Joins: [<A>[AB]]; [<[<A>C][C]>[<[AB]>C]]
Meets: [<AB>A]; [<[<C>A][<C>B]>[<C>(AB)]]

[<A>B] ≗ AB=A ≗ [AB]=B ≗ A=□ ≗ [<><A>]

Relative complements:
C<AB> ≗ [(C<A>)(C)]
C<[AB]> ≗ C<A>
C<B<A>> ≗ [(AC)(C)]
[(B<A>)]C ≗ [([(BC)]<A>)] ≗ [(B[(C<A>)])]
[(B<A>)C] ≗ [BC]<A<C>>
A<A> ≗ □
□<A> ≗ □
A<□> ≗ A
B<A> ≗ <A>B
<B<A>> ≗ [A]
[o]<A> ≗ <A>
A<[o]> ≗ □

De Morgan's types 3

2008-01-28T08:10:00.000+02:00

In a comment to Brandon's post about subalternation and existential import I said:

"difference between traditional and modern understanding of the relation between A and I propositions is in a sense similar to the difference between deduction and implication. In deduction we assume premises and are able to deduce conclusion, in implication we don't assume antecedent."

In De Morgan's spicular notation, modified as noted in my previous post (ie. '.' marks particularity), this idea is easy to express. Difference between modern and traditional interpretations of A proposition is "a matter of dot":

S.))P -- traditional interpretation
S))P -- modern interpretation

Dot makes explicit the enthymetical assumption that there are S-s we are talking about. It doesn't matter whether they are real or nominal only.

De Morgan's types 2

2008-01-27T18:40:00.000+02:00

I'll try now to write down syllogisms contained in De Morgan's zodiac in modified spicular notation, where dot does not signify negation but particularity. I'll present each universal syllogism together with it's opposing particular syllogisms.

X))Y))Z=X))Z
Y((X(.(Z=Y(.(Z
X(.(Z((Y=X(.(Y

Z((Y((X=Z((X
Y))Z).)X=Y).)X
Z).)X))Y=Z).)Y

X))Y)(Z=X)(Z
Y((X(.)Z=Y(.)Z
X(.)Z)(Y=X(.(Y

Z)(Y((X=Z)(X
Y)(Z(.)X=Y).)X
Z(.)X))Y=Z(.)Y

X)(Y()Z=X))Z
Y)(X(.(Z=Y).(Z
X(.(Z()Y=X(.)Y

Z()Y)(X=Z((X
Y()Z).)X=Y(.)X
Z).)X)(Y=Z).(Y

X()Y))Z=X()Z
Y()X).(Z=Y(.(Z
X).(Z((Y=X).(Y

Z((Y()X=Z()X
Y))Z).(X=Y).(X
Z).(X()Y=Z).)Y

Now, this was a piece of cake. Iconicity of the spicular notation is a great help.

De Morgan's categorical types

2008-01-25T22:21:00.000+02:00

In a comment to the last post, Brandon asked what would be the equivalent of De Morgan's "zodiac" (section 48 of the "Syllabus of a Proposed System of Logic") in my notation?

Here it goes:
1. Let's enumerate symbols in zodiac with clock numerals, so that 12 is in the top, 3 on right, 6 in bottom, 9 on left.
1 )( 2 () 3 )) 4 ).) 5 ).) 6 ).( 7 () 8 )( 9 (.) 10 ).) 11 ).) 12 ))

Corresponding formulas in my notation (for coherence I'll use De Morgan's letters):
A. Read clockwise
1 (-X-Y) 2 (+X+Y) 3 [-X+Y] 4 (-X+Y) 5 (-X+Y) 6 [-X-Y]
7 (+X+Y) 8 (-X-Y) 9 [+X+Y] 10 (-X+Y) 11 (-X+Y) 12 [-X+Y]

B. Read counterclockwise (ie. assymmetric formulas change)
1 (-X-Y) 2 (+X+Y) 3 [+X-Y] 4 (+X-Y) 5 (+X-Y) 6 [-X-Y]
7 (+X+Y) 8 (-X-Y) 9 [+X+Y] 10 (+X-Y) 11 (+X-Y) 12 [+X-Y]

Universal syllogisms are composed of universal premises at 12, 3, 6 and 9, read either clockwise or counterclockwise:

a1) 12[-X+Y]+3[-Y+Z]=[-X+Z]
a2) 3[+Z-Y]+12[+Y-X]=[+Z-X]
b1) 3[-X+Y]+6[-Y-Z]=[-X-Z]
b2) 6[-Z-Y]+3[+Y-X]=[-Z-X]
c1) 6[-X-Y]+9[+Y+Z]=[-X+Z]
c2) 9[+Z+Y]+6[-Y-X]=[+Z-X]
d1) 9[+X+Y]+12[-Y+Z]=[+X+Z]
d2) 12[+Z-Y]+9[+Y+X]=[+Z+X]

For every universal syllogism there are two opposed syllogisms, constructed of one universal premise retained from universal syllogism and of a particular premise adjacent to it in zodiac but "external", which is contradictory to the conclusion of the corresponding universal syllogism. Premises are read in opposite direction to the universal syllogism they oppose, and retained universal premise is converted. I understand that adjacent "external" formulas in comparison to eg. 12 and 3 are correspondingly 11 and 4. Hence:

a11) 12[+Y-X]+11(+X-Z)=(+Y-Z)
a12) 4(+X-Z)+3[+Z-Y]=(+X-Y)
a21) 3[-Y+Z]+4(-Z+X)=(-Y+X)
a22) 11(-Z+X)+12[-X+Y]=(-Z+Y)
b11) 3[+Y-X]+2(+X+Z)=(+Y+Z)
b12) 7(+X+Z)+6[-Z-Y]=(+X-Y)
b21) 6[-Y-Z]+7(+Z+X)=(-Y+X)
b22) 2(+Z+X)+3[-X+Y]=(+Z+Y)
c11) 6[-Y-X]+5(+X-Z)=(-Y-Z)
c12) 10(+X-Z)+9[+Z+Y]=(+X+Y)
c21) 9[+Y+Z]+10(-Z+X)=(+Y+X)
c22) 5(-Z+X)+6[-X-Y]=(-Z-Y)
d11) 9[+Y+X]+8(-X-Z)=(+Y-Z)
d12) 1(-X-Z)+12[+Z-Y]=(-X-Y)
d21) 12[-Y+Z]+1(-Z-X)=(-Y-X)
d22) 8(-Z-X)+9[+X+Y]=(-Z+Y)

Wow! That was not easy but it works. Was that a memory-device?

Missing types 7

2008-01-23T22:29:00.000+02:00

To be uneducated is a great misery. Today I discovered De Morgan's eight types of categorical propositions (in "Syllabus of Proposed System of Logic"), which match my types, even if he seems not to have recognized the same natural language expressions for new types. His notation is interesting:

Universal propositions:
[-X+Y]: X))Y All Xs are some Ys
[-X-Y]: X).(Y All Xs are not (all) Ys
[+X+Y]: X(.)Y Everything is either some X or some Y (or both)
[+X-Y]: X((Y Some Xs are all Ys

Particular propositions:
(+X-Y): X(.(Y Some Xs are not (all) Ys
(+X+Y): X()Y Some Xs are some Ys
(-X-Y): X)(Y Some things are not either (all) Xs nor (all) Ys
(-X+Y): X).)Y All Xs are not some Ys

'X)' and '(X' is read '(all) X'
'X(' and ')X' is read 'some X'
'.' is negative copula
'X' is positive term, 'x' negative.

For me, this notation would be clearer, if dot would always mark particular proposition. Then the meaning of symbols would be almost iconically given. E.g.:

Universal propositions:
X))Y All Xs are only Ys
X)(Y All Xs are not any Ys
X()Y Only Xs are not only Ys
X((Y Only Xs are any Ys

Particular propositions:
X(.(Y Some Xs are not any Ys
X(.)Y Some Xs are some Ys
X).(Y Some things are neither any Xs nor any Ys
X).)Y All Xs are not some Ys

He calls this notation spicular, borrowing the name from Sir Hamilton, who characterized it as "horrent with mysterious spiculae."

On algebra of logic 4

2008-01-18T23:31:00.001+02:00

In a comment, Brandon asked about how to represent in my version of logic algebra ordinary propositional formulas among modal formulas. This question has bothered me for some time and I have thought of some possibilities to introduce needed distinctions. As it stands now, we can interpret brackets and parentheses either with quantification or without it, while retaining in both cases their disjunctive versus conjunctive character. This makes it impossible to explicitly specify whether we have quantification or not. But to introduce new special symbols moves us away from the unity of notation between different levels of logic, which I consider a feature with great value, and natural to this kind of algebraic representation. As I noted in my response to Brandon, at present I prefer to introduce the distinction between quantificational and non-quantificational contexts with minimal altering of existing symbols -- quantificational contexts will be bolded and non-quantificational contexts will be plain.

Thus, we can differentiate between universal, particular and singular categorical propositions, and between propositional and modal formulas, using for all purposes the same syntax:

Categorical formulas:
[-S+P] Every S is P Everything is either not S or is P	[-S-P] No S is P Everything is either not S or not P	[+S-P] Only S is P Everything is either S or not P	[+S+P] Only S is not P Everything is either S or P
[-S+P] This is either not S or is P	[-S-P] This is either not S or not P	[+S-P] This is either S or not P	[+S+P] This is either S or P
([-S+P]) Something is either not S or P	([-S-P]) Something is either not S or not P	([+S-P]) Something is either S or not P	([+S+P]) Something is either S or P
[(-S+P)] Everything is not S but B	[(-S-P)] Everything is neither S nor P	[(+S-P)] Everything is S but not P	[(+S+P)] Everything is S and P
(-S+P) This is not S but P	(-S-P) This is neither S nor P	(+S-P) This S is not P This is S but not P	(+S+P) This S is P This is S and P
(-S+P) Not only S is P Something is not S but is P	(-S-P) Not only S is not P Something is neither S nor P	(+S-P) Some S is not P Something is S but not P	(+S+P) Some S is P Something is S and P
Predicate/modal formulas:
[-p+q] Necessarily, if p, then q	[-p-q] Necessarily, if p, then not q	[+p-q] Necessarily, only if p, q	[+p+q] Necessarily, either p or q
[-p+q] If p, then q	[-p-q] If p, then not q	[+p-q] Only if p, q	[+p+q] Either p or q
([-p+q]) Possibly, if p, then q	([-p-q]) Possibly, if p, then not q	([+p-q]) Possibly, only if p, q	([+p+q]) Possibly, either p or q
[(-p+q)] Necessarily, not p but q	[(-p-q)] Necessarily, neither p nor q	[(+p-q)] Necessarily, p but not q	[(+p+q)] Necessarily, both p and q
(-p+q) Not p but q	(-p-q) Neither p nor q	(+p-q) p but not q	(+p+q) Both p and q
(-p+q) Possibly, not p but q	(-p-q) Possibly, neither p nor q	(+p-q) Possibly, p but not q	(+p+q) Possibly, both p and q

With this alteration of symbols, Brandon translated axioms of different modal systems. Formulas may be simplified by dropping outer brackets, according to the principle that by default, formulas starting with '-', are disjunctive, formulas starting with '+', conjunctive (conforming to SETL algebra). But let for now the outer context be explicitly stated:

K: [-[-p+q]+[-[p]+[q]]]
D: [-[p]+(p)]
M: [-[p]+p]
4: [-[p]+[[p]]]
B: [-p+[(p)]]
5: [-(p)+[(p)]]

I add the rest from SEP article, no.8:

CD: [-(p)+[p]]
[M]: [-[p]+p]
C4: [-[[p]]+[p]]
C: [-([p])+[(p)]]

On algebra of logic 3

2008-01-09T00:14:00.001+02:00

Introducing Domains

I'd like expressions of the algebra to refer to some domain. To bring this about, let's think of these formulas as containing a hidden variable referring to the domain, let's say A for actual world. (To mark domain I'll use capitalized italic letters.) Then to say Everything is P in actual world A we should write [-A+P], which might be read also as Everything either is not in actual world or is P. In more general way [P] refers by default to some domain, where everything is P. To make this explicit we write it [-D+P], where D stands for the specific domain, and brackets quantify over every individual object. In this sense D is the "top predicate" in given domain. Every predicate that refers to anything at all in given domain refers to something to which D refers. It is different from other predicates that might also refer to all individuals in given domain in that D refers to nothing outside given domain, while other predicates might refer to something in other domains also. When the domain is contextually clear, we may hide the domain letter, but when it is unclear or we are explicitly reasoning over multiple domains, we should explicate the domain, to which given expression is referring. In case of particular propositions the domain is by default marked positively, ie. (P) with domain explicated is (+D+P), meaning Something in domain D is P, or Something is in domain D and is P.

Possibility of making domains explicit broadens significantly the use of algebra. We can now express that some predicate refers to something only in given domain, eg. [+D-P], read as Something is P only in domain D, or Only in domain D is anything P. Of course we can as well express that Something is P not only in domain D -- (-D+P).

We can have empty domains -- [-D] (Everything is not in domain D) as well as -(D) (Nothing is in domain D).

We can reason about relations of domains -- eg.:
[-E-F]: Domains E and F are disjoint, having no common individuals.
[-E+F]: Domain E is included in (/is subdomain of) domain F; ie. all individuals in domain E (if it is not empty) are included in domain F.
[+E-F]: Only domain E includes domain F; ie. domain E is superdomain of F; only individuals of domain E are included in domain F.
[+E+F]: Only domain E is not in domain F; ie. any individual that is not in domain E is in domain F.
(-E-F): Something is neither in domain E nor in domain F.
(-E+F): Something that's not in domain E is in domain F.
(+E-F): Something in domain E is not in domain F.
(+E+F): Something is both in domain E and in domain F.

And we can define universal domain, including (all individuals from) all domains: [U], meaning everything is in U, or it is not at all, in any way, really or virtually, or in whatever form. U is top-domain, domain of all domains. U cannot be empty: (U).

Existential import

Every predicate is instantiated in some domain, but they can be applied in domains where they are not instantiated. Thus (1) Every A is B in D is true even if there are no A-s in D. And (2) Some A is B in D can be true only if there is some A in D. So (1) implies (2) only if there are A-s in D. This corresponds to standard modern interpretation of existential import. But even if there are no A-s in D, A has to be instantiated in some domain (real or imaginary or abstract or whatever), to be counted as predicate, ie. to be meaningful. Hence, globally it cannot be said that there are no A-s, if A is applied in any domain at all as predicate. This seems to be the position of those who insist that both universal and particular propositions have existential import (eg. J.P.N. Land, in the article "Brentano's Logical Innovations" in Edward Bruckner's Logic Museum, says: "In an ordinary proposition the subject is necessarily admitted to exist, either in the real or in some imaginary world assumed for the nonce."). And finally we may distinguish one domain, usually (but not necessarily) the actual world, in relation to which neither universal nor particular propositions need to have existential import, which seems to have been the position in logica antiqua. Consider for example Aristotle: "Take the proposition 'Homer is so-and-so', say 'a poet'; does it follow that Homer is, or does it not? The verb 'is' is here used of Homer only incidentally, the proposition being that Homer is a poet, not that he is, in the independent sense of the word." (De Interpretatione, 11). For last two positions universal propositions unconditionally imply particular ones. (See also this article in Logic Museum)

I think these three positions differ in that in logica antiqua existence was granted to real essences only but talk was meaningful about nominal essences also; in Land's position existence is broadly granted to real as well as nominal essences; but for fathers of modern logic nominal essences were largely meaningless mumblings. At least this is my impression. If domains are not explicated, then logical, physical and metaphysical existence is undistinguished and confusion results.

ADDENDUM

We can use domains with propositional logic as well.
[-E+p] In domain E proposition p is true
[-E-p+q] In domain E, if p is true, then q is true
...

On algebra of logic 2

2008-01-06T14:22:00.000+02:00

Brandon's thoughts on my version of logic algebra lead me to think anew about propositional version of it. And I came to conclusion that in addition to thinking of the propositional version of it as simple transformation where term letters are replaced by proposition letters, we can interpret it also as modal propositional logic. And it reveals amazing unity between different "layers" of logic.

As said, we can move to ordinary propositional logic (in singleton universe as Brandon calls it) by simply replacing terms by propositions. Then we should interpret parentheses simply as conjunction and brackets as disjunction. But this in not fully equivalent translation of quantified term logic. Rather, it is equivalent to term logic with single individual.

But if we transfer quantification also, then we have modal propositional logic. Parentheses, as usual, set up particular context, which should be read as In some case... Brackets, as usual, set up universal context, read as In every case... (Ordinary propositional logic instead presumes the preamble It is the case that...)

[-p+q] In every case (/necessarily), if p is true, q is also
[-p-q] In every case (/necessarily), if p is true, q is not
[+p-q] In every case (/necessarily), if p is true, q may be true
[+p+q] In every case (/necessarily), if p is true, q may be not true
(-p+q) In some case (/possibly), p is not true, but q is
(-p-q) In some case (/possibly), neither p nor q is true
(+p-q) In some case (/possibly), p is true but not q
(+p+q) In some case (/possibly), both p and q are true

([+p+q]) In some case (/possibly), either p or q is true
[(+p+q)] In every case (/necessarily), both p and q are true
...

Further, [p] is read as Necessarily p, ie. In every case p is true
[-p] Necessarily -p; p is impossible; In every case p is not true
(p) Possibly p; In some case p is true
(-p) Possibly -p; In some case p is not true

In singleton universe, or monotonic propositional logic, (p) is read as just p is true, or It is the case that p (contextually), and it is the same as [p], read as It is the case that p (universally), with all contextual references (places, times, persons...) replaced with absolute references.