Wednesday, April 18, 2012

Venn diagrams for modal logic 5

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:

1. A B C   2. A B 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]

3. A B C   4. A B 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:

5. A B C   6. A B 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])

7. A B C   8. A B 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.

9. A B C


Wednesday, April 11, 2012

Venn diagrams for modal logic 4

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). 

4 1. A B C  4 2. A B C

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).

4 3. A B C

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.

4 4. A B C  4 5. A B C

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:

4 A B C 4*.

Venn diagrams for modal logic 3

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.

1. !Barbara P M S   2. !Darii P M S
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

3. !Celarent P M S   4. !Ferio P M S
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.

Wednesday, April 4, 2012

Venn diagrams for modal logic 2

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.

Sunday, April 1, 2012

Venn diagrams for modal logic

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:

□A □-A ◊A ◊-A

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

Thursday, March 29, 2012

Venn diagrams of opposing terms

A Ā

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:


A Ā E Ē 1 2 3 4 5 6 7 8 9 10

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

Wednesday, March 21, 2012

Relations in Venn diagrams

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: