Talk:Square of opposition
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- 1 References or Footnotes
- 2 Rewrite
- 3 Subaltern
- 4 Some extra information to think about
- 5 In contemporary terms...
- 6 I really don't want to get further into another topic but seriously,
- 7 The article logical hexagon created by Gregbard
- 8 Concise remarks on De interpretatione as being at the origin of the logical square and of modal logic
- 9 About the logical particulars I and O wrongly identified with the particulars of natural language 'Some S are P' and 'Some S are not P'
- 10 Spelling
- 11 Some more on the imperfection of the traditional presentation of the square of opposition
- 12 On the origin of the square of opposition
References or Footnotes
The "References" section of this article seems to contain a list of notes, not actual references. I need to check WP:STYLE in order to figure out how this should be formatted, but I'm thinking about coming back to work on this. J Crow (talk) 22:14, 7 January 2011 (UTC)
How do I do tables? Dbuckner 07:42, 1 August 2006 (UTC)
Does Aristotle even define the subaltern in De Interpretatione? I can't find it. You might define it as truth of one implies the other but not vice verse, or falsity of one implies falsity of the other but not vice versa, or make both requirements. These 3 definitions will all be equivalent in the traditional square, but will diverge for statements that cannot be put into standard form, and I can't find which def is used. Maybe this is a later Aristotelian twiddle. Likewise, I always portray the traditional square with subaltern lines with arrows for the truth goes up and falsity goes down parts of the subaltern inferences. Bmorton3 15:26, 3 August 2006 (UTC)
- Good point. I checked and I think you are right (though always much harder to establish someone didn't say something than conversely). I will alter it at some point. Clearly the subaltern relation follows from contrariety and contradiction).
- Brian: I don't suppose you would be interested in getting this one in shape for FAC? Or has the other thing put you off? Dbuckner 18:44, 6 August 2006 (UTC)
- It's short, and not very "pressing," let's get it to GA, and then see how any of the FA stuff works out. See my comments on well-written vs well-edited at the guideline for reviewers site. I suspect I lack the skills required to push an article from GA to FA, and am instead interested in trying to get stuff to GA. Look at Nature (philosophy) and Naturalism (philosophy) those sites are much higher traffic, more pressing and badly need work jsut to get to GA. Bmorton3 15:55, 7 August 2006 (UTC)
I used to have a great list of like half a dozen different medieval compromises on existential import, but I can't find it. So I can't reference the "lots of compromises" claims, but not I ws thinking much earlier than Keynes. Also Spade asserts that Aristotle has no term for subaltern, is it Porphyrian maybe? Bmorton3 20:41, 3 August 2006 (UTC)
- Boethius explicitly calls it that in his commentary. Dbuckner 18:45, 6 August 2006 (UTC)
My pre-Spade notes claim a modified Aristotelian position of all propositions have EI in subject only (which ditches conversion) in the early middle ages (but who? I don't know), Only affirmitive statements have EI in some late middle age texts (damn who?), and Boole and Ferre's positions. For what it is worth, Spade argues that Aristotle is committed to EI on all terms, not from his discussion of O forms, but from the very head of Prior Analytics where he says our aim is only to explore demonstration and demostrative science, look also at the posterior analytics "We suppose ourselves to possess unqualified scientific knowledge of a thing." The Demonstrative is only for knowledge of things, not of that which is not (although of course there are other mannners of knowing, and non-demonstrative forms of argumentation). But again lots of what the medievals get is coming from Theophrastus, Porphyry, Alexander Aphrodisias, etc. Bmorton3 21:03, 3 August 2006 (UTC)
Uhm some of my edit got squashed, if you allow EI on A and I but not on E and O, subcontriety fails as does half of subalternation, and there might be other problems too. Bmorton3 19:10, 5 September 2006 (UTC)
- Look at the Stanford Encyclopedia of Philosophy article (top external link). It shows in gory detail how you get all the traditional inferences if you put EI on AI but not EO. Namely, if the subject term is empty, then I is false, E is true, A is false, O is true. This satisfies subcontriety and the "negative half" of subalternation
- Well, crap, I hate feeling stupid. You are right, these would save the square doctrines. It would force some revision on how obversion and contraposition work, but it would save the square. Bmorton3 16:35, 6 September 2006 (UTC)
What is with the following sentence in this article?
"For example, if 'every man is just' is true, its contrary 'no man is white' is false."
Not sure this is entirely accurate....
Woodzie 23:44, 23 April 2007 (UTC)
Some extra information to think about
I'm researching an article on the Square at the moment – looking at what the 'traditional logicians' actually wrote, in Latin, as opposed to what some modern logicians, writing with hindsight and with the benefit of the assumptions behind modern predicate calculus, say that the traditional logicians said, in English.
E.g. Parsons writes "For most of this history, logicians assumed that negative particular propositions ("Some S is not P") are vacuously true if their subjects are empty." First of all, a huge sweeping assertion, apparently covering logicians as diverse as Aristotle, Apuleius, Boethius, Abelard, Ockham, Buridan, Zabarella, Sanderson, Hamilton, Mill, Brentano, and many in between. When you come to what such logicians actually wrote, of course you find that they are arguing and quarreling among themselves, proposing new theories, disputing old ones – pretty much the same as today. It's like characterising all logic between 1880 and 2006 as involving the same assumptions, which of course it doesn't.
In any case, most of the logicians Parsons is thinking about are the medieval ones, who wrote in Latin. Did they say anything that suggests they assumed what Parsons characterises iin English as 'assumed that … their subjects are empty'? It turns out that logicians before the 13C couldn't really express this assumption at all, since Parsons is talking about terms that stand for or denote nothing, and they had no word for that. The idea of 'standing for something' (supponens pro aliquo) was an innovation of the late 12C whose origins are mysterious. William of Sherwood mentions it, Ockham and Buridan write a great deal about it.
Before the 13c logicians talked about subjects 'not existing' (non existens). Thus the term 'chimera' does stand for something, as it were, namely a chimera, a thing that does not exist in reality (in rerum natura, in rebus, in rebus materialibus &c). Thus they are commited to the truth of "some things do not exist in reality". But the sentence "some things do not exist in reality" is of course an O proposition, and it is clear that if the A proposition 'all things exist in reality" is false, "some things do not exist in reality" is true, and the traditional relationships hold, but for very different reasons than Parsons claims.
Now there is a nominalistic tradition, of whom Ockham and Buridan are notable exponents, according to which there isn't a world of impossibilia and imaginary things. But this is a separate tradition, and it died out after the 14C. The tradition that survived is the Thomistic one which continues the pre-13C tradition of non-existent things. This survives until the nineteenth century and beyond (in the hands of the neo-thomistic logicians such as Joyce and Wade).
However, this probably counts as OR –just letting you know that the position is a little more complex than anything in this articles makes out. I'm still working on the paper. If I finish it and it is published, perhaps you can refer to it! Best. Dbuckner 10:40, 29 September 2006 (UTC)
- Yeah, I have lots of worries here, but it is largely on the edges of my competence. First, when we think of an empty subject do we mean one that lacks 'standing for something' (supponens pro aliquo) or 'refering to something' (significans pro aliquo). Peter of Spain already wants to talk about the difference between signification and supposition. The doctrine of the difference between supposition proper and improper is all about this difference. In improper supposition a term might supposit, even though the term cannot significate. Consider the term "the cup I drank yesterday" in the claim "the cup I drank yesterday was poisoned." I didn't drink ANY cup yesterday, I drank the wine in it. So the term "the cup I drank yesterday" can't refer to anything, but it does nonetheless supposit for the liquid in the cup which I drank yesterday. Here is another wrinkle from Buridan's Sophismata Chap 2. For a affirmative statement to be true, it requires something of which the subject term can be truly affirmed. But in a claim like "Hominem esse animal est verum", ther is nothing for the subject term to be about because of the infinitive. In Paul Spade's word's there is nothing you can point to and say "this is for a man to be an animal" So the term can't be "taken significatively." There's lots of other problems but at minimum you have to figure out how the signification and supposition talk is supposed to map onto modern sensibilities, what was it for, what was it doing? Terms can be empty in lots of ways. A term can fail to mean anything. It can fail to refer to anything. I can fail to be used to mean anything. It can mean something other than what it is used to mean. It can be used to mean something other than what it refers to. It can even (one of my old papers) be used to mean something while failing to mean anything! Moderns step blithely between issues, medievals made fine distinctions on, like the difference between different kinds of ways that a term can be empty. Bmorton3 15:14, 29 September 2006 (UTC)
In contemporary terms...
Is it possible to make the following pairings:
subalternate: → (IMPLIES/IF)
I really don't want to get further into another topic but seriously,
"Particular statements are subcontraries. 'Some man is just' and 'some man is not just' cannot be false together"
- This is just terrible English. I cannot suppose this is how Aristotle would have worded his beliefs, because if so my opinion of him is going to rapidly drop. I completely appreciate that not everybody in the world is a native English speaker, and/or numerous other reasons for explaining the use of man rather than men and so on (such as the forming of sub contraries into one word is 'wrong'), but it seems to me to further highlight a problem Wikipedia is displaying. It is losing relevance through poor maintenance, cross referencing etc.. Especially on articles of such GREAT importance as these on concepts/methods of thinking which are so valuable to our future being a positive one! BoredextraWorkvidid (talk) 10:00, 20 September 2010 (UTC)
The article logical hexagon created by Gregbard
I have created an article for Logical hexagon and refactored a large amount of material contributed by User:Jean KemperNN. The material is wonderful, but I think it is more appropriate in its own article.Greg Bard (talk) 22:59, 14 November 2010 (UTC) (Jean KemperN (talk) 06:48, 3 January 2011 (UTC)) http://www.grammar-and-logic.com/dossiers.php — Preceding unsigned comment added by 184.108.40.206 (talk) 21:19, 3 February 2013 (UTC) (220.127.116.11 (talk) 16:11, 9 December 2013 (UTC)) mindnewcontinent
Concise remarks on De interpretatione as being at the origin of the logical square and of modal logic
The logical square, also called square of opposition or square of Apuleius has its origin in the four marked sentences to be employed in syllogistic reasoning: Every man is white, the universal affirmative and its negation Not every man is white (or Some men are not white), the particular negative on the one hand, Some men are white, the particular affirmative and its negation No man is white, the universal negative on the other. Robert Blanché published with Vrin his Structures intellectuelles in 1966 and since then many scholars think that the logical square representing four values should be replaced by the logical hexagon which by representing six values is a more potent figure because it has the power to explain more things about logic and natural language. The study of the four propositions constituting the square is found in Chapter 7 and its appendix Chapter 8. Most important also is the immediately following Chapter 9 dealing with the problem of future contingents. This chapter and the subsequent ones are at the origin of modal logic. Perhaps Blanché's hexagon is particularly useful in the domain of modal logic in so far as it explains clearly the nature and importance of the bilateral possible. The notion of bilateral possible is crucially important to understand both logic and natural language when applied to modal values. (Jean KemperN (talk) 06:42, 3 January 2011 (UTC))
http://erssab.u-bordeaux3.fr here http://www.grammar-and-logic.com/dossiers.php (Jean KemperN (talk) 13:05, 5 January 2011 (UTC)) (18.104.22.168 (talk) 23:49, 12 January 2011 (UTC))(cf. here) (22.214.171.124 (talk) 08:59, 28 January 2011 (UTC)) (126.96.36.199 (talk) 00:22, 23 November 2011 (UTC)) Article : " Du nouveau sur Aristote. Remarques sur deux traductions arabes du De Interpretatione", L'Enseignement philosohique, 53e année - n° 4, mars-avril 2003 (format PDF - 14 pages)
About the logical particulars I and O wrongly identified with the particulars of natural language 'Some S are P' and 'Some S are not P'
These are the four propositions which are at the origin of the square of opposition and are to be found in De Interpretatione, Chapter 7, (the De Interpretatione, Peri Hermeneias in Greek is the second book of Aristotle's Oganon):
- The so-called 'A' proposition, the universal affirmative (universalis affirmativa), whose form in Latin is 'omne S est P', usually translated as 'every S is a P'.
- The 'E' proposition, the universal negative (universalis negativa), Latin form 'nullum S est P', usually translated as 'no S are P'.
- The 'I' proposition, the particular affirmative (particularis affirmativa), Latin 'quoddam S est P', usually translated as 'Some S are P'.
- The 'O' proposition, the particular negative (particularis negativa), Latin 'quoddam S non est P', usually translated as 'Some S are not P'.
Let Some S are P be represented by Some men are white and Some S are not P by Some men S are not white, these example sentences coming from Aristotle's On interpretation (or De interpretatione, the second book of the Organon, more precisely coming from the seventh chapter thereof. Obviously,the two natural particulars Some men are white and Some men are not white cannot be identified with the logical particulars I and O. Demonstration to follow pretty soon http://erssab.u-bordeaux3.fr here http://www.grammar-and-logic.com/dossiers.php
- Please make clearer what you quote and what you say, and if you think that something in the article should be changed. Lipedia (talk) 13:58, 15 July 2011 (UTC)
(188.8.131.52 (talk) 20:21, 13 August 2012 (UTC))Dear Sir, type Mindnewcontinent. If you want a fruitful dialogue, this is my email adress: firstname.lastname@example.org. Jean-François Monteil (184.108.40.206 (talk) 09:21, 12 September 2012 (UTC))Dear Sir, I don't think that the article should be changed now. It is excellent to the extent that it represents perfectly well a present state of human knowledge. That the reason why I content myself with introducing some suggestions only into the talk page. Consider the sentence Some men are white. I call it the affirmative particular proposition of human language. To me, its sense and that of the affirmative particular proposition of logic are different. The latter means At least one member of mankind is white. We have to note that this logical affirmative particular does not exclude the fact that all men are white while the natural affirmative particular Some men are white excludes not only the fact apprehended by No man is white but also the fact apprehended by the affirmative universal of human language All men are white. TO BE CONTINUED
Some more on the imperfection of the traditional presentation of the square of opposition
The article and particularly what I read concerning the square of opposition and the propositions used in the syllogistic reasoning are pretty good. My purpose is to suggest that the form in which appear the four propositions constituting the square induces some errors concerning the relation between logic and natural language. Two examples: the content of the logical proposition A corresponds to the referent apprehended not only by the sentence All Men are white but also by the sentence Men are white. In natural language, there is not one affirmative universal but two. The imperfection of the examples from the linguist's viewpoint is still more evident when one envisages the particulars. The affirmative particular Some men are white of English, for instance,does not correspond to the content of I, the affirmative particular of logic. The logical proposition I means At least one man is white. So, At least one man is white does not exclude the content of All men are white whereas Some men are white excludes both the content of No man is white and the content of All men are white. The person who says Some men are white implies that some other men are not white.
I advise to read
1KNOLmnc 1 From the deficient square of opposition to Blanché’s hexagon. The triangle of Indian logic as a simplification of the latter. The rationalization of the scholastic symbolization.
2 KNOLmnc 1 Gist of the question . The essentials in seven pages.
3 KNOLmnc 0 Diffusion
On the origin of the square of opposition
The De Interpretatione is at the origin of the logical square, which must be replaced by the logical hexagon presented by Robert Blanché in Structures intellectuelles (1966). Jean-François Monteil reads Greek and Arabic. One of his specialities is the Arab handing down of Aristotle. He particularly knows the problems attached to the chapter 7 of On interpretation, a founding text since it is at the origin of the logical square, called square of Apuleius.
Professor Paul Gohlke is the only translator to fully respect Aristotle’s own conception of indeterminates. He was the first to perceive the linguistic problem raised by the indeterminate negative. All the other translators of De interpretatione mistakenly render Aristotle’s indeterminates, which are particulars, as universals. The origin of this mistake lies in one of the two Arab translations.
In Peri Hermeneias, chapter 7, Aristotle mainly studies the four marked natural propositions, which are at the origin of the logical square:
-A Pas anthropos leukos Everyman (is) white
-E Oudeis anthropos leukos No man (is) white
- I Esti tis anthropos leukos Some man (is) white
-O Ou pas anthropos leukos Not everyman (is) white.
He also studies two propositions said to be indeterminate in so far as they are devoid of quantifying morphemes like Tis-Some or Pas-Every : Esti anthropos leukos, Ouk esti anthropos leukos. Since Aristotle explicitly says that they are both true, he implicitly says that they are particular propositions. Therefore, the affirmative one Esti anthropos leukos means There are white men, the negative one Ouk esti anthropos leukos means There are non-white men. Semantically, the indeterminate propositions of On interpretation, chapter 7, are equivalent to the marked particulars Some men are white and Not everyman is white (or Some men are not white).
But an attentive Hellenist cannot but recognize that the form of the negative indeterminate Ouk esti anthropos leukos with the negative adverb ouk bearing on an esti signifying there is conveys the meaning of a negative universal proposition There is not any white man, that is, No man is white. Aristotle himself confesses it. For the indeterminate negative to be interpreted as a particular negative, one must mentally replace Ouk esti anthropos leukos by Esti ou leukos anthropos There is a non-white man with the negation ou bearing on leukos and not on the verb esti there is.
Gohlke’s attitude is exemplary. On the one hand, he renders the indeterminates by particulars as Aristotle wants. On the other, he courageously says that we have to do with a “fait du Prince” since Aristotle enjoins us to interpret a sentence which obviously has the meaning of a universal negative as if it were a particular negative. All translators, save Gohlke, render the indeterminate propositions of chapter 7, which, according to Aristotle, are particular propositions, that are true by universal propositions, that are false. The unmarked natural universal Man is white L’homme est blanc is used to translate a sentence There are white men Il y a des hommes blancs that means Some men are white Certains hommes sont blancs. In the same way, the universal Man is not white, L’homme n’est pas blanc, which is the unmarked universal negative and has therefore the same referent as the marked universal negative No man is white Aucun homme n’est blanc, is used to render a particular negative signifying that only some men are not white.
Our papers explain this aberration. There exists a system of three pairs of natural contradictories:
pair a Men are white versus Men are not white
pair b All men are white versus Some men are not white
pair c Some men are white versus No man is white.
First act of the drama: Aristotle removes Pair a where two natural unmarked universals oppose one another contradictorily: Men are white (or Man is white Ho anthropos esti leukos) versus Men are not white (or Man is not white Ho anthropos ouk esti leukos).
Second act: to render the indeterminates, one of the two translators mentioned by I. Pollak, much embarrassed by the indeterminate negative, thinks it judicious to make use of the two unmarked universals removed by Aristotle and for this reason available.
The mutilation has disastrous consequences for linguistics and logic. Aristotle’s indeterminates are useless. However the problematic indeterminate negative may be interpreted, they add nothing to the four marked propositions of chapter 7. Therefore, an erroneous translation of them is of little importance in itself. But the drawback is that the incredible rendering hides the fact that Aristotle mutilates the system of natural language. How could people be aware that he eliminates Ho anthropos esti leukos Man is white and Ho anthropos ouk esti leukos Man is not white, when they see these sentences used to translate Aristotle’s indeterminates?
So, in Chapter 7 of On Interpretation, Aristotle alters a system of three pairs of natural mutually contradictory propositions, in that he eliminates the pair where two natural universals Men are white ( Man is white) and Men are not white (Man is not white) oppose each other contradictorily (see the first diagram below). This alteration has serious consequences. These are the two natural pairs that Aristotle considers exclusively:
– All men are white ( Everyman is white) versus Some men are not white (Not all men are white Not everyman is white)
– Some men are white versus No man is white
They are illegitimately identified with the two pairs of logical contradictories constituting the logical square: A versus O and I versus E :
– A (x)f(x) → g(x) Whatever x may be, if x is man, then x is white versus O (∃x)f(x)&~g(x)At least one member of mankind is non-white
– I (∃x)f(x) & g(x). At least one member of mankind is white versus E(∃x)f(x) →~g(x) Whatever x may be, if x is man, then x is non-white
Thus, the level of natural language and that of logic are confused.
- A is the logical universal affirmative Whatever x may be, if x is man, then x is white Whatever a member of mankind may be, he is white (x)f(x) → g(x). This logical universal affirmative is mistakenly confused with what I call the natural marked universal affirmative All men are white Everyman is white. In fact, A represents the referent not only of All men are white Everyman is white but also of what I call the natural unmarked universal affirmative Men are white Man is white.
- E is the logical universal negative Whatever x may be, if x is man, then x is non-white, Whatever a member of mankind may be, he is non-white (x) f(x) → ~g(x). This logical universal negative is mistakenly confused with what I call the natural marked universal negative No man is white. In fact, E represents the referent not only of No man is white but also of what I call the natural unmarked universal negative Men are not white Man is not white.
- I is the logical particular affirmative There exists at least one x that is man on the one hand and is white on the other There exists at least one member of mankind that is white (∃x)f(x)&g(x). This logical particular affirmative is peccaminously confused with what I call the natural particular affirmative Some men are white. There is more information in the natural than in the logical one. The logical particular At least one member of mankind is white merely excludes the content of E whereas the natural particular Some men are white excludes both the content of E and that of A. For when you say that some men are white, you imply that some are not.
- O is the logical particular negative There exists at least one x that is man on the one hand and is non-white on the other There exists at least one member of mankind that is non-white (∃x)f(x)&~g(x). This logical particular negative is peccaminously confused with what I call the natural particular negative Some men are not white. There is more information in the natural than in the logical one. The logical particular At least one member of mankind is non-white merely excludes the content of A, whereas the natural particular Some men are not white excludes both the content of A and that of E. For when you say that some men are not white, you imply that some are.
The reading of the logical propositions is naturally simplified : A becomes Whatever the member of mankind may be, he is white ; E becomes Whatever the member of mankind may be, he is non-white ; I becomes At least one member of mankind is white ; O becomes At least one member of mankind is non-white. These readings purposely remain unnatural. Thus, the logical universal affirmative A will not be confused with All men are white Everyman is white, the logical universal negative E will not be confused with with No man is white, the logical particular affirmative I will not be confused with with Some men are white, the logical particular negative O will not be confused with with Some men are not white Not all men are white Not everyman is white.
The exact representation of the four propositions of the logical square x
A-Whatever a member of mankind may be, he is white E-Whatever a member of mankind, he is non-white
I-At least one member of mankind is white O-At least one member of mankind is non-white
The fallacious representation of the four propositions of the logical square
A All men are white E No man is white
I Some men are white O Some men are not white
The pair of logical propositions mutually contradictory: A versus O, Whatever the member of mankind may be, he is white versus At least one member of mankind is non-white must not be confused with the pair of natural propositions mutually contradictory All men are white versus Some men are not white.
The pair of logical propositions mutually contradictory: I versus E, At least one member of mankind is white versus Whatever the member of mankind may be, he is non-white must not be confused with the pair of natural propositions mutually contradictory Some men are whiteversus No man is white.
The unfortunate Aristotelian alteration is concealed by the translation of the propositions mentioned above and known as indeterminate propositions): Esti leukos anthropos, There are white men and Ouk esti leukos anthropos, There are non-white men, (so must be rendered the latter, if we want to conform to Aristotle’s intention). To translate these, which, semantically, are particulars, all scholars, save Paul Gohlke, employ the natural universals excluded by the master Men are white (Man is white) and Men are not white (Man is not white)! The work of Isidor Pollak, published in Leipzig in 1913, probably reveals the origin of this nearly universal translation mistake: the Arab version upon which Al-Farabi unfortunately bases his commentary. The logical hexagon of Robert Blanché adds the meanings Y and U to the four ones of the square. Thanks to these additions, an understanding of the manner in which the logical system and the natural one are linked becomes possible.
Two informative papers to be be read :
-paper 1 Paul Gohlke-Two. “A German exception: the translation of On Interpretation by Professor Gohlke. His tenth note on indeterminate propositions.(published in la Revue des Etudes anciennes 2001-Numéro 3-4)”
-paper 2 “From the logical square to the logical hexagon.The logical square of Aristotle or square of Apuleius.The logical hexagon of Robert Blanché in Structures intellectuelles. The triangle of Indian logic mentioned by J.M Bochenski