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I am familiar with many of the surveys of many valued logic referenced in the SEP article on many valued logic, such as Ackermann, Rescher, Rosser and Turquette, Bolc and Borowic, and Malinowski. It is asserted in the article that "Many-valued logic as a separate subject was created by the Polish logician and philosopher Łukasiewicz (1920), and developed first in Poland. His first intention was to use a third, additional truth value for “possible”, and to model in this way the modalities “it is necessary that” and “it is possible that”. This intended application to modal logic did not materialize. " My question is, why didn't it?

Edited: to move summarized known objections to an answer.

Martin Sleziak
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Confutus
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4 Answers4

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I found a reference to a paper that proves that it is impossible (haven't seen the article myself) Dugundji, James. Note on a Property of Matrices for Lewis and Langford's Calculi of Propositions. Journal of Symbolic Logic 5 (1940), no. 4, 150--151. jstor, doi: 10.2307/2268175

From "many valued logic" by Reshner (1969). (Gregg Revivals page 192) "There exist no finitely many valued logic that is characteristic of any of the Lewis systems S1 to S5 , because any finitly many-valued logic will contain tautologies that are not theorems of S5 (and fortiory not of S1 to S4 either)

Martin Sleziak
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Willemien
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  • The 3 valued modal logic is not one of the Lewis systems. It can be developed into a system formally analogous to S5, but discarding the Law of the Excluded middle and the different definition of implication make it different enough that the proof by Dugundji doesn't apply. – Confutus Mar 06 '14 at 21:58
  • It is the other way around S5 is a Lewis system. 3 valued logic cannot developed in an system formally analogous to S5, S5 has an infinite amount of truthvalues (true/false in an infinite amount of worlds) In S5 the values in a world of $ p, \lnot p, \square p , \square \lnot p , \lnot \square p , \lnot \square \lnot p $ can all be different so you need even in one world for this 6 different truthvalues and in 3 valued logic you are alleady 3 short of what is needed here.

    An other maybe simpler proof is that intuitionistic logic (equivalent to S4) has infinite truthvalues.

     – Willemien Mar 07 '14 at 09:21
  • How do you know that the 3 valued logic cannot be developed in a formally analogous system? Have you tried substituting a strict Lukasiewicz (NCpq) conditional for the Lewis strict conditional in the axioms for S5 and evaluating the resulting formulas? – Confutus Mar 07 '14 at 16:54
  • That's LCpq. Also M for and L for – Confutus Mar 07 '14 at 17:47
  • That is not the point you can use polish notation ( http://en.wikipedia.org/wiki/Polish_notation ) for any logic. It is independent from 3- valued logic. (and I am a bit a fan of polish notation myself) The problem is in the 3-valued-ness, How do your (3 valued) truthtables look? And is every tautology (from these truthtables) a theorem of S5 and every theorem of S5 a tautology (from these truthtables)? in Lukasiewicz 3 valued logic $ ( P \to ( Q \to R)) \to ((P \to Q) \to (P \to R)) $ is not a tautology but it s a theorem in S5. so itdoesn't fit. – Willemien Mar 08 '14 at 07:58
  • The truth tables are the same as for those of Lukasiewicz (http://en.wikipedia.org/wiki/Three-valued_logic) except for the conditional. Use a modified conditional L(P->Q) instead, so that t → u and u → f are f instead of u. (P→(Q→R))→((P→Q)→(P→R)) does hold with this definition of the conditional. Not every tautology of S5 is a tautology, but many are, and not all of these tautologies are theorems of S5, but many are. Analogous does not mean identical. – Confutus Mar 08 '14 at 18:44
  • You cannot pick and mix like that, you need to have a 3 valued logic that is identical/ equivalent to S5 (or another Modal logic), something a bit analogous (whatever that may mean) is just not good enough. literally every theorem of your modal logic should be a tautology of your many valued logic and every tautology of your many valued logic should be a theorem of your modal logic, there is no space for a bit of fuzz here. (and especially not in both directions) – Willemien Mar 08 '14 at 20:40
  • I didn't think the Lewis Systems were Holy Writ. The tautologies of 3-valued modal logic are theorems of that logic. They just aren't identical to the theorems of S5, S4, K, or any other accepted system of modal logic. – Confutus Mar 08 '14 at 21:49
  • the problem is that they re not exactly the theorems of any modal logic (or you cannot fabricate a modal logic that is equivalent to them) – Willemien Mar 08 '14 at 21:55
  • It seems that "modal logic" is an undefined term here, and so are what properties a system needs to be have in order to be "good enough" to qualify. – Confutus Mar 08 '14 at 22:43
  • one of the properties is that all theorems of classical logic are tautologies, another that it is some way captures the meaning of neccesity and maybe you can come up with some more – Willemien Mar 09 '14 at 16:03
  • The law of the excluded middle doesn't hold in 3-valued logic, nor do any theorems equivalent to it. However, the theorems of 3VL do reduce to those those of classical logic in the cases where the third value can be excluded. – Confutus Mar 09 '14 at 19:08
  • ps there are many different 3 value logics don't treat tham as all the same, and yes if all connectives have only a 2 value outcomes it becomes a bi value logic – Willemien Mar 12 '14 at 08:23
  • There are two forms of the law of the excluded middle: an object-language one (OLEM) and a metalanguage one (MLEM). OLEM is: "For each model M and sentence A, M satisfies (A v ~A)." MLEM is: "For each model M and sentence A, M satisfies A or M satisfies ~A." Classical logic supports both OLEM and MLEM. There might be systems which support OLEM but not MLEM. The Lewis systems are defined proof-theoretically. It may be possible to devise a model theory for, say, S5 which supports OLEM but not MLEM. Is that a 3-valued system? Yes and no. – MikeC Mar 13 '14 at 15:54
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Have you seen in SEP Many-Valued Logic ? There is a reference to Melvin Fitting, Many-valued modal logics (I,II) (Fundamenta Informaticae, 15 and 17, 1991/92) : "considers systems that define such modalities by merging modal and many-valued logic, with intended applications to problems of Artificial Intelligence".

Also :

Osamu Morikawa, Some modal logics based on a three-valued logic, Notre Dame Journal of Formal Logic (1988)

and

KRISTER SEGERBERG, Some Modal Logics based on a Three-valued Logic, Theoria (1967).

Martin Sleziak
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Mauro ALLEGRANZA
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  • I am familiar with many of the surveys of many valued logic referenced: Ackermann, Rescher, Rosser and Turquette, Bolc and Borowic, and Malinowski. It is asserted that "Many-valued logic as a separate subject was created by the Polish logician and philosopher Łukasiewicz (1920), and developed first in Poland. His first intention was to use a third, additional truth value for “possible”, and to model in this way the modalities “it is necessary that” and “it is possible that”. This intended application to modal logic did not materialize. " My question is, why didn't it? – Confutus Mar 02 '14 at 17:35
  • Have you seen Jan Lukasiewicz, Aristotle Syllogistic From the Standpoint of Modern Formal Logic (1957), Ch.VII : The System of Modal Logic; in §49 The four-valued system of modal logic, page 166, he says : "Every system of modal logic ought to include as a proper part basic modal logic, i.e. ought to have among its theses both the M-axioms $CpMp$, $CMpp$, and $Mp$, and the L-axioms $CLpp$, $CpLp$, and $NLp$. [...] The functors $M$ and $L$ have no interpretation in two-valued logic. Hence any system of modal logic must be many-valued." – Mauro ALLEGRANZA Mar 02 '14 at 17:54
  • I've encountered it but not explored it in detail, because it was inconsistent with what I had found in the three valued case. I have speculated that one of the reasons the application was not successful was that he moved on to infinite-valued and the four valued modal system before fully understanding the simpler three valued case. – Confutus Mar 02 '14 at 19:33
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Suppose the value of a sentence at a model is one of {t, f, u}. Suppose further that a disjunction (p v q) has the value t if & only if at least one disjunct does, while a conjunction (p & q) has the value t iff both conjuncts do. Suppose Mp is t if p is either t or u, and f otherwise, while Lp is t if p is t, and f otherwise. Lastly ~p is t if p is f, u if u, and t if f. These are natural 3-valued semantics.

Next consider the formula:

L(p v q) & M~p & M~q

This says at least one of the two sentences { p, q } must be true, but either one of the two may be false. Given our understanding of natural language etc. this seems to be a formula which should be true at some models in our logic. But on the "natural 3-valued semantics" given above, no assignment gives this formula the value t.

This is a serious obstacle to a 3-valued modal logic. (Though in fact I believe this idea in general is promising; it's this particular version of it that fails.)

MikeC
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  • This is a disguised form of trying to discard the principle of the excluded middle on the one hand and keep it on the other. Let p and q be mutually exclusive, so that q = ~p. While M~p & Mp asserts that the principle of the excluded middle does not hold, L(p v ~p) is an assertion that it does. – Confutus Mar 03 '14 at 05:42
  • So it is an obstacle, or not? – MikeC Mar 03 '14 at 17:03
  • Trying to insist that contradictory statements are both true is indeed an obstacle. If p stands for "it's raining" and the value u means we don't know that it is, it's reasonable to assume that we also don't know that it isn't raining. If p means "wanda is a fish and q means "wanda is a bicycle", a consistent interpretation requires that we remain noncommittal about about "wanda is both a fish and a bicycle" and refrain from insisting that she cannot possibly be both, – Confutus Mar 03 '14 at 17:53
  • Upon reflection, it appears that the obstacles are more conceptual than technical, and some form of the question may be better posed on the Philosophy SE than here. – Confutus Mar 04 '14 at 08:08
  • Russell was certainly worried about sentences that were neither true nor false. His "theory of definite descriptions" was designed to make them go away, as I remember. See you on Philosophy SE. – MikeC Mar 04 '14 at 14:21
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I have identified a few obstacles.

  1. The prefix notation developed and used by Łukasiewicz is far less popular than the infix type of notation used in conventional and boolean algebra and by most logicians, where the "box" and "diamond" notation used by C.I. Lewis for modal logic is typically not used in investigation of Łukasiewicz logic.

This objection is one of familiarity.

  1. The 3-valued Łukasiewicz logic can be used to distinguish between necessary and impossible propositions on the one hand, and contingent, doubtful, or equivocal propositions on the other, but does not distinguish between contingently true and contingently false.

It would need require at least a four valued logic to make this distinction, but it is not so clear why this is essential to a first step. There is some reason to suspect that it would be useful to clarify the principles of the the three valued case first.

  1. The definitions of "possibility" and "necessity" in Łukasiewicz differ from those conventionally used in modal logic. In the usual semantics of modal logic, the formula:

A: L(p v q) & M~p & M~q

is desirable. This would assert that the disjunction of two statments may be necessarily true, while the individual statements themselves are not necessarily true. As an example, this might be interpreted as a claim that it is necessarily the case that Wanda is a fish or Wanda is a bicycle, but it is possible that Wanda is not a fish and Wanda is not a Bicycle. But in the Łukasiewicz semantics, no assignment gives this formula the value t. Similarly, the formula B: (Mp & Mq) & L~(p&q) is also thought sometimes desirable. This would claim that "It is possible that Wanda is a fish and Wanda is a bicycle, but it is necessary not the case that Wanda is both fish and bicycle".

This objection may be partly addressed by noting that if p and q are mutually exclusive so that q = ~p, this amounts to L(p v ~p) on the left side hand of expression A, and M~p & Mp on the right. Likewise, we have Mp & M~p on the left of expression B and L~(p&~p) on the other right. The Lukasewicz-consistent semantics would have it that that this is equivalent to asserting the law of the excluded middle on the one hand and denying it on the other, and furthermore, that this is a contradiction.

This objection has to do with a conflict between the dictates of the truth functional de Morgan algebra of the logic, and the requirements of conventionalized intution. It also has to do with a strong tendency to adhere to the law of the excluded middle, in spite of the numerous and venerable challenges to it. Although the conventional interpretations of modal logicians are long-standing and even well formalized, it would not be the first time that the algebra has been more consistent than intuition. It might be useful to compare and contrast intuitive objections in more detail.

  1. Another objection, raised by C.I. Lewis, is that in Łukasiewicz logic, the conditional Cpq does not support the most common and valuable rules of inference such as modus ponens and transitivity of the conditional.

This objection can be addressed by noting that this is because Cpq allows doubtful conditionals (those that have the value of u), and that these should should not be allowed in a system of valid deduction. Łukasiewicz himself overlooked this, and so did his critics. Adopting a definite or strict conditional LCpq is a simple definition that more than repairs the deficiency. It is apparently much too simple to be believed or investigated.

  1. It has been noted that that there is a proof that none of the Lewis systems of Modal logic can be reduced to a system of three valued logic: Dugundji, James. Note on a Property of Matrices for Lewis and Langford's Calculi of Propositions. Journal of Symbolic Logic 5 (1940), no. 4, 150--151. jstor, doi: 10.2307/2268175

From "many valued logic" by Reshner (1969). (Gregg Revivals page 192) "There exist no finitely many valued logic that is characteristic of any of the Lewis systems S1 to S5 , because any finitly many-valued logic will contain tautologies that are not theorems of S5 (and fortiory not of S1 to S4 either)

This objection is specious. The system of modal logic based on Łukasiewicz logic is not one of the Lewis systems. It is truth functional, which the Lewis systems are not, it rejects the law of the excluded middle as a universally truth, while the Lewis systems accept it, and it uses a different conditional than the Lewis systems. None of the Lewis systems reduce to it, it is not "characteristic of any of the lewis systems S1-S5", and it does contain tautologies that are not theorems of S5.

Martin Sleziak
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Confutus
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