Talk:Principle of bivalence

message: The story custed by the contingent. A boy caught a butterfly and asked the teacher to answer whether it is alife keeping the insect between his hands. If teacher tells "It is dead", he will let the butterfly to fly away. If teacher tells "It is alife" he will imperceptibly squash the poor insect to fool the teacher otherwise. The wise teacher got to the core of the trap and answered: "Everything is in your hands". Excuse me if it is inappropriate for bivalence. Peahaps it links to the liar paradox, since present state manipulates the future. --Javalenok 11:03, 15 September 2006 (UTC)
thread_title:  A story about Clever Teacher 

message: I've done much fixing of the article, but the vagueness section is still much in need of repair. — Charles Stewart (talk) 19:58, 30 June 2009 (UTC)
thread_title: Vagueness

message: From my recent revert:
*There is no n-ary generalisation of bivalence worth noting.  Bivalence is not about truth tables, it is about truth.  To apply bivalence to a many-valued logic, one identifies some set of propositional values as true, and the others as not true.
*"Gap theory" means Gap creationism (20 000 ghits vs 46).  I don't think it can be right to talk of a particular theory or family of theories identified only by rejection of bivalence.  But the content here is worth rescuing:
 A theory that rejects bivalence is sometimes called `a gap theory'. Gap theories allow that some sentences may lack a truth value, or that there are more than two semantic values. Some motivations for rejecting bivalence are:
 * Intuitionism: If truth is grounded in verification or proof conditions, then since some sentences are undecidable or undecided they are neither true (provable) nor false (refutable).
 * Irreferential (non-denoting) singular terms: If a sentence involves an irreferential singular term like 'Pegasus', it is truth-valueless ("gappy");
 * Future contingents (see next section);
 * Paradoxes: E.g. Kripke's theory of truth deems the liar sentence gappy;
 * Vagueness (see below).
*The discussion in "Bivalance and non-contradiction" seems useful to me.  I don't see why nearly all of it should be deleted.
The point about Boolean subalgebras is a good one.  It may be worth having a section on algebraic semantics that explains why Stone duality has led some to reject bivalence. — Charles Stewart (talk) 08:06, 3 July 2009 (UTC)
thread_title:  Some issues with semantics of bivalence & gappiness 

message: Bivalent logic redirects here - what does it mean? Is it correct to say bivalent logic describes the set of logic systems that have only two values, and binary logic would be an exemplary member of this set? Are bivalent logic and two-valued logic synonymous? --Abdull (talk) 21:55, 8 February 2010 (UTC)
replies: text: Yes, bivalent and two-valued are synonymous. Binary is also a synonym, but "binary logic" is usually used in digital circuit engineering, so it usually means propositional two-value logic. Perhaps that one should be redirected elsewhere, like to logic gate. Tijfo098 (talk) 10:12, 28 March 2011 (UTC)

text: I have made binary logic a dab. Tijfo098 (talk) 10:50, 28 March 2011 (UTC)
thread_title:  Bivalent logic 

message: Principle of bivalence →  — Occurs twice as often in google books compared to the current title. Tijfo098 (talk) 06:32, 28 March 2011 (UTC)
replies: text: You're right, my proposal would inadvertently narrow the scope of the article to the more technical issues. So I'm withdrawing the move request. Tijfo098 (talk) 10:46, 30 March 2011 (UTC)
thread_title:  Rename request 

message: Principle of bivalence → 
* There are quite a few Google books hits which talk of this topic without "principle of" or using "law of" instead etc. So it seems more natural unprefixed. (Compare with extensionality, which also has "principle of" appearances). Tijfo098 (talk) 11:06, 30 March 2011 (UTC)
replies: text: My dictionary (Merriam-Webster's 9th Collegiate) had only "bivalent" -- the chemical defintion above. If anyone has access to the full Oxford English Dictionary (I don't) we'd probably have to accept their findings as definitive. A quick look at the on-line available version agrees with Tijfo098. Bill 23:38, 9 May 2011 (UTC)
replies: text:  It's not hard to derive the noun with that meaning, and one scientific dictionary does that. So, it's perhaps best to leave the article at the current name which is unambiguous, even if "bivalence" is seldom used with the chem meaning. Tijfo098 (talk) 02:55, 10 May 2011 (UTC)
thread_title:  New request 

message: * Betti Arianna (2002) The Incomplete Story of Łukasiewicz and Bivalence in T. Childers (ed.) The Logica 2002 Yearbook, Prague: The Czech Academy of Sciences—Filosofia, pp. 21-26
* .
* Jean-Yves Béziau (2003) "Bivalence, excluded middle and non contradiction", in The Logica Yearbook 2003, L.Behounek (ed), Academy of Sciences, Prague, pp.73-84.
-- Tijfo098 (talk) 20:37, 14 April 2011 (UTC)
thread_title:  A few other interesting readings 

message: Neither philosophy nor mathematics are fixed in time; as knowledge advances, certain things that had been previously "obvious" and taken for granted, such as "bivalence==excluded middle", are eventually seen to not be the case, and the subtle differences are articulated. The proper approach is probably a historical one: describe what was thought, when, by whom, and why (and why they didn't see that they were wrong).  For example, a new, modern 20th century conception is autoepistemic logic, which rejects the law of the excluded middle, but keeping bivalence: a third possibility is that the truth of a proposition is unknown.  In such systems, the LoEM can be asserted as a fact, but only if one wishes -- it is not built into the system, a-priori, by default.  Aristotle would have understood the arguments for this, but wouldn't have appreciated the significance for computer science and chip design :-) linas (talk) 16:53, 10 June 2011 (UTC)
thread_title: A historical approach is best?!

message: For those that feel that the discussion is not formal enough, there's an abstract algebraic logic framing of many principles related to sentential logics. This are indeed more complicated if one considers non-truth-functional operators. See formalization  and turn the page for quite a list of few principles! Tijfo098 (talk) 19:05, 15 April 2011 (UTC)
thread_title:  AAL to the rescue (or not) 

message: In response to my example above of "It is raining" with Paris and London being the possible worlds, User:Tijfo098 responded with "When you start speaking of interpreting a proposition in Paris or London you are already talking of possible worlds, thus modal logic (of some kind)" and asked about incorporating those worlds into the language, using examples like "Alice knows it's raining in Paris" and "Bob knows it's not raining in London".  However modal logic does not work that way, it does not name worlds.  In a multimodal logic like dynamic logic it may name relations, but by default a model of modal logic is a Kripke structure, meaning a set W of possible worlds and a single binary relation R on W of accessibility between the worlds, which is named implicitly only by the modalities \Box and \Diamond.
A standard translation of intuitionistic logic into modal logic renders conjunction P∧Q and disjunction P∨Q and their units 1 and 0 classically, i.e. without modification, but interprets implication P→Q as \Box(P→Q), meaning "In all worlds accessible from here, P→Q holds".  (Think of accessible worlds as conceivable worlds and read intuitionistic P→Q as "necessarily P→Q.")  The intuitionistic meaning of negation ¬P, which merely abbreviates P→0, then becomes \Box(P→0), that is, "P is inconceivable" (since the P→0 inside the box is interpreted classically).
Now if (Paris,London) is in the relation R but not (London,Paris), this means that Londoners find Paris inconceivable but not conversely.  More specifically with regard to rain, Parisians could understand that even though it is not raining in Paris it could be raining in London.  Londoners however cannot imagine that it could be raining in Paris but not in London, because Paris is not a world they can contemplate as an alternative to their own.
It follows that in London, either it is raining or it is inconceivable that it is raining.  In Paris however it is not the case that either it is raining or it is inconceivable that it is raining, because Parisians can conceive of rain falling on London even though it is (classically) not raining in Paris.
Hence we have found a world in which LoEM fails, namely Paris.  LoEM does hold in London assuming this R, but one world is sufficient to rule out a law as a universal truth.
But if Parisians lost their ability to imagine London as an alternative world to their own, so that R became a discrete relation connecting no two distinct worlds (I'm assuming R is at least reflexive, and usually transitivity is assumed too, i.e. an S4 modal logic), then LoEM would become a universal truth, that is, either P holds or P is inconceivable (the meaning of intuitionistic-not-P), because as a "predicate transformer" \Box would then be the identity operation on predicates, so that the translation of intuitionistic P→Q to modal \Box(P→Q) would be semantically equivalent to classical P→Q.
One thing to be aware of is that no finite number of worlds can capture exactly intuitionistic logic in this way, but only something intermediate between classical (Boolean) logic and intuitionistic logic.  Only an infinite Kripke structure can give the exact semantics of intuitionistic logic via the translation of intuitionistic P→Q to modal \Box(P→Q).  However two worlds are enough to find counterexamples to LoEM.
This is food for thought about the relationship between LoEM and bivalent logic.  With the S4 translation of intuitionistic logic, whether LoEM holds or not can depend purely on the accessibility relation without changing any other detail of the logic.  --Vaughan Pratt (talk) 05:09, 17 April 2011 (UTC)
thread_title:  Remark on modal logic 

message: Making a note here to mention that topi in the vagueness section . Tijfo098 (talk) 07:05, 22 April 2011 (UTC)
thread_title:  Degrees or truth preservation 

message: The Logical Foundations of Metaphysics discusses bivalence. Kiefer.Wolfowitz 01:49, 15 November 2012 (UTC)
thread_title:  Dummett: Not just tarot