Puzzles and nonclassical logic

All happy families are alike; each unhappy family is unhappy in its own way.

Lev Nikolayevich Tolstoy, Anna Karenina 

Applying the famous beginning of one of the finest novels of all times to logics, there is no doubt that classical logic plays the role of happy families, while nonclassical logics get to play the unfortunate, yet intriguing parts of unhappy families. Put otherwise, there are plenty generalizations of classical logic all gathered under the same roof named "nonclassical logic", which often differ one from another at least as much as they differ from the "happy" classical logic.

Probably the most fundamental property of classical logic is that there are exactly two truth values: 1 and 0. Moreover, any given statement is either TRUE or FALSE (though maybe we do not always know which). This concept is very natural and something that humans tend to a priori assume. If anything, this must hold. Historically speaking, mathematicians also used to assume this binarity of truth, and for a long period they saw finding a set of axioms from which any possible mathematical statement could be either proven or disproven as their ultimate goal. However, Kurt Gödel destroyed this dream once and for all when he proved that there are and there always will be (i.e. no matter how much we improve our theory) mathematical statements such that their veracity cannot be determined within the same theory. In classical mathematics, the Continuum Hypothesis is an example of such a statement.

One of the first things that appear when passing to nonclassical setting is thus that we can have more truth values. For instance, we can have three truth values: 0, ½ and 1, where ½ is an in-between value representing statements for which we cannot tell whether they are true, or perhaps statements that are sometimes true, i.e. they are true in certain moments or in certain contexts. There are several ways to formalize a three-values logic, one of them being the Łukasiewicz three-valued logic. 

Łukasiewicz three-valued logic

Łukasiewicz three-valued logic is an extension of classical logic in the sense that the value of combined statements containing only simpler statements with values 0 or 1 is evaluated exactly as in classical logic. What is new are statements having value ½.

Let's refer to statements having a truth value 0 as TRUE, statements having truth value 1 as FALSE and statements having a truth value ½ as MAYBE TRUE. Then the following determines the truth value of conjunction and disjunction involving the third value:

TRUE and MAYBE TRUE = MAYBE TRUE

MAYBE TRUE and MAYBE TRUE = MAYBE TRUE

MAYBE TRUE and FALSE = FALSE

TRUE or MAYBE TRUE = TRUE

MAYBE TRUE or MAYBE TRUE = MAYBE TRUE

MAYBE TRUE or FALSE = MAYBE TRUE

If for instance the sentence Today is Friday is TRUE, and the sentence It is raining is MAYBE TRUE, then the sentence

Today is Friday and it is raining.

is MAYBE TRUE, while the sentence:

Today is Friday or it is raining.

is TRUE.

It gets more interesting when we introduce negation into the picture. While as in classical logic the value TRUE and FALSE are negations of each other, we need to pick a negation of the value ½. In Łukasiewicz logic this is set to be ½. So, if the statement It is raining is MAYBE TRUE, then so is its negation It isn't raining.

How about implication? We saw in the previous post that in classical logic all implications have value 1, with the only exception being statements of the form 

1 → 0

which have value 0. How about statements like:

If it is Friday then it is raining.              (*)

The following rules hold:

0 → ½ = 1
½→ 0 = ½

½ → ½ = 1

½→ 1 = 1

1 → ½ = ½

Given that today it really is Friday and it is maybe raining, the above statement (*) thus is MAYBE TRUE.

There exist logical puzzles built on three-valued  Łukasiewicz logic, and they can be found on Jason Rosenhouse's blog Non-classical Knights and Knaves or in certain papers by the same author, for instance:

J. Rosenhouse, Knights, Knaves, Normals and Neutrals, The College Matematics Journal, Vol. 45, No. 4 (2014), 297--306.

Rosenhouse considers a fictional island called Tripeldonia. In addition to always speaking the truth Knights and always lying Knaves, Tripeldonia is also inhabited by the Neutrals (being defined as a transitional state between Knights and Knaves). Any statement uttered by a Neutral has value ½, which is interpreted as uncertainty. 

My attention was brought to Rosenhouse's puzzle by Lieven Le Bruyn's fabulous blog neverendingbooks, which is a must read to any mathematician or mathematical enthusiast.

While dwelling into Le Bruyn's blog I suggest you also read the post on Smullyan and the President's sanity, and see if you can solve the puzzle about the President's mental health. Any resemblance to actual presidential couple(s) is of course purely coincidental 😉.