Why did many valued logic fail in describing modal logic?

Question

The SEP article on many valued logic makes the following statement:

The introduction of systems of MVL by Łukasiewicz (1920) was initially guided by the (finally unsuccessful) idea of understanding the notion of possibility, i.e. modal logic, in a 3-valued way.

Why was this unsuccessful? It seems almost intuitive to assign a value x such that 0 < x <1 to describe an event that is possible but not true or certain. Isn't this what is being done in probability theory already, when assign probabilities to events from the [0,1] interval?

Answer 1

So the three valued logic of Łukasiewicz has three truth values {1,i,0}. Łukasiewicz was trying to solve the problem of future contigents with this logic.

His view is that statements about the past and present have an unalterable truth value, so if they are true they are necessarily true, if they are false they are necessarily false. Future contingents are assigned the value i, those statement are possible. Statement that are true are also of course possible. If we add a modal operator ◇ to the language we can formalize these ideas by the following truth tables.

╔═══╦═══╦
║ A ║◇A ║ 
╠═══╬═══╬
║ 1 ║ 1 ║ 
║ i ║ 1 ║ 
║ 0 ║ 0 ║ 
╚═══╩═══╩

Now defining ◻A, as ¬◇¬A you get the table

╦═══╦═══╔
║ A ║◻A ║ 
╠═══╬═══╬
║ 1 ║ 1 ║ 
║ i ║ 0 ║ 
║ 0 ║ 0 ║ 
╚═══╩═══╩

One reason why this is considered a failure is that it has strange consequences for a modal logic. For example in this logic you can show (after the other truth tables and validity have been defined) that ◇A,◇B ⊨ ◇(A & B). That is not intuitively acceptable for a logic of possibility.

See An Introduction to Non-Classical Logic (Priest) chapter 7 from where my answer was taken almost verbatim. It also contains a proof of the claim mentioned by Bumble.

Answer 2

The specific quotation you gave about Łukasiewicz refers to the fact there was an attempt to understand intuitionistic logic as a many-valued logic, but this failed because Gödel proved in the early 1930s that intuitionistic logic is not n-valent for any n.

To address your last paragraph, truth and certainty are quite different things. To ask, to what degree is this true? is different from asking, how likely is it that this is true? The problem is that these are often conflated: people think that because there are some propositions such that they don't know whether they are true or not, a three-valued logic can be used with values true, false, and unknown. Enter the confusion. Unknown (or undetermined, or indeterminate) is not a truth value: it is an epistemic status. Undefined is not a truth value either, but a lack of one.

The confusion can be avoided by treating degree of truth and degree of certainty separately. One of the popular approaches to degree of truth is fuzzy logic, which has been used to account for vague predicates and to explain Sorites problems. Degree of certainty, on the other hand is already covered by probability, provided we understand probability epistemically as a measure on information.

Answer 3

It was unsuccessful because nobody invested enough time and thought to develop things to a point where they could produce useful results. And why did nobody do that? Maybe it was because everybody who took a close look came to the conclusion that it is a dead end (or that success is very unlikely).

You came up with the very very first step: Assign a value 0 < x < 1. Now what's the second step? How would you for example connect the statements "X is a man" (not sure about that), "X is married" (not sure about that either) and "X has a wife" (which doesn't follow completely from the first statements, which would surely make this a suitable situation for many-followed logic), but using rules that don't require you to examine the statements, but only their truth values?

Answer 4

There are several possible reasons.

1) There is considerable resistance to rejection of the "laws" of bivalence and the excluded middle. For instance, an early and persuasive objection to the use of the third logical value for the representation of future contingents depends on whether the law of the excluded middle is accepted or not.

2) The technical requirements of possibility and necessity are at odds with those traditionally recognized in modal logic and philosophy. There are other difficulties of interpretation of the system. The Lewis systems and their kin have achieved such acceptance that an alternative has to demonstrate substantial superiority.

3) In Lukasewicz logic, various rules of inference of classical logic, notably modus ponens and the rule of transitivity are not tautologies, which seriously inhibits the practical usefulness of this logic. An acceptable theory of deductive inference using this logic has not been worked out.

4) It has been proven that a 3-valued logic is incompatible with both intuitionism and modal logic as they have been developed.

I have reason to believe that none of these objections is inherently insuperable, but collectively they represent formidable psychological barriers to serious consideration of modal logic based on Lukasiewicz 3-valued logic.