6

The description of Product systems in the SEP entry on Many-Valued Logic uses Dunn/Belnap's 4-valued system as an example of a product system:

In this way, the truth degrees of Dunn/Belnap's 4-valued system can be considered as evaluating two aspects of a state of affairs (SOA) related to a database:

  1. whether there is positive information about the truth of this SOA or not, and
  2. whether there is positive information about the falsity of this SOA or not.

Both aspects can use standard truth values for this evaluation.

(Edit: After reading Schiphol's answer, I can understand why it makes sense that negation leaves the degrees {⊥ , ⊤} and ∅ fixed for this interpretation. I even got the impression that "set values" model this "whether there is" type of information quite naturally.)

In this case, the conjunction, disjunction, and negation of Dunn/Belnap's 4-valued system are componentwise definable by conjunction, disjunction, or negation, respectively, of classical logic, i.e. this 4-valued system is a product of two copies of classical two-valued logic.

It seems to me that for this semantics, the negation should exchange the degrees {⊥ , ⊤} and ∅. However, in the description of Dunn/Belnap's 4-valued system in the same article a conflicting definition of negation is given:

A negation is, in a natural way, determined by a truth degree function which exchanges the degrees {⊥ } and {⊤}, and which leaves the degrees {⊥ , ⊤} and ∅ fixed.

A search with google seems to indicate that this is indeed the correct definition of negation for this system. But then, how can this system be a product system?

3

the ... negation of Dunn/Belnap's 4-valued system [is] componentwise definable by negation ... of classical logic, i.e. this 4-valued system is a product of two copies of classical two-valued logic. [my emphasis]

This means that, in order to apply negation to the different truth values, you apply it (classically) to each of its components:

  • When you apply it to {⊤}, you end up with {~⊤}, that is, {⊥}
  • When you apply it to {⊥}, you end up with {~⊥}, that is, {⊤}
  • When you apply it to {⊤,⊥}, you end up with {~⊤, ~⊥}, that is, {⊤,⊥} (remember that these are sets, not ordered duples).
  • When you apply it to ∅, you apply it to all of its components (that is, none), which leaves the empty set as it is.

EDIT: The above is how one can see that negation as suggested by Belnap and Dunn is reasonable. But I agree with the questioner that it is unclear that the logic, so described, is a product system.

A bit more detail on how to see Dunn/Belnap's system (DB henceforth) as a product system:

In DB we have four truth degrees:

  1. ∅ -- this means: there is no information concerning a certain state of affairs [SOA henceforth]
  2. {⊤} -- this means: there is information saying that the SOA obtains
  3. {⊥} -- this means: there is information saying that the SOA fails
  4. {⊤,⊥} -- this means: there is information saying that the SOA fails and information saying that it obtains.

What the author of the SEP article claims (Siegfried Gottwald, incidentally the author of a popular Treatise on Many-Valued Logics) is that BP can be profitably understood as a product system: we take BP's truth degrees as evaluating whether

a. There is positive information about the truth of the SOA.

b. There is positive information about the falsity of the SOA.

We can track these two aspects with an ordered duple: ; a will be true [t] iff there is positive information about the truth of the SOA, and false [f] otherwise. Mutatis mutandis for b. If so, there is the following mappings of BP truth degrees onto the ordered pairs (represented with square brackets and not the habitual corner brackets; mysteries of the SE markup) of our product system:

  • ∅ maps onto [f,f] -- no positive information about the truth of the SOA, nor about its falsity.
  • {⊤} maps onto [t,f] -- positive information about the truth of the SOA, no positive information about its falsity.
  • {⊥} maps onto [f,t]
  • {⊤,⊥} maps onto [t,t]

But, of course, if this is the intended mapping and, e.g., negation is to be calculated as classical negation over the two components, ~ ∅ is {⊤,⊥} and vice versa, as the question rightly claims.

| improve this answer | |
  • Thanks for the answer. I'm not convinced however. For example, the language isn't right. The description of Product system talks about k-tuples and components, but your interpretation refers to sets and elements. This set formalism is also a nice notion (and it even seems to match well with the given interpretation of the Dunn/... system), but is this really (one of) the intended meaning(s) of Product system? – Thomas Klimpel Jul 22 '12 at 8:44
  • I added (Edit: After reading Schiphol's answer, I can understand why it makes sense that negation leaves the degrees {⊥ , ⊤} and ∅ fixed for this interpretation. I even got the impression that "set values" model this "whether there is" type of information quite naturally.) to my question at the appropriate place. So I understand the Dunn/... system better now, but my question remains whether this is really a good example for a Product system. – Thomas Klimpel Jul 22 '12 at 9:04
  • 1
    @ThomasKlimpel, I actually think you are right that there is something unclear here. I provide a bit more detail on the terse remarks in the SEP article. Maybe we can figure this out together! – Schiphol Jul 22 '12 at 20:21

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.