# Is Belnap's four-valued logic technically a relevance logic?

Belnap, the American Logician, constructed a four-valued logic which is a form of relavance logic; interestingly the truth-values it takes are:

• true

• false

• both true & false

• neither true nor false

This, of course, reflects the Buddhist tetralemma or the positive configuration of the Catuskoti (चतुष्कोटि);

Its semantics

is designed to cope with multiple information sources such that if only true is found then true is assigned, if only false is found then false is assigned, if some sources say true and others say false then both is assigned, and if no information is given by any information source then neither is assigned.

Now, in what way is this technically a relevance logic - is it substructural or modal for example? To what extent are the usual (boolean) laws of logic preserved?

Its also worth asking here what are the connections between these information sources and the truth-values that it determines; is the best way to show this explicitly through truth-tables?

• If some information sources are found true, and some are found false, shouldn't 'false' be assigned? Isn't the statement "the sky is blue and the grass is blue" false? Jul 2 '14 at 0:01
• @matt: Well that is true for classical logic; but this is a heterodox logic; I'm still puzzling over the connection between information sources and truth-values myself - its probably useful to add that in as a sub-question. Thanks for picking up on that. Jul 2 '14 at 0:08
• @MoziburUllah I know you are talking about MU-SHI MU-SUM, but I do not know exactly what you are trying to ask about it. acidharma.org/aci/online/_media/text/course1/C01Notes.pdf 3rd page down. If you could rephrase or reword your question, I might be able to answer. Jul 17 '15 at 0:44
• @hellyale: I'm not asking about how the cautoskoti is used in Buddhism, at least not here; I'm just noting the parallel here with Belnaps logic; the question is really about this logic, and why it's called a relevance logic. Jul 17 '15 at 10:46
• (a comment, because I don't trust myself enough to make an answer without a discussion first) It does not appear to be a substructural logic because if you consider the subset of the logic dealing only with true and false, all of the normal structures are present. The "reference logic" approach might be possible. Consider that conflicting statements retain some connection to their original sources of information. Dec 30 '15 at 19:38

To what extent are the usual (boolean) laws of logic preserved? Its also worth asking here what are the connections between these information sources and the truth-values that it determines; is the best way to show this explicitly through truth-tables?

It can exist as normal, propositional logic if you look at it in the appropriate context:

"...if some sources say true and others say false then both is assigned, and if no information is given by any information source then neither is assigned..."

If separate variables are assigned for each source, for example:

``````Source X says "it is day"
Source Y says "it is night"

D = it is day
N = it is night
``````

Then they can be evaluated using normal rules of logic because in the context of X, D is true and in the context of Y, N is true. These no longer contradict. Yet, if evaluating it like so:

``````All sources = A, which is composed of X and Y:
source X says "it is day,"
source Y says "it is night"
D = it is day
N = it is night
``````

In this case, for A, both D and N are simultaneously both true and both false. This is nonsense and cannot be properly evaluated using normal logic. If desiring to show that sources contradict, A could be said to fall under those two categories. But, when actually applying the rules of logic, it is necessary to break down A into the component parts X and Y, so it can be properly evaluated.

Most of my experience is in propositional logic, but from the articles here on Relevance Logic and Modal Logic:

-- it feels more like Relevance Logic to me. The "it is necessary..." "it is possible..." etc., don't feel like they relate as much to the logical system you outlined. The notion of multiple worlds, as mentioned in the article on Relevance logic, seems more conducive to reconciling things:

"Like the semantics of modal logic, the semantics of relevance logic relativises truth of formulae to worlds." (from the article mentioned above)