In classical logic you have truthtables like:
& | T | F ---|---|--- T | T | F F | F | F
In many valued logic you have tables like:
(this one is of four valued Łukasiewicz logic)
Cpq q= | 0 | 1 | 2 | 3 || Np || Designated --------|---|---|---|---||----||----- p = 0 | 3 | 3 | 3 | 3 || 3 || no p = 1 | 2 | 3 | 3 | 3 || 2 || no p = 2 | 1 | 2 | 3 | 3 || 1 || yes p = 3 | 0 | 1 | 2 | 3 || 0 || yes
Gottwald in ( http://plato.stanford.edu/entries/logic-manyvalued/ )
calls them "Standard Logical Matrices", and I also have seen names like Cayley matrices, and many valued truthtables.
But i was wondering is there an other name for them and which branch of mathematics does study them in depth?
In short the only criteria for these tables are:
- a set $S$ of values (four in this case)
- a (proper) subset of $S$ called the designated values
an (unrestricted) number of functions:
$S \to S$
$S \times S \to S$
and maybe even $S^n \to S$
That are all fundamental criteria,
(for Standard Logical Matrices there is an extra truthp reservation criteria but can be overlooked here)
I had a look into abstract algebra / universal algebra/ algebraic structures but all these fields seems only to look into systems where the matrices have to submit to some strict criteria, or only allow a very limited amount of functions.
Is there a branch of mathematics that studies these matrices in a more general way? (and what is a good introduction to this branch?)
in many valued logic you study only one (set of) Standard Logical Matrices at a time, but i am wondering is there a branch of mathematics that study them in a more general way?