# which branch of maths studies Standard Logical Matrices

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:

1. a set $S$ of values (four in this case)
2. a (proper) subset of $S$ called the designated values
3. 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?

• You can get certain particularly important kinds of generalized truth tables from Heyting algebras. These are related to intuitionistic logic, pointless topology, and topos theory. This part of mathematics is a mix of logic, topology, and category theory. Another keyword here is "subobject classifier." – Qiaochu Yuan Aug 5 '13 at 13:51
• Anyway, keep in mind that if you don't require anything then you probably can't prove anything. A set $S$ equipped with a binary operation $S \times S \to S$ satisfying no other requirements is called a magma ( en.wikipedia.org/wiki/Magma_(algebra) ), for example, but there is really not much to say about these. – Qiaochu Yuan Aug 5 '13 at 14:03
• @QiaochuYuan There exists 16 magmas of order 2. Up to isomorphism, there exist 10 magmas of 2 elements. Conjecture: The family of magmas M of a given finite order has more elements than M up to isomorphism. Conjecture: For any monomorphism M from a magma A-1 of a finite order n to a magma A-2 of order (n+k), where (n+k)>n, there exists a finite number of magmas of order (n+k) M is also a map from A-1. We can in principle say much more. – Doug Spoonwood Aug 5 '13 at 15:03
• There's a volume called "Theory of Logical Calculi: Basic Theory of Consequence Operations" by Ryszard Wojcicki. It has a 50+ page chapter on "Logical Matrices". I think it reaches the level of generality that you seek, but I can't quite tell, in part because I haven't read the entire book and what I've scanned/ quarter-read, I'm not sure I've understood to a meaningful extent. – Doug Spoonwood Aug 7 '13 at 16:02
• @DougSpoonwood (Generalized) Logical Matrices are studied in a field called Abstract Algebraic Logic. The author of the question might want to check this survey by Font, Jansana and Pigozzi. – J Marcos Aug 19 '13 at 16:51