categories: SYNTAX vs SEMANTICS ??

["F. William Lawvere" <wlawvere@hotmail.com>]

Several have asked whether categorical algebra can clarify the relation 
between syntax and semantics, two terms often paired in theoretical 
linguistics. The hope is that, as in previous work,  focusing on  
mathematical experience will give information that can be investigated 
further and which is more richly detailed than abstract speculation about 
cognition in general has been able to provide. Whatever serious discussion 
we can have here will be limited to mathematics in particular, although the 
mathematics may suggest  generalizations to other fields.

Semantics has been previously analyzed as the contravariant 2-functor which 
relates the abstract general and the concrete general aspects of each 
general concept which is admissible in a given doctrine. In some
doctrines semantics has an adjoint, called structure, a paradigmatic example 
of which is the structure that the system of cohomology operations naturally 
has. This adjoint does not have much directly to do with syntax, except 
insofar as the structure of something, being an abstract general, is in need 
of presentations to permit many kinds of calculation and reasoning about it.

The word SYNTAX has the same Greek root has TACTICS . That suggests thinking 
of syntax as the tactics for manipulating symbols , specifically the symbols 
involved in presentations of abstract generals. More generally we might also 
include the word problems associated to individual algebraic categories 
relative to underlying set functors, but that case of 1-categories is 
distinct from the 2-categorical case I am emphasizing here.

Presentations themselves are often objectified for mathematical study, and 
in all known cases that involves a choice of a further category with an 
adjoint pair connecting the latter with the category of objects to be 
presented. For example, in the doctrine where abstract generals are 
identified with single-sorted algebraic theories
a standard choice is the category of sequences of sets, often called 
"signatures", with the functor assigning to each theory its sequence of ( 
)-ary operations; the left adjoint is then determined and hence a monad T on 
the chosen category. That monad has a crucial further role, in the 
construction of the third category Pres(T) of presentations wherein the 
signatures also play a further role as AXIOMS : namely this "syntactical" 
category has as objects pairs G,R of "signatures" equipped with a pair of 
maps from R to T(G) . The presentation functor applies the left adjoint, 
then takes the coequalizer in the first category; this presentation functor 
might also be considered part of syntax.

Even if one considered (as was done for many years) that semantics was 
really the functor going all the way from syntax to the concrete generals, 
the fact that it has a preferred factorization would not remain concealed 
forever from mathematicians: the discovery of groups
revealed a rich content beyond permutation symbols, and calculations
with characteristic polynomials of linear transformations are clarified by, 
as well as present, the "abstract" rings whose representations are involved 
(There is a SLNM by Lambek about Linear Semantics).

In these terms the choice of home for the notion "theory" can be clarified . 
There are apparently FOUR reasonable possibilities :
1) The presentations of abstract generals. This syntactical emphasis
was the one most used for many decades by logicians.
2) The abstract generals themselves. This choice, exemplified by the term 
"algebraic theory", emphasizes that theories should be those objects which 
play the pivotal role of forming the one category which
is functorially linked to both syntax and semantics. (Note that there is 
usually no way of getting from concrete generals to syntax).
3) The concrete generals themselves. The word "theory" has only occasionally 
been used in this sense, but note that is philosophically analogous to the 
use of "homology theory" to signify an objective functor.
4) But when a group theorist refers to "group theory" he does not usually 
mean either 1) or 2), but more like
     Presentations of his Concrete General !
For 3) has also an algebraic structure, but in a 2- dimensional sense; 
specifically, in the doctrine of algebraic theories,etc., the concrete 
generals all have the natural 2-structure presented by filtered colimits, 
reflexive coequalizers, and all small limits. Thus we could start with any 
small family of groups and apply any composite of those operations, apply 
also any other such composite functor, and compare the results 
homomorphically.  2-syntax as "theory"?



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