[FOM] "Global" versus "local" dimensions in logic

[Joao Marcos]

> This is a good opportunity to mention an important fact that almost
> all textbooks I know hide or ignore. In my paper "Simple Consequence
> Relations" (Information and Computation 92, 105-139, 1991), as well
> as in my contribution to the Volume "What is a Logical System (edited
> by D. Gabbay) I emphasized (though I dont claim any priority here,
> of course) that there is no single "First Order Logic",
> but there are TWO different ones. They share the same class of languages,
> and have the same set of logically valid formulas, but they differ
> in their consequence relations. According to one (which I have called
> the "truth consequence relation"), a formula A follows from a theory T
> if every pair <M,v> (where M is a structure for the corresponding
> language, and v is an assignment in the domain of M) which satisfies all
> the formulas in T  satisfies also A. According to the other
> (which I have called the "validity consequence relation"),
> a formula A follows from a theory T if A is valid in every structure M
> (i.e.: satisfied in M by every assignment v) in which all formulas
> of T are valid.

In the philosophical literature, truth-preserving rules are often called
"inference rules", and contrasted to to validity-preserving rules, known as
"deduction rules".  A more widely adopted denomination nowadays calls the
former "local rules" and the latter "global rules", and a local / global
consequence relation is one defined exclusively by collecting local / global
rules.  These are like 2 dimensions of a logic, but there might of course be
more, depending on which logical features you want to quantify over.

There is a wealth of papers in which the difference between local and global
consequence relations are explicitly stated, among others all the papers
produced in the last few years by the "Portuguese school" that investigates
combinations of logics (check for the word "fibring" at
http://wslc.math.ist.utl.pt/s84.www/cs/clc/publist/publist.html).  Arnon asked
though about BOOKS that do mention that.  There are at least 2 that I can
remember of, namely, Blackburn et al's "Modal Logic" and Rybakov's
"Admissibility of logical inference rules".


>   It is worth noting that these two consequence relations are
> not peculiar to FOL, and the difference is not
> due to the presence of quantifierss, but to the use
> of variables inside the formal language.

I do not see in fact why this distinction would be a privilege of first-order
logic.  Arnon's examples about the deduction theorem and the uniform
substitution, for instance, clearly show the usefulness of such a
discrimination already at the propositional level.  What would be the
"variables" in the deduction theorem, or in the necessitation rule in modal
logic (a rule that holds globally but not locally)??


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Why is it that such a distinction between local and global consequence
relations is in general overlooked in the literature?  I believe there might
be several reasons for that, including:

(1) the coincidence between the set of local rules and the set of global rules
in classical propositional logic, and in a few other more usual non-classical
logics

(2) the fact that most logics people work with are compact and respect some
form of the deduction theorem, what allows for the fundamental logical notion
of *inference* to be substituted by the poorer notion of *theoremhood* (the
local and the global consequence relations as defined in Arnon's message
coincide over the set of theorems of a logic)

(3) the fact that sequents for usual logics are most of the time finitary and
often can count on some form of the deduction theorem, reducing this to reason
(2)

(4) the more comprehensive fact that, while the local and the global
dimensions are clearly distinguishable in a Hilbert-style axiomatization, they
become transparent in sequent-style systems

(5) that the distinction is often emphasized for syntactical reasons, but
makes more sense from a semantical perspective

Perhaps other people from the list will see other reasons?  Perhaps they will
see reasons to disagree that this is a relevant issue?


Joao Marcos
http://www.geocities.com/jm_logica/