[FOM] Question of the Day: What is a Logic?

[Thomas Forster]

The thread seems to have veered off the point of Erik's question, and it 
might be an idea to return to it.

I was brought up to believe that a theory was a set of formulae closed
under logical consequence and a logic is a theory closed under uniform
substitution. Thus a logic that says that a binary relation R is
transitive myst say that all binary relations are transitive.  A theory is
intended to capture *truth* (set of allegations true in a given structure
is a standard example of a theory) whereas a logic is intended to capture
validity.

    I am not quite sure who taught me this - and i know there are people 
who don' like this characterisation (My colleague Martin Hyland for one) - 
and I would be interested to know where this picture somes from.

On Thu, 5 Oct 2006, Erik Douglas wrote:

> 
> I have two questions for the erudite members of this list from the
> foundations of logic.  Some of the answers may extend outside the range of
> topics appropriate to this list, but I would be very grateful all the same
> for every response; I am a philosopher by trade, and so I am fearless if not
> also a bit foolish.  My email is erik at temporality.org and I will also
> eventually compile the answers in an informal discussion on the web (and
> possibly more).
> 
> 1. What are the qualifications on a (formal) language to be a logic? 
> 
> In general, a logic appears to belong to a subclass of formalized languages,
> in a larger category of symbolic systems used to communicate.  I do not mean
> to ask the rather obscure philosophical questions about what it means to
> communicate, or even in what symbolic systems consist.  Rather, I accept
> languages as primitive, and moreover that some are *formal* (explicit and
> unambiguous in their application).  Also, it seems important that such
> formal languages have both syntactic and semantic dimensions, that we can
> identify something like *expressions* in the syntax, as well as some minimal
> distinction between *semantic* kinds amongst those expressions (e.g., TRUE
> and FALSE). 
> 
> The question is, given these minimal criteria, what further conditions
> qualify, generally, a particular formal language as a logic?   There is a
> surprising paucity in the literature addressing this question (I have found
> some references, but I would also be grateful for any and all in your
> responses).  At some level, the matter appears to be conventional or
> metaphysical (depending significantly on your religious inclinations).
> However, in either case, I would be very grateful to know your operative
> (and/or cherished) beliefs here.
> 
> The second question (below) then turns on what is often assumed as such a
> fundamental condition, the law of non-contradiction.  It seems other laws
> that were at one time accepted as etched in stone, such as the law of
> excluded middles, have given way to intuitionist and fuzzy logics.  In
> recent years, we have seen the construction of several variety of
> *paraconsistent* logics which do prima facie deny the law of
> non-contradiction.  So my question is as follows, and I have in mind that
> many of the folks on this list are mathematicians:
> 
> 2. Suppose I present a paraconsistent logic that I claim suffices as a
> foundation for mathematics (or some significant part thereof), what (other)
> properties must it have for you to accept it as such?
> 
> Note on my motivation:
> 
> I am in the business of attempting to construct alternative logics, notably
> paraconsistent logics, with an eye to addressing certain issues that sit
> uneasily these days between philosophy, metaphysics, mathematics, and
> physics, especially those that turn on how we understand *time* and *mind*.
> There are other applications in the development of a universal logic that
> also motivate my investigation here.
> 
> Cheers,
> Erik
> 
> 
> 
> 
> 
> 
> 
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