# [FOM] More on first-order logic with operation constants

A.P. Hazen a.hazen at philosophy.unimelb.edu.au
Sun Dec 5 03:38:16 EST 2004

```John Corcoran has asked a series of questions about the history of

>In Hilbert-Ackermann 1938 and in Church 1956, addition is is treated as
>a 3-place relation. Kleene 1952 and Mendelson 1956, on the other hand,
>treat addition as a 2-place operation.
>Q1.When did the logic community, as opposed to the algebra community,
>come to recognize the "legitimacy" of taking operation symbols as
>primitive symbols?

---> Did they ever NOT recognize the legitimacy of function symbols?
To give one example, Herbrand's 1931 paper (the one that comes AFTER
Gödel in Van Heijenoort's "From Frege to Gödel") gives a logical
treatment of an  axiomatization of arithmetic with = as the only
predicate but with assorted function symbols.

>Q2.What was the motivation, philosophical or otherwise, of treating
>operations as if they were "really" relations?

---> CONJECTURE: The primary motivation was just to simplify things.
essential increase in expressive power.  (Valuable economies in terms
of length and readability of formulas, but "in principle...".)
Quine, and Russell for that matter, had philosophical reasons
for thinking it was interesting and important that you COULD make do
with predicates: avoiding the semantic puzzles of negative
existentials.  But I'd bet that these weren't the primary motivations
for (e.g.) Church to focus on logic formulated with predicates rather
than function symbols.  (The discoverer of lambda-calculus and the
functional version of simple type theory couldn't have thought there
was THAT much wrong with function symbols!)

>Q3.Was there ever any discussion in the mathematical or philosophical
>literature of whether the operation had the same ontological or logical
>status and legitimacy as the relation, or whether operations are
>"really" relations or are in some other way inferior or improper?

---> The only thing that comes to mind is Quine's discussion of the
"elimination of singular terms": cf. his "Word and Object" and the
exchange with Strawson in Davidson & Hintikka, eds., "Words and
Objections."  But this is far too late!  If there was a discussion in
the "formative" period of modern mathematical logic (1920s-1950s), I
don't know it.

>Q4.Do the above issues relate to the issue of whether it is more
>"proper" to call first-order logic a "predicate calculus" or "functional
>calculus"?
>Q5.Has anyone of note expressed in print reasons for prefering one of
>the six expressions'logic'[unmodified], 'predicate logic', 'predicate
>calculus', 'functional logic', 'functional calculus', 'quantification
>theory'to be modified by 'first-order'as opposed to the others?

---,---> Church 1956 uses predicates, but calls the system a
"functional calculus": cf. section 49, especially fn. 458 and the
text it is a note to.  Basically, he wanted to restrict the word
"predicate" to the linguistic expression, but took the logical system
to be a formal theory about the, umm, THINGS that predicates express
or stand for.  And these he called functions: "propositional
functions," a term going back to Russell.  Church's views on proper
nomenclature are admirably clear (and forcefully expressed).  They
presuppose a certain degree of Platonism: he took it for granted that
there ARE, umm, things expressed by predicates.  I have a feeling
that this may have led OTHER people to prefer the term "predicate
logic," as not presupposing that there are any propositional
functions for logical theory to be "about."  (I think this comes out
in a review-- by ???Max Black???-- of some logic book in the
??1950s??.)

>Q6.Do you have opinions on these issues or do you regard them as
>subjective matters of taste? If the later, what would good taste
>dictate?
--->  Call it "first order logic": this will help identify the
students who are really interested (they'll ask what "second order
logic" would be).  ...  My experience, trying to teach the unscreened
mine-run of humanities students who take logic courses in philosophy
departments is that it's hard to get very much over in a semester:
the simplicity of FOL with only predicates recommends itself.
(Particularly since the NON-mathematical examples given of functional
expressions-- "father of" and the like-- are simply WRONG.)

oooo----> Sorry.  As you can see, I don't have much to say about
John's  historical questions.  I think they are interesting, and real
answers to them would help give a picture of logical thinking in the
period just before the emergence of modern mathematical logic.  Maybe
by raving on this way I can provoke someone else into giving