FOM: Existential commitments in logic
J.J.Ketland at lse.ac.uk
Fri Oct 15 13:04:24 EDT 1999
I think that Frege is simply pointing out to Husserl the well-known fact
(nowadays), which we teach to undergraduates, but which troubled generations
of thinkers from Aristotle to Frege, that
"All Fs are Gs" (x)(F(x) --> G(x))
does not imply the existential claim
"There exists an F" Ex(F(x))
Of course, we have
(x)F(x) |- ExF(x) (*)
in ordinary classical first-order logic, because we always consider
non-empty models with at least one element. In particular, Ex(x = x) counts
as a logical truth. Another reason is that
F(t) |- ExF(x), (**)
where t is any term. So, first-order logic says "there is at least one
object", and that allows logical theory (especially the rules for
instantiation and generalization on arbitrary terms) to be developed
*smoothly*. But this is all really a matter of convention, as Quine has
often pointed out. See:
Quine, WV 1993: "Comment on Orenstein", in R. Barrett and R. Gibson (eds.)
Perspectives on Quine (Oxford: Blackwell), pp. 271-272.
"The exclusion of the empty universe from predicate logic is purely
a matter of efficiency. If you leave the empty universe out of account,
you gain a lot of convenient laws that you can use in your deductions
in applied logic and which would not hold if you included the empty
universe. .... If you want to allow for the possibility that your
universe of discourse might prove empty, you can still handle the empty
case separately. It is trivial, in that all formulas there admit of an
immediate decision procedure: just mark the existentials as false and the
universals as true and resolve out truth-functionally. So there is
nothing philosophical about the empty universe in predicate logic"
In contrast, natural languages contain terms (like "Pegasus", "God", "social
justice in the USA", etc.) which designate nothing. One could, in principle,
adjust first-order logic so that no existential statements count as logical
truths (and inferences like (*) and (**) are not valid). Any such resulting
system is called a "(existence) free logic".
The early workers in this field were (I think) Lambert and van Fraassen.
Lambert, K. 1965: "Existential Import Revisted", Notre Dame J. of Formal
Logic 6: 135-141.
---- 1967: "Free Logic and the concept of existence", Notre Dame J. of
Formal Logic 8: 133-142.
van Fraassen, B.C. 1969: "Presupposition, Supervaluations and Free Logic",
in K. Lambert (ed.) The Logical Way of Doing Things (New Haven: Yale
Some of the ideas behind such free logics are discussed in:
Mark Sainsbury 1991: Logical Forms (Oxford: Blackwell), Chapter 4, Section
Stephen Read 1995 (I think): Thinking About Logic (Blackwell, I think).
Philosophy, London School of Economics
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