[FOM] Why Do we Need a New Notion of Logical Content
Ken Gemes
K.Gemes at bbk.ac.uk
Tue Sep 25 04:51:48 EDT 2007
Thanks for the off-forum posts about my query on logical content.
While there have been some suggestions for an answer I will only post a
summary of replies, as some have requested, if I feel a worked out
solution is available.
Meanwhile to those who think that logical content of a wff is to be
identified with the class of the wffs logical consequences, or perhaps
the class of its non-tautologous consequences (as per Carnap in his
The Logical Syntax of Language) and hence
there is no need for a new notion of logical content:
If we count every (non-tautologous) logical consequence as part of the
content of a wff, then where Fa and Fb are atomic wffs, we have to say
that
(1) Fa&Fb and ~Fa&~Fb share common content, namely Fav~Fb
(2) ~Fa conclusively confirms part of Fa&Fb since it logically entails
its content part ~FavFb
(3) We cannot make sense of the notion of Fa confirming all the content
of (x)Fx since part of the content of (x)Fx is ~FavFb - clearly ~Fa
entails and hence confirms ~FavFb, and presumably it cannot be that both
Fa and ~Fa confirms ~FavFb.
(4) Where Fa is true we cannot say that Fa has more verisimilitude than
~Fa, where verisimilitude is, ala Popper, explained in terms of truth
content inclusion: If Fb is true than ~FavFb is part of the truth
content of ~Fa but is not part of the truth content of Fa; on the other
hand, if ~Fb is true then ~Fav~Fb is part of the truth content of ~Fa
but not part of the truth content of Fa.
If we have a notion of logical content along the lines I have suggested,
where not every logical consequence of a wff counts as part of the
logical content of that wff none of the results (1) to (4) follows.
Furthermore, we can then explicate a notion of natural axiomatization
that allows for clear definitions of empirical significance and
Hypothetico-deductive confirmation that are not open to known
counter-examples.
A quick and hopefully not too brief illustration of the point: Consider
the two claims that the nothings nothings (symbolized by the atomic
sentence 'Nn') and the claim that Sydney has a harbour bridge
(symbolized by 'Bs'). Now consider the following axiomatization of the
theory that simply claims that Sydney has a harbour bridge and the
Nothing nothings:
Axiom 1: Nn
Axiom 2: Bs
It is tempting to claim here that Nn is not empirically significant for
the theory that is here axiomatized since removing Axiom 1 from the
Axiom set {Axiom 1, Axiom 2} does not diminish the empirical import of
the Axiom set; But note, a logically equivalent axiomatization of the
same theory is
Axiom 1*: Nn
Axiom 2*: Nn -> Bs. (i.e. Nn materially implies Bs)
In this case removing Axiom 1* from the set {Axiom 1*, Axiom 2*) leads
to an axiom set that does not have the same empirical consequences as
the original set - this is the kind of point Isaiah Berlin famously
raised against A.J.Ayer's initial attempts to define verifiability. But
note, if in determining empirical significance we restrict ourselves to
looking only at natural axiomatizations of the theory - where a natural
axiomatization of a theory only contains axioms that are content parts
of the theory - then no such problem arises. Nn-> Bs is not part of
the content of a theory that entails Nn&Bs since that relevant model of
Nn-> Bs that assigns F to both Nn and Bs cannot be extended to a model
of Nn&Bs.
Links to articles covering this ground in more detail are on my web page
indicated below.
Ken Gemes
Reader in Philosophy
School of Philosophy
Birkbeck College
University of London
Malet Street
London, WC1E 7HX
Reader in Philosophy
Philosophy
School of Humanities
University of Southampton
Southampton, S017 1BJ
web page: http://www.bbk.ac.uk/phil/staff/academics/gemes/
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