[FOM] Inconsistensy, contrasistency, contradicion and complementarity

Frode Bjørdal frode.bjordal at ifikk.uio.no
Tue Sep 17 11:39:08 EDT 2013

In response to recent quests from Harvey Friedman concerning systems that
approach naïve comprehension in ways that are inconsistent or nearly so I
relate some on these matters as they pan out in my librationist system £
which I have written about earlier on the fom list. Much of what I relate
are known by some already as I have lectured about these issues for the
Oslo Logic Group and at UNILOG 2013 in Rio in April and also circulated
material to some including the editorial board of fom. More material
including the detailed manuscripts Reality is Incoherent as well as A
Conditional Sober Librationist Interpretation of ZF are available upon
request. The long paper Librationist Closures of the Paradoxes is published
in Logic and Logical Philosophy Volume 21 (4), p. 323-361, 2012, and is
available for free online. It is doubtful that the following is accessible
without also studying my paper or more material available.

Let me emphasize that I completely agree with Friedman and others that
inconsistencies and contradictions are not to be accepted. I think,
however, that there are some intricacies in explaining how it is that these
are not to be accommodated in dealing with paradoxical phenomena. Hopefully
the following will help explain these and further relevant matters.

I use slightly non-standard or rare names for connectives, and also
introduce some other new terminology.

Let us say that a system T is contrasistent iff for some sentence A it is
the case that A is a thesis of T and it is also the case that the
negjunction -A of A is a thesis of T. A system T is inconsistent iff for
some sentence A the adjunction A&-A is a thesis of A. Typically
inconsistent systems are also contrasistent, and contrasistent systems are
inconsistent e.g. if they have adjunction as an inference rule.

£ is, as I now see it, an alternative set theory and its language may be
taken as just that of the language of set theory including set abstracts
(on account of the essential failure of extensionality in £) plus two set
constants for “truth” and “enumeration”; I will not dwell on the nature of
truth and enumeration here.

£ is super formal in that (i) it supports infinitary demonstration
principles as the omega rule and (ii) noemata, which correspond to what is
normally taken as free variables, are names. In consequence of (ii) we have
what we take as a nominality interpretation of what is usually taken as
open sentences and not what Kripke dubbed a generality interpretation of
them; it follows that all formulas of £ are sentences of £.

£ is contrasistent and consistent. The regulations (corresponding to
“inference rules”) of £ are novel and modus ponens is not a regulation. The
regulations of £ along with the prescriptions (axiom schemas) and
prescripts (proper axioms) of £ are justified by a Herzbergerian style semi
inductive semantics which is also related to Infinite Time Turing Machines.

£ is contentual in the sense that all formulas as per above have a content,
and we focus upon one intended model so that soundness and completeness
considerations are extraneous. The contentuality of £ is reflected in that
it is negjunction complete in that for any sentence (formula) A of £ either
A is a thesis of £ or the negjunction -A is a thesis of £.

Arguably, £ is not a logic and it is not recursively axiomatizable.
However, £ is eminently a mathematical system. Let us say that A is an anti
thesis of a system T iff the negjunction -A is a thesis of T. We say that a
system T extends system U soberly iff all theses of U are theses of T and
no thesis of T is an anti thesis of U. £ is a sober extension of classical
logic. A system T soberly interprets system U iff T interprets U and for no
interpretans A(I) of T of an interpretandum A(i) of U is it the case that T
has A(I) as a thesis and also the case that A(I) is an anti thesis of T. By
the results of A Conditional Sober Librationist Interpretation of ZF £
soberly interprets ZF if SO =S+”there are omega inaccessible cardinals” has
a well founded model; here S is Friedman's 1973 system ZF minus
extensionality with collection and a weak power set notion which he showed
interprets ZF. The proof invokes Theorem 1 of Friedman's The Consistency of
Classical Set Theory Relative to a Set Theory with Intuitionistic Logic, J.
of Symbolic Logic, Vol. 38, No. 2, (1973), pp. 315-319, and makes use of a
fixed point construction I call manifestation point (which goes back to
Andrea Cantini in Logical Frameworks for Truth and Abstraction, Elsevier
1996 with roots in earlier work by Albert Visser) upon a definability
notion going back to Gödel which elaborates upon an exposition by Kenneth

Naive abstraction does not hold in £, but the implicitly defined
librationist comprehension approaches naïve abstraction very closely indeed
(and as it were from above) and e.g. both (A=>xe{x:A}) is a thesis of £ as
well as ({xe{x:A}=>A} is a thesis of £.

A thesis A of £ is a maxim of £ if A is not an anti thesis of £. A thesis A
of £ is a minor of £ if it is not a maxim of £. Regulations of £ are
sensitive as to whether assumptions are maxims or minors and e.g. modus
maximus is the regulation that if A is a maxim of £ and also (A=>B) is a
maxim of £ then B is a maxim of £ as well.

In the statement of regulations and prescriptions of £ it is useful to use
an auxiliary truth operator T defined by TA as short for (Ey)(ye{y:A})
where y is not a noema of A. In the following considerations concerning
Curry phenomena we invoke modus maximus as per above, alethic comprehension
which is the maxim ae{x:A}<=>TA, T(out) which is the thesis schema (which
may have minor instances) TA=>A, modus subiunctionis which is the
regulation that B is a thesis if A is a thesis and (A=B) is a maxim, modus
ascendens (maximus, minor) which is the regulation that TA is a thesis
(maxim, minor) of £ if A is a thesis of £, modus descendens (maximus,
minor) which is the regulation that A is a thesis (maxim, minor) of £ if TA
is a thesis (maxim, minor) of £, modus scandens (maximus, minor) which is
the regulation that TA is a thesis (maxim, minor) of £ if -T-A is a thesis
(maxim, minor) of £ and classicality which is the principle that all
classical logical theses are maxims of £.

We now explore the Curry condition C(A)={x:xex=>A} for arbitrary A. T(out),
alethic comprehension and classicality gives us C(A)eC(A)=>(C(A)eC(A)=>A).
By classicality (C(A)eC(A)=>(C(A)eC(A)=>A))=>(C(A)eC(A)=>A) is a maxim of £
so by modus subiunctionis C(A)eC(A)=>A is a thesis of £. By modus ascendens
T(C(A)eC(A)=>A) is a thesis of £, so by alethic comprehension C(A)eC(A) is
a thesis of £. Suppose C(A)eC(A) is a maxim of £: By modus maximus and
alethic comprehension then T( C(A)eC(A)=>A) is a maxim of £ so that by
modus descendens maximus C(A)eC(A)=>A is a maxim of £ so that by modus
maximus A is a maxim of £. Suppose contrary wise that C(A)eC(A) is a minor
of £: In this case also -C(A)eC(A) is a thesis of £ so that by modus
subiunctionis and alethic comprehension -T(C(A)eC(A)=>A) is a thesis of £.
By modus descendens, modus descendens and classicality C(A)eC(A)&-A is a
thesis of £ . By classicality and modus subiunctionis thence -A is a thesis
of £. As A was arbitrary it follows by parity of reasoning that for any
formula A either A is a maxim of £, -A is a maxim of £ or A is a minor
thesis of £. But this only reminds that £ is negjunction complete.

In reasoning about £ from the outside, or at a meta level if one insists, I
take the valency of a formula to be the set of ordinals where it holds
according to the semantical set up. The valor of a sentence is the least
upper bound of its valency. A sentence is true iff its valor is the closure
ordinal Koppa of the semantics. A formula is false iff its negjunction is
true. The contravalancy of a formula A is Koppa minus the valency of A. Two
sentences A and B are isovalent iff they have the same valency. Two
sentences A and B have opposite valencies iff B is isovalent with the
negjunction -A of A. The ambovalence of A and B is the intersection of the
valency of A and B. The subvalence of A under B is the union of the valency
of -A with that of B and the homovalence of A and B is the union of the
valency of A&B with that of -A&-B.

Connectives in £ are valency functional, and in case formulas are maxims
valency functionality and truth functionality coincide.

We take a formula to say its valor and the valency of the formula is taken
to be the way it says its valor. Let r=[x:-xex} be Russell's set and let
rer be Russell's sentence. According to £ Russell's sentence is a minor so
that both Russell's sentence is a thesis of £ and Russell's sentence is an
anti thesis of £. However, we take rer to say the same thing as -rer, viz.
Koppa. Nevertheless, Russell's sentence says Koppa in a way which is
opposite to the way in which its negjunction says Koppa.

We take a theory to be contradictory iff it has theses which say different
things. Two sentences are complementary iff they are both true though have
opposite valencies. £ is not a contradictory theory by these standards, but
it is a complementary theory as it has theses which are complementary.

We suggest that the failure of adjunctivity of £ should be regarded as an
incoherency phenomenon concerning paradoxical formulas. One rightly holds
it against someone if she says A and thence -A if she is meant to elucidate
a coherent topic. However, our diagnosis is that paradoxicalities are
incoherent, and so the incoherency of £ should be taken as a fulfillment of
an important adequacy condition for dealing with the paradoxes.

Professor Dr. Frode Bjørdal
Universitetet i Oslo Universidade Federal do Rio Grande do Norte
quicumque vult hinc potest accedere ad paginam virtualem
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