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Њ = # p .7Բ # SORITES # X
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An International Digital Journal of Analytical Philosophy
j d Indexed and Abstracted in THE PHILOSOPHER'S INDEX
. ISSN 1135-1349
! G
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,H P# Editor: Lorenzo Pe9a
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Associate Editor: Txetxu Aus1n
+
[ ! ; (Spanish Institute for Advanced Studies)
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t JeanYves B)ziau, Enrique Alonso, GuillermoHurtado, ManuelLiz,
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P n) ) t Issue #17 " October 2006 n) =p-p-p-
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SORITES ($2"($)
. y ISSN 1135-1349
) Issue #17 " October 2006
" Copyright by SORITES and the authors
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Main InterNet Access:
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b 8 M X` hp x (#%'0*,.8135@8: 5 SORITES
1 ISSN 1135-1349
H 0 Roll of Referees
M b Rainer Bornp9!1 JohannesKepler Universitaet Linz (Austria)
Amedeo Contep9!A University of Pavia (Italy)
Newton C.A. da Costap9!< University of SMo Paulo (Brazil)
Marcelo Dascalp9!= University of Tel Aviv (Israel)
Dorothy Edgingtonp|9!@ Birbeck College (London, UK)
Graeme Forbesp9!- Tulane University (New Orleans, Louisiana, USA)
Manuel Garc1a-Carpinterop 9!= University of Barcelona (Spain)
Laurence Goldsteinp!9!9 University of Hong Kong (Hong Kong)
Jorge Graciap_9!1 State University of New York, Buffalo (USA)
Nicholas Griffinp9!- McMaster University (Hamilton, Ontario, Canada)
Rudolf Hallerp'9!3 KarlFranzensUniversitaet Graz (Austria)
Terence Horganp 9!6 University of Memphis (Tennessee, USA)
Victoria Iturraldep9!* Univ. of the Basque Country (San Sebastian, Spain)
Tomis E. Kapitanp9!: Northern Illinois University (USA)
Manuel Lizp(9!- University of La Laguna (Canary Islands, Spain)
Peter MenziespT9!( Australian National University (Canberra, Australia)
Carlos Moyap9!> University of Valencia (Spain)
Kevin Mulliganp9!: University of Geneva (Switzerland)
JesCs PadillaGlvezp9!1 JohannesKepler Universitaet Linz (Austria)
Philip PettitpT9!( Australian National University (Canberra, Australia)
Graham Priestp9!. University of Queensland (Brisbane, Australia)
Eduardo Rabossip9!6 University of Buenos Aires (Argentina)
DavidHillel Rubenp9! School of Oriental and African Studies, University of London
Mark Sainsburyp9!A King's College (London, UK)
Daniel Schulthesspn9!7 University of Neuch tel (Switzerland)
Peter Simonsp9!= University of Leeds (Leeds, UK)
Ernest Sosap_9!, Brown University (Providence, Rhode Island, USA)
Friedrich Stadlerp9!5 Institut Wien Kreis
(Vienna, Austria) & = p-p-p- `h
p x( !0$& b/
jcM"z *!#%2(*, b # -t \ P C 'qP# =
/ g 5 SORITES
1 ISSN 11351349
, Issue #17 " October 2006
v m 0 Table of Contents
a b
o bb " Abstracts of the Papers9!pi 9!Z 03
o bb " About Properties of LInconsistent Theories
by Vyacheslav Moiseyev9!pi 9!Z 07
o bb " Paraconsistent logic! (A reply to Slater)
by Jean-Yves B)ziau9!pi 9!Z 17
o bb " The Logic of Lying
by Moses @k/9!pi 9!Z 27
o bb " Sparse Parts
by Kristie Miller9!pi 9!Z 31
o bb " Are Functional Properties Causally Potent?
by Peter Alward9!pi 9!Z 49
o bb " Subcontraries and the Meaning of `If8Then'
by Ronald A. Cordero9!pi 9!Z 56
o bb " Does Frege's Definition of Existence Invalidate the Ontological
Argument?
by Piotr Labenz9!pi 9!Z 68
o bb " Why Prisoners' Dilemma Is Not A Newcomb Problem
by P. A.
Woodward9!pi 9!Z 81
o bb " A Paradox Concerning Science and Knowledge
by Margaret Cuonzo9!pi 9!Z 85
o bb " Between Platonism and Pragmatism: An alternative reading of Plato's
Z | Theaetetus
by Paul F. Johnson9!pi 9!Z 95
o bb " Blob Theory: Nadic Properties Do Not Exist
by Jeffrey Grupp9!p 9!Y 104
Z L o bb " The SORITES Charter9!p 9!Y 132
o bb " Release Notice9!p 9!Y 137 = p-p-p- b # 'V 2 P kC 4P# X01 Í Í. X01 Í Í. 4 X 1 Í . X* Í . 4 # 4o \ P C +XP# = 5 a v 1 )
W B a " SORITES Issue #17 " October 2006issnĠ1135-1349`!%"] ă
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W B a " SORITES ĠIssue #17 " October 2006. issnĠ1135-1349`!%"] ă
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A d d x @ ---- " y M ( SORITES ($2"($), ISSN 1135-1349
- http://www.sorites.org
( Issue #17 " October 2006. Pp. 36
- W Abstracts of the Papers
% Copyright by SORITES and the authors -- , a 5 a v
s * ] Abstracts of the Papers
+
X B
About Properties of LInconsistent Theories
X * A by Vyacheslav Moiseyev
, a In the paper a new type of the formal theory, Linconsistent theory
, is constructed and
some properties of such theories are investigated. First a theory T* is defined as a set of
limiting sequences of formulas from a theory T with a language L. A limiting sequence
X { An }n=1 of the formulas from T is said to be a theorem of the theory T* if there exists an
X m0 such that for any nm the formula An of the language L is a theorem of the theory T.
X T is embeded into T*. Then, a theorem of T* is called an Lcontradiction if the limit of this
X theorem equals B U B , where B is a formula of the language L. Finally, the theory T* is said
X o to be an Linconsistent theory if there exists an Lcontradiction in T*. It is proved that the
theory T* is consistent, complete, etc., iff the theory T is consistent, complete, etc. However,
T* contains more theorems and inferences than T (see Theorems 9 and 10). Linconsistent
theory T* can be presented as a new approach to the Philosophical Logic, dealing with an
extension of Method of Limits to thinking. Namely some philosophical antinomies, for
example Kantian ones, could be presented as Lcontradictions in an Linconsistent theory.
a+ ك
X a! Paraconsistent logic! (A reply to Slater)
X a, by Jean-Yves B)ziau ă
We answer Slater's argument according to which paraconsistent logic is a result of a
verbal confusion between contradictories
and subcontraries
. We show that if such notions
are understood within classical logic, the argument is invalid, due to the fact that most
paraconsistent logics cannot be translated into classical logic. However we prove that if such
notions are understood from the point of view of a particular logic, a contradictory forming
function in this logic is necessarily a classical negation. In view of this result, Slater's
argument sounds rather tautological.
a+ ك
X % , The Logic of Lying
X
' / by Moses @k/
) a By definition, a lie is a dishonestly made statement. It is a wilful misrepresentation, in
one's statement, of one's beliefs. Both a truthful person and a liar could hold false beliefs. We
should not uncritically regard an untruthfully made statement as an untrue statement, or a
truthfully made statement as a true statement. The only instance when a lie is necessarily false '+ = p-p-p- is when the liar's corresponding belief that was distorted was true. In other instances, the lie
could be either true or false. We conclude that a lie is not necessarily a false statement.
a+ ك
X ;a/ Sparse Parts
X a- by Kristie Miller ă
Four dimensionalism, the thesis that persisting objects are four dimensional and thus
extended in time as well as space, has become a serious contender as an account of
persistence. While many four dimensionalists are mereological universalists, there are those
who find mereological universalism both counterintuitive and ontologically profligate. It would
be nice then, if there was a coherent and plausible version of four dimensionalism that was
nonuniversalist in nature. I argue that unfortunately there is not. By its very nature four
dimensionalism embraces theses about the nature of objects and their borders that make any
version of nonuniversalist four dimensionalism either incoherent or at least highly
implausible.
a+ ك
X k a Are Functional Properties Causally Potent?
X a. by Peter Alward ă
Kim has defended a solution to the exclusion problem which deploys the causal
inheritance principle
and the identification of instantiations of mental properties with
instantiations of their realizing physical properties. I wish to argue that Kim's putative solution
to the exclusion problem rests on an equivocation between instantiations of properties as
X bearers of properties and instantiations as property instances. On the former understanding,
the causal inheritance principle is too weak to confer causal efficacy upon mental properties.
And on the latter understanding, the identification of mental and physical instantiations is
simply untenable.
a+ ك
X H Subcontraries and the Meaning of If8Then
X + by Ronald A. Cordero
) a In this paper I maintain that useful, assertable conditional statements with subcontrary
antecedents and consequents do actually occur. I consider the paradoxical results of applying
rules of inference like Transposition in such cases and argue that paradox can be avoided
through an interpretation of conditionals as claims that the truth of one statement would permit
a sound inference to the truth of another.
a+ ك
X h% a Does Frege's Definition of Existence Invalidate the Ontological Argument?
X & a. by Piotr Labenz ă
It is a well-known remark of Frege's that his definition of existence invalidated the
ontological argument for the existence of God. That has subsequently often been taken for
granted. This paper attempts to investigate, whether rightly so. For this purpose, both Frege's
ontological doctrine and the ontological argument are outlined. * p-++!! Ԍ Arguments in favour and against both are considered, and reduced to five specific
questions. It is argued that whether Frege's remark was right depends on what the answers to
these questions are, and that for the seemingly most plausible ones " it was not.
a+ ك
X &
a Why Prisoners' Dilemma Is Not A Newcomb Problem
X a- by P. A. Woodward ă
David Lewis has argued that we can gain helpful insight to the (all too common)
Prisoners' Dilemmas that we face from the fact that Newcomb's Problems are easy to solve,
and the fact that Prisoners' Dilemmas are nothing other than two Newcomb Problems side by
side. The present paper shows that the (all too common) Prisoners' Dilemmas that we face are
significantly different from Newcomb Problems in that the former are iterated while the latter
are not. Thus Lewis's hope that we can get insight into the former from the latter is illusory.
a+ ك
X Y a A Paradox Concerning Science and Knowledge
X a, by Margaret Cuonzo ă
Quine's and Duhem's problem regarding the laying of blame
that occurs when an
experimental result conflicts with a scientific hypothesis can be put in the form of a standard
X ) philosophical paradox. According to one definition, a philosophical paradox is an argument
with seemingly true premises, employing apparently correct reasoning, with an obviously false
or contradictory conclusion. The Quine/Duhem problem, put in the form of a paradox, is a
special case of the skeptical paradox. I argue that both the Quine/Duhem paradox and the
skeptical paradox enjoy the same type of solution. Both paradoxes have the kind of restricted
solution that Stephen Schiffer calls mildly unhappyface
solutions. Although there can be
no solution to these two paradoxes that gives an accurate account of the relevant notions (e.g.,
knowledge), replacement notions are given for the ones that lead to the paradoxes.
a+ ك
X H ~a Between Platonism and Pragmatism: An alternative reading of Plato's Theaetetusă
X a, by Paul F. Johnson ă
In a letter to his friend Drury, Wittgenstein claims to have been working on the same
X problems that Plato was working on in the Theaetetus. In this paper I try to say what that
problem might have been. In the alternative reading of the dialogue that I construct here,
attention is drawn to Socrates' frequent appeal in the course of discussion to the ordinary ways
of speaking that he, and Theaetetus, and everyone else in Athens at the time engaged in. The
more abstruse theories of Heraclitus and Protagoras which Socrates and Theaetetus are
discussing are found to do violence to these ordinary ways of talking, and found seriously
wanting as a result. A case is made that the conventions and presuppositions of ordinary
conversational speech are inherently normative, and constitute a valid standard against which
philosophical theories may be measured. Lines of affinity are drawn between these claims
advanced by Plato and the recent work of contemporary neopragmatists, and Robert
Brandom's work in particular.
a+ ك k* p-++!! Ԍ X a Blob Theory: Nadic Properties Do Not Exist
X _ ga- by Jeffrey Grupp ă
I argue for blob theory: the philosophic position that nadic properties do not exist. I
discuss hitherto unnoticed problems to do with the theories of property possession in the
ontological theories of ordinary objects: the bundle theory of objects and substance theories
of objects. Specifically, I argue that theories of property possession involved with the bundle
theory and substance theories of objects are contradictory, and the best theory we have been
given by metaphysical realists is a theory that reality is propertyless. K p-++!! 5 a v 7 * H2 v W B a " About Properties of LInconsistent Theories
by Vyacheslav Moiseyev`!%"] ă
y I d d d y 7 =
a a @
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a d d x @ ---- " y M ( SORITES ($2"($), ISSN 1135-1349
- http://www.sorites.org
' Issue #17 " October 2006. Pp. 716
# About Properties of LInconsistent Theories
!
Copyright by Vyacheslav Moiseyev and SORITES -- , a 5 a v
s o About Properties of LInconsistent Theories
X , Vyacheslav Moiseyev
+
a Apparently, there have been two traditions in the history of logic, these are Line of
Parmenide and Line of Heraclitus. Former is originated from the ideas of ParmenideAristotle
and is based on the Law of Identity. This line constitutes formal logic. Latter is originated
from the ideas of HeraclitusPlato and has been expressed itself in the ideas of dialectics, or
dialectical logic. Contemporary mathematical logic is the worthy result of the development of
the first line. Possibility of good precision and clear procedures of justification is the most
strong side of this line. On the other hand, dialectics always have been trying to deny the
meaning of Law of Identity. Dialectical ideal have been expressed itself in the idea of
contradiction. But a very big problem have been subsisted here. This is the problem which we
X shall call Problem of Logical Demarcation (PLD). Breafly speaking, essence of the problem
is in the following idea. Mistakes are contradictions too and if dialectics does not want to be
simply mistaken reasoning, then it must show a criterion with the help of which we could to
separate contradictionsmistakes from dialectical contradictions (antinomies). We shall call
X x such criterion as Criterion of Logical Demarcation (CLD). Although dialectics has not been
able to show CLD but there have been many interesting attempts to find the Criterion. One
can refer here to Plato, Nicholas from Cusa, Russian Philosophy of AllUnity, etc.
It seems to us that one of the interesting ideas here is the idea of some connection
between CLD and concept of limit. For example, Nicholas from Cusa tried to express idea of
God in the image of a straight line which is limit for the infinite sequence of tangent
circumferences. Our paper is an attempt to extend this trend and to formulate a version of
CLD, where dialectical contradictions (antinomies) can be expressed as limits of ifinite
sequences of formulas in a formal language. Main new idea is here in the technique of work
X ! with the limiting sequences of formulas, not terms. This idea is fully correlated with the
method of extension of rational numbers by irrational ones in mathematical analysis. As is
well known, every irrational number can be represented by a limiting sequence of rational
numbers. Then we can represent rational numbers itselves as a particular case of limiting
sequences, i.e., as stationary sequences. Thus we are passing to a new type of objects and we
can define operations with these objects generalizing of operations on rational numbers. The
same approach is demonstrated below but in the logical sphere.
Basic task here is to define limiting sequences of formulas. Separate formulas in a formal
language can be considered as analogues of rational numbers in analysis. Stationary sequences
of formulas must be a particular case of the definition of limiting sequence. We shall carry
out the task of limiting sequence of formulas definition by use of limiting sequences of terms.
Let us see the following simple example. Let 1/n = 1/n and 1/n c 1/n+1 be formulas in a
theory T generalizing theory of real numbers. Then for every n = 1, 2, 3, we can prove that
the corresponding formulas are theorems. Let us propose that we can prove also formula z, = p-p-p- X lim(1/n) = 0 in T (I shall mean the limit of sequence of terms an as n under the symbol
X lim(an)), i.e., limit of sequence of numbers 1/n is zero. (1/n = 1/n U 1/n c 1/n+1) is also
formula in T and we can consider the following infinite sequence of formulas
(1/1 = 1/1 U 1/1 c 1/2), (1/2 = 1/2 U 1/2 c 1/3), (1/3 = 1/3 U 1/3 c 1/4),
Every element of the sequence is formed as the result of substitution of constants 1/1, 1/2,
1/3, etc., for the places of variables in formula (x=x U x c y). For example, first formula can
X z be represented as (x=x U x c y) x,y [1/1, 1/2], i.e., as the result of substitution of constants 1/1
and 1/2 for variables x and y respectively. Hence we can rewrite the sequence of formulas in
the following form
X (x=x U x c y) x,y [1/1, 1/2], (x=x U x c y) x,y [1/2, 1/3], (x=x U x c y) x,y [1/3, 1/4],
It permits to us to use the following designation for the expression of this sequence
X u {(x=x U x c y) x,y [1/n, 1/n+1]}
Let us define the limit of this sequence as the result of substitution of limits of sequences
of terms for the variables. In our case we receive
X " lim((x=x U x c y) x,y [1/n, 1/n+1]) =Df (x=x U x c y) x,y [lim(1/n), lim(1/n+1)]
Since lim(1/n) = lim(1/n+1) = 0, we finally receive
X lim((x=x U x c y) x,y [1/n, 1/n+1]) = ((x=x U x c y) x,y [0, 0]) = (0 = 0 U0 c 0),
i.e., contradiction.
However, though limit of sequence of formulas is contradiction, every formula from the
sequence is theorem of T. Such consequence of formulas plays a role similar to role of
consequence of rational numbers which limit is absent between rational numbers, i.e., is an
X e irrational number. We shall call consequences of theorems which limit is contradiction as L X P contradiction, i.e., limit contradiction. Instead of working with contradiction we can work with
limiting consequence of formulas which limit is the contradiction. Logic of limiting
consequences of formulas is not poorer than logic of formulas since the last is generalized by
the former on the basis of stationary sequences.
Finally, we can formulate CLD with the help of the idea of Lcontradiction. Namely,
X U contradiction AUA is called an antinomy (dialectical contradiction) relatively consistent
X D theory T if AUA is formula of the language of T and there exists an extension of the theory
T to a theory T* of limiting consequences of formulas from T such that there exists an Lcontradiction from T* which limit equals AUA. Therefore it is clear that satisfactory decision
of CLD and PLD is the consequence of satisfactory formulation of the theory T* and its
properties. Below we shall investigate namely this problem.
Suppose T is a formal theory with a language L such that there exist formulas in L, which
X <% can be represented in the metalanguage as expressions of the form lim ( an ) = a
, where an
,
X %& a
are names for terms from L, n
is name for natural number n, and these formulas can
X ' be interpreted in a model M of the theory T as the equality of the limit of a sequence {an}
with individuals from M to an individual a from M. We shall say that such theory T is called
X ( tlimiting theory ( t
is used from term
). A theory of sets and theory of real numbers are
examples of tlimiting theories. )
p-++!! Ԍ X Let T be a tlimiting theory with a language L, where An is a formula from L such that
X _ (*) An = A x1, x2, , xm [ a1n p1, a2n p2, , amn pm ],
X where pj N ,
X ) N is set of natural numbers,
and j=1,,m.
X In other words, the formula An is the result of the substitution of terms a1n p1, a2n p2, ,
X amn pm for free entrances of the variables x1, x2, , xm into a formula A , where each of the
X terms ajn pj is an element of an infinite sequence { ajk } (k is variable of sequence here), and
theorems of the form
X limajn = aj
are deduced in the theory T for every j.
X
The sequence { An }n=1 is defined for the formulas An of the sort (*). By definition, put
X 0 A = lim An = A x1, x2, , xm [ lim ( a1n ), lim ( a2n ) ,, lim ( amn )].
X This definition allows us to reduce a concept of formula limit to limits of terms, which
are included into a formula.
X o DEFINITION 1. Sequences { An }n=1 of the formulas An of the sort (*) and also stationary
X sequences of the formulas from L are called limiting sequences of the formulas from L.%"
Let a language L* be the set of all the terms from L and also the set of all limiting
sequences of the formulas from L. The language L can be embeded into the language L* with
X the help of the injective map E*: L L* such that if ! is a term from L, then E*( ! )= ! , if A
X is a formula from L, then E*( A ) is the stationary sequence of the formulas A .
X I o DEFINITION 2. Limiting sequences of the formulas { An }n=1 are called formulas of the
X 4 language L*.%"
Thus the languages L and L* do not differ between themselves by the alphabets and sets
of terms but only sets of the formulas.
X o DEFINITION 3. We say that two formulas { An }n=1 and { Bn }n=1 from L* are called equal
X and this is denoted by { An }n=1 = { Bn }n=1
if the formula lim An (i.e. formula, which
X is limit of sequence { An }n=1) can be obtained from the formula lim Bn by right renaming
of bound variables.%"
X ! o DEFINITION 4. A formula { An }n=1 of the language L* is said to be a metatheorem of the
X " theory T if there exists an m0 such that for any nm the formula An of the language L
is a theorem of the theory T.%"
X 2% o DEFINITION 5. A limiting sequence of the formulas { An }n=1, which is a metatheorem of the
X & theory T, is called an Lcontradiction ( L
from limit
) if the limit of this sequence,
X ' lim An , equals B U B , where B is a formula of the language L.%"
X k( o DEFINITION 6. The set of metatheorems of the theory T is called a theory T*.%"
X ) In this case, the metatheorems of the theory T can be called also theorems of the theory
X * T*. The language L* is the language of the theory T*. We shall say that T* is called a ft * p-++!! ԫ X limiting theory ( f
from formula
). The approach, circumscribed above, can be considered
as a methodology of buildingup of ftlimiting theories on the basis of tlimiting theories. The
X theory T* is said to be an Linconsistent theory if there exists an Lcontradiction in T*.
X 5 o DEFINITION 7. The theory T* is called consistent if not all formulas from L* are theorems
of the theory T* (see also Theorem 20).%"
o THEOREM 1. If the theory T is consistent, then the theory T* is consistent.%"
X o PROOF. Suppose the theory T is consistent; then there exists a formula A from the language
X L such that A is not a theorem of the theory T. Let { An }n=1 be the stationary sequence,
X where for any n An is A . The sequence { An }n=1 is a formula from L* but it is not a
theorem of the theory T*. Therefore the theory T* is consistent.%"
o THEOREM 2. If the theory T* is consistent, then the theory T is consistent.%"
o PROOF. Assume the converse. Then the theory T* is consistent and the theory T is not. If T
is nonconsistent, then any formula of the theory T is the theorem of this theory. If T* is
X ) consistent, then there exists a formula { An }n=1 from L* such that { An }n=1 is not a
theorem of the theory T*. Hence for any m0 there exists an nm such that the formula
X An is not a theorem of the theory T. This contradiction proves the theorem.%"
o THEOREM 3. The theory T is consistent iff the theory T* is consistent.%"
o PROOF. See Theorems 1 and 2.%"
o DEFINITION 8. Let M be a structure for the language L. We shall say that a formula
X { An }n=1 from the language L* is valid in M if there exists an m0 such that for any nm
X the formula An is valid in M.%"
X U o DEFINITION 9. A structure M for the language L is called a model of the theory T* if any
theorem of the theory T* is valid in M.%"
X We shall say that a structure M for the language L is called a structure for the language
X L*.
o THEOREM 4. Let M be a model of the theory T; then M is a model of the theory T*.%"
X J o PROOF. Let a formula { An }n=1 from the language L* be a theorem of the theory T*. Then
X 3 there exists an m0 such that for any nm the formula An is a theorem of the theory T,
X " i.e., An is valid in the model M of the theory T. Therefore the formula { An }n=1 is valid
in M. Hence M is a model of the theory T*.%"
o THEOREM 5. Let M be a model of the theory T*; then M is a model of the theory T.%"
X " o PROOF. Let M be a model of the theory T* and A be a theorem of the theory T. Suppose
X # { An }n=1 is the stationary sequence, where An = A for any n; then { An }n=1 is a theorem
X $ of the theory T* and { An }n=1 is valid in M, i.e., there exists an m0 such that for any
X % nm the formula An is valid in M. Since An is A , we see that the formula A is valid in
M. Therefore M is a model of the theory T.%"
o THEOREM 6. M is model of the theory T iff M is model of the theory T*.%"
o PROOF. See Theorems 4 and 5.%" 7) p-++!! Ԍ X o DEFINITION 10. A formula { An }n=1 from the language L* is said to be an axiom of the
X theory T* if there exists an m0 such that for any nm the formula An is an axiom of the
theory T.%"
X 9 o DEFINITION 11. Suppose n is a set of formulas from L and there exists an m0 such that
X ( for any nm n An is an inference of an formula An from n in the theory T. Let the
X sequence of the sets {n}n=1 have the limit and the sequence of the formulas { An }n=1
X also have the limit. Then the object {n An }n=1 (i.e. sequence of inferences n An ,
X where n=1,2,3,) is called an inference in the theory T*. Denote by {n }n=1 T*
X { An }n=1
, or {n}n=1 { An }n=1
, any inference {n An }n=1.%"
X I o DEFINITION 12. An inference {n An }n=1 in the theory T* is said to be a proof in the
X 8
theory T* if there exists an m0 such that for any nm n is a set of axioms of the theory
X ' T or n is empty.%"
X o DEFINITION 13. A sequence of the sets {n}n=1 is called regular if {n}n=1 = {Bk=1N{ Akn }}
X {
(here {Bk=1N{ Akn }} is sequence, where n=1,2,3,, of unions Bk=1N{ Akn } of oneelement
X j sets { Akn }), while N is a finite natural number or infinity, and { Akn }n=1 (here { Akn } is
sequence of formulas of L, where n=1,2,3,) is a formula from L*. In this case, denote
X B by {{ Akn }n=1} k=1N
any {n }n=1 and denote by {{ Akn }n=1} k=1N { An }n=1
any
X 1 inference {n}n=1 { An }n=1. We shall say that the formula { An }n=1 is deduced from the
X set {{ Akn }n=1} k=1N of the formulas { Akn }n=1 in the theory T*.%"
X o THEOREM 7. If { An }n=1 is a theorem of the theory T*, then { An }n=1 is deduced in the
theory T* from axioms of the theory T*.%"
X o PROOF. Let { An }n=1 be a theorem and not be an axiom of the theory T*; then there exists
X an m0 such that for any nm the formula An is a theorem and is not an axiom of the
X theory T, i.e., there exists an inference Bn1 , Bn2 , , Bnk ( n ) An in the theory T, where Bn1 ,
X Bn2 , , Bnk ( n ) are axioms of the theory T. Here n = { Bn1 , Bn2 , , Bnk ( n )}. Further, if n
X } An , then n An , where n = B k=1n k. The sequence {Bk=1n k}n=1 has the limit, this
X l limit equals = Bk=1 k, and, for any nm, we have An . can be represented
X [ as { B1 , B2 , , BN }, where N is a finite number or infinity, and B1 , B2 , , BN are axioms
X J of the theory T. Let {*n}n=1 be the new sequence, where, for any n, *n = . Further,
X 9 for any nm, we have *n An and the sequences {*n}n=1, { An }n=1 have the limits.
X ( Hence the inference {*n}n=1 { An }n=1 is defined in the theory T*. Besides, the
X sequence {*n}n=1 is regular. Indeed, {*n}n=1 = {Bk=1N{ Bkn }}n=1, where Bkn = Bk for
X any k. It follows that {*n}n=1 = {{ Bkn }n=1} k=1N, where the stationary sequences of the
X axioms from T { Bkn }n=1 are the axioms of the theory T*. In other words, if { An }n=1 is
X ! a theorem and not an axiom of the theory T*, then there exists the inference {{ Bkn }n=1}
X " k=1N { An }n=1 of the theorem { An }n=1 of the theory T* from axioms of the theory T*.
X # If { An }n=1 is an axiom of the theory T*, then there exists the inference { An }n=1 of the
X $ theorem { An }n=1 of the theory T* from the empty set of axioms of the theory T*.%"
Let Thm* be the set of all the theorems of the theory T*, Thm be the set of all the
theorems of the theory T.
o THEOREM 9. If the theory T is consistent and the theory T* contains an Lcontradiction, then
there does not exist a map h: Thm* Thm such that%"
1) h is a bijection, *
p-++!! Ԍ X 2) h({ An }n=1) is a theorem of the theory T iff { An }n=1 is a theorem of the theory T *,
X _ 3) if { An }n=1 is a stationary sequence, then h({ An }n=1) = An .
o PROOF. Assume the converse, i.e., there exists a map h with properties 1, 2 and 3. It follows
X that if { An }n=1 is a theorem of the theory T*, then there exists the theorem A from T
X such that h({ An }n=1) = A . Let { Bn }n=1 be the stationary sequence such that Bn is A for
X y any n. Therefore, we have h({ An }n=1) = A = h({ Bn }n=1). Since h is bijection, we obtain
X b { An }n=1 = { Bn }n=1, i.e., any theorem of the theory T* equals some stationary sequence
X K of theorems from the T. On the other hand, let { Cn }n=1 be an Lcontradiction, i.e.,
X 4 Cn Ġ= C , where { Cn }n=1 is a theorem from T* and C is a contradiction. By assumption,
X { Cn }n=1= { Dn }n=1, where { Dn }n=1 is a stationary sequence of the theorems from T, i.e.,
X
for any n, Dn is D and D is a theorem from T. This implies that Cn is Dn but Cn is C , and
X
C is a contradiction, Dn is D , and D is a theorem from T. Since the theory T is consistent,
X we see that C can not be a theorem of T, i.e., C can not be equal to D . This contradiction
proves the theorem.%"
X o DEFINITION 15. Let the map take each formula { An }n=1 from L* to lim An and take each
X term ! from L* to ! . The map : L* L is called a natural embedding of the language
X L* into the language L. On the other hand, let the map * take each formula A from L
X to { An }n=1, where An is A , and take each term ! from L to ! . The map *: L L* is
X called a natural embedding of the language L into the language L*.%"
X = Obviously, if A is a theorem of the theory T, then *( A ) is also a theorem of the theory
T*. The return relation, as follows from Theorem 9, is not correct.
Maps and * can be extended to the set of inferences in the theories T* and T. Namely
X z if an inference {n An }n=1 is given in the theory T*, then we can define the object ({n
X i An }n=1) = lim{n An } = limn lim An (in accordance with Theorem 10, the object limn
X X lim An is not always an inference of the theory T. In this case, the sign
is used as a
X A formal character). On the other hand, if an inference A is given in the theory T, then by
X 0 definition, put *( A ) ={n An }n=1, where n is and An is A for any n. Obviously, if
X A is an inference in the theory T, then *( A ) is an inference in the theory T*. The
return relation is not always correct (see below).
o THEOREM 10. If the theory T is a consistent theory and the theory T* contains an L X V contradiction, then there exists an inference {n An }n=1 in the theory T* such that
X E ({n An }n=1) is not an inference of the theory T.%"
X o PROOF. Let {n An }n=1 be an inference in the theory T*, where for any n n = H, i.e., the
X ! inference is a proof, and { An }n=1 is an Lcontradiction. In this case, ({n An }n=1) =
X " lim{n An } = limn lim An = lim An , where lim An is a contradiction. Since the
X w# theory T is consistent, we see that the object lim An can not be an inference in the theory
T.%"
X % o DEFINITION 16. Let the theory T be a theory with axiom schemes and { An }n=1 be an axiom
X & of the theory T*. If there exists an m0 such that for any nm An belongs to one axiom
X ' scheme A, then we say that { An }n=1 belongs to the axiom scheme A.%"
X ( The theory T* is called a theory with axiom schemes if the theory T is a theory with
X ) axiom schemes and for any axiom { An }n=1 in T* there exists an axiom scheme A such that
X * { An }n=1 belongs to A. * p-++!! Ԍ o THEOREM 11. Let T and T* be theories with axiom schemes and axioms of different
schemes be mutually independent in the theory T; then axioms of different schemes in
the theory T* are mutually independent.%"
X 1 o PROOF. Assume the converse. Therefore if A , B are axioms of different schemes in the
X theory T, then there does not exist an inference A B in the theory T. Besides, there
X exist axioms of different schemes { An }n=1 and { Bn }n=1 in the theory T* such that
X { An }n=1 and { Bn }n=1 are not mutually independent, i.e., there exists an inference { Bn }n=1
X { An }n=1 in the theory T*. It follows that there exists an m0 such that for any nm
X there exists an inference Bn An in the theory T, where Bn and An are axioms of
different schemes in the theory T. This contradiction proves the theorem.%"
o THEOREM 12. Let the theories T and T* be theories with axiom schemes. If any two axioms
of different schemes in the theory T* are mutually independent, then any two axioms of
different schemes in the theory T are mutually independent.%"
o PROOF. Assume the converse, i.e., any two axioms of different axiom schemes of the theory
X & T* are mutually independent and there exist axioms A and B of different schemes in the
X theory T such that there exists an inference A B in the theory T. Then there exists the
X inference { Bn }n=1 { An }n=1 in the theory T*, where An is A and Bn is B for any n.
X Besides, the stationary sequences { An }n=1 and { Bn }n=1 are axioms of different schemes
in the theory T*. This contradiction proves the theorem.%"
o THEOREM 13. Let the theories T and T* be theories with axiom schemes. Then axioms of
different schemes in the theory T are mutually independent iff axioms of different
schemes in the theory T* are mutually independent.%"
o PROOF. This follows from Theorems 11 and 12.%"
o THEOREM 14. Let the theories T and T* be theories with axiom schemes. If there does not
exist an axiom of the theory T, which can be deduced from any finite set of axioms of
another schemes of the theory T, then this is correct for axioms of the theory T*.%"
o PROOF. Assume the converse, i.e., the condition of the theorem is true and there exist an
X axiom { An }n=1 and a set of axioms of another schemes {{ Bkn }n=1}k=1N in the theory T*
X such that there exists an inference {{ Bkn }n=1}k=1N { An }n=1 in the theory T* and N is
X a finite number. The expression {{ Bkn }n=1}k=1N
guesses that the set of the axioms
X {{ Bkn }n=1}k=1N is regular, i.e., {{ Bkn }n=1}k=1N = {{ Bkn }k=1N}n=1 and there exists an m0
X } such that for any nm there exists an inference { Bkn } k=1N An in the theory T, where the
X l formulas Bkn and An are axioms of different schemes in the theory T. This contradiction
proves the theorem.%"
o THEOREM 15. Let the theories T and T* be theories with axiom schemes. If there does not
exist an axiom of the theory T*, which can be deduced from any finite set of axioms of
another schemes of the theory T*, then this is correct for axioms of the theory T.%"
o PROOF. Assume the converse, i.e., the condition of the theorem is true and there exist an
X & axiom A and axioms B1 , B2 , , Bn , which are of another schemes than A , such that there
X ' exists an inference B1 , B2 , , Bn A in the theory T. Let {{ Bki }i=1}k=1 { Ai }i=1 be the
X ( inference of the theory T*, where Ai is A and Bki is Bk for any i. This inference will be
X ) an inference of the axiom { Ai }i=1 from the axioms { Bki }i=1 of another schemes. This
contradiction proves the theorem.%" ~* p-++!! Ԍ o THEOREM 16. Let the theories T and T* be theories with axiom schemes. There does not
exist an axiom of the theory T*, which can be deduced from any finite set of axioms of
another schemes of the theory T* iff this is correct for axioms of the theory T.%"
o PROOF. See Theorems 14 and 15.%"
X o DEFINITION 17. The theory T (T*) is called semantically complete if any formula from the
language L (L*), which is valid in any model M of the theory T (T*), is a theorem of the
theory T (T*).%"
o THEOREM 17. If the theory T is semantically complete, then the theory T* is semantically
complete.%"
o PROOF. Assume the converse, i.e. the theory T is semantically complete and the theory T*
X
is not. It follows that there exists a formula { An }n=1 of the language L* such that
X { An }n=1 is valid in any model M of the theory T* and at the same time { An }n=1 is not
X a theorem of the theory T*. If { An }n=1 is valid in a model M of the theory T*, then there
X
exists an m0 such that for any nm the formula An is valid in the model M. Since,
according to Theorem 6, the model M of the theory T* is simultaneously a model of the
X theory T, we see that for any nm the formula An is valid in the model M of the theory
T. Since M is any model of the theory T and the theory T is semantically complete, we
X _ see that for any nm the formula An is a theorem of the theory T, i.e., the formula
X N { An }n=1 from the language L* is a theorem of the theory T *. This contradiction proves
the theorem.%"
o THEOREM 18. If the theory T* is semantically complete, then the theory T is semantically
complete.%"
o PROOF. Assume the converse, i.e., the theory T* is semantically complete and the theory T
X is not. It follows that there exists a formula A of the language L such that A is valid in
X any model M of the theory T and at the same time A is not a theorem of the theory T.
X Let { An }n=1 be the formula of the language L* such that An is A for any n. The model
M, according to Theorem 6, is simultaneously a model of the theory T*. Then the
X k formula { An }n=1 is valid in any model M of the theory T*. Since the theory T* is
X T semantically complete, we see that the formula { An }n=1 is a theorem of the theory T*,
X = i.e., the formula A is a theorem of the theory T. This contradiction proves the theorem.%"
o THEOREM 19. The theory T* is semantically complete iff the theory T is semantically
complete.%"
o PROOF. See Theorems 17 and 18.%"
X C" o THEOREM 20. Let a formula { An }n=1 of the language L* be a formula, for which there
X ,# exists an m0 such that for any nm An is B U B , where B is a formula of the language
X $ L. Then for any formula { Cn }n=1 from the language L* there exists an inference in the
X % theory T* { An }n=1 { Cn }n=1.%"
X c& o PROOF. According to the definition, an inference { An }n=1 { Cn }n=1 is an object { An
X L' Cn }n=1, where there exists an p0 such that for any np An Cn is an inference of the
X ;( theory T. If there exists an m0 such that for any nm An is B U B , then an inference
X *) An Cn is always well defined in the theory T for any nm, i.e., p=m.%" *) p-++!! Ԍ After the technical side of the theory we would like to return to more philosophical
aspects of the problem. Below we try to show an example of representation of concrete
philosophical antinomy as an Lcontradiction.
Let S be a consistent formal set theory with the posibility to prove in S a theorem of
X existence of set theoretical universes U0, U1, U2, , where Ui E Ui+1, Ui c Ui+1 for any i, and
X if X E Ui and X Ui, then XUi+1.
X n Let then Ri+1 = {XUi: XX} be an (i+1)Russell set. We can prove that Ri+1 Ui, Ri+1
X ] Ri+1, Ri+1E Ri+2 and Ri+1 Ri+2 are theorems in S for any i. Let S be tlimiting theory, i.e., limit
of infinite sequence of sets is defined in ordinary sense. In particular, limit of infinite sequence
X 5 of sets {Xn}, where XnEXn+1 for any n, equals infinite unification B{Xn}. From here we
X $
receive that infinite sequence {Ri+1}i=0 of iRussell Sets has limit. I shall denote this limit as
X
R. Therefore there exists Linconsistent theory S*, where the infinite consequence of
X formulas {Ri+1 Ri+1 U Ri+1 Ri+2}i=0 is Lcontradiction. Really limit of this consequence is
X contradiction R R U R R. If this contradiction is considered as analogue of Russell
paradox, then we receive the proof that this paradox is antinomy, not mistake.
X 3 On the basis of limiting consequence {Ri+1}i=0 and limiting consequences of formulas we
could to try to interprete some philosophical antinomies, for example, first antinomy of Kant'
Critique. This antinomy asserts that Universe is and at the same time is not limited in space X time. Let XX be axiom of the theory S. Then Ri+1 = Ui+1. Hence iRussell set equals i X Universe, where i=1,2,3,. If X Ri, then we can say that X is ilimited. Thus assertion that
X X is not ilimited is expressed itself in the formula XRi. The first Kantian antinomy can be
interpreted not so much separate proposition as limiting consequence of propositions, i.e., as
X Lcontradiction {Ri+1 Ri+1 U Ri+1 Ri+2}i=0, or {Ui+1 Ui+1 U Ui+1 Ui+2}i=0. Therefore Russell
paradox has obvious connection with the dialectical tradition of philosophical logic and his
formulation today is a certain manifestation of outgrowing of Line of Parmenides in
contemporary logical thinking.
By the same way we can express another philosophical antinomies, for example Hegel
X antinomy of being which is also nonbeing. Formula X Ui can be expressed the idea that X
X is ibeing. Let {Xi}i=0 be a sequence of sets, where Xi+1Ui and Xi+1Ui+1 for any i=0,1,, and
X there exists limit X of sequence {Xi}i=0. For example, Xi+1 = Ri+1. Then sequence {Xi}i=0
can be an expression of principle, which is being and also nonbeing. Really we have L X ] contradiction {Xi+1Ui U Xi+1Ui+1}i=0. Let U be an operation, where {Xi+1Ui}i=0 U
X L {Xi+1Ui+1}i=0 equals by definition {Xi+1Ui U Xi+1Ui+1}i=0. We can understand U
as
metaconjunction such that metaconjunction of two sequences of formulas is sequence of
conjunctions of formulas (by the similar way, we can define another logical operations in logic
X
" of limiting sequences of formulas). Then left member {Xi+1Ui}i=0 of metaconjunction can
X " be read as X is nonbeing
, in accordance with limit lim(Xi+1Ui) = XU. Accordingly,
X # right member {Xi+1Ui+1}i=0 of metaconjunction can be read as X is being
, in accordance
X $ with the limit lim(Xi+1Ui+1) = XU. Finally we can interprete Lcontradiction {Xi+1Ui U
X % Xi+1Ui+1}i=0 as Hegel antinomy there exists a principle which is being and nonbeing
.
In our opinion, by the similar way another philosophical and religious antinomies may
be interpreted in suitable Linconsistent theories. Taking into account the analogy between
method of construction of mathematical continuum and method of Linconsistent theories
formation, one may conclude that numerous antinomies, constantly have been reproduced in
the history of human thinking, are examples of logical irrationalities
. These are antinomies * p-++!! of all the limiting concepts of philosophy, for example, World
, Being
, Consciousness
,
Will
, Freedom
, Personality
, etc. And just as there exists common method of
mathematical irrationalities expression, there could be a common method of logical
irrationalities representation. Author hopes that ideas of this paper could to help us to come
nearer to this method.
b L v Vyacheslav Moiseyev
7 Moscow Medical Stomatological University
Q p-++!! N %" 5 a 5 H2 W B
a " Paraconsistent logic! (A reply to Slater)
by JeanYves B)ziau`!%"] ă
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