DebateLabKIT/deepa2-corpus · Datasets at Hugging Face

text stringlengths 0 1.71k
all cases (logical pluralism, hypothesis to be reduced).
C1. Since they are different collections of valid inferences, there is an
inference X => Y belonging to one of the collections but not to the other
( modus ponens , P4, P5).
C2. If X => Y is a valid inference, then it holds in all cases (equivalence,
simplifi cation, P2).
C3. If X => Y is not a valid inference, then it does not hold in all cases
(equivalence, simplifi cation, P2 ′ ).
C4. X => Y holds in all cases ( modus ponens , C1, C2).
C5. X => Y does not hold in all cases ( modus ponens , C1, C3).
C6. X => Y holds in all cases and X => Y does not hold in all cases (conjunction,
C4, C5).
C7. There are not even two collections of inferences that are different
and hold in all cases ( reductio , P5 – C6).
C8. There is exactly one collection of inferences holding in all cases
(disjunctive syllogism, P3, C7).
29
The Maximality Paradox
Nicola Ciprotti
Adams , Robert . “ Theories of Actuality , ” No û s 8 ( 1974 ): 211 – 31 . Reprinted
in The Possible and the Actual. Readings in the Metaphysics of Modality ,
edited by Michael Loux , 190 – 209 . Ithaca, NY : Cornell University Press ,
1979 . (All subsequent references are to this edition.)
Chihara , Charles . The Worlds of Possibility: Modal Realism and the Semantics
of Modal Logic . Oxford : Clarendon Press , 1998 .
Davies , Martin . Meaning, Quantifi cation, Necessity: Themes in Philosophical
Logic . London : Routledge & Kegan Paul , 1981 .
Divers , John . Possible Worlds . London : Routledge , 2002 .
Grim , Patrick . The Incomplete Universe. Totality, Knowledge, and Truth .
Cambridge, MA : The MIT Press , 1991 .
The suggested label for the argument to follow, the “ maximality paradox, ”
is tentative. As a matter of fact, there currently is no consensus as to what
the most appropriate label might be; what ’ s more, there is not even consensus
as to who fi rst formulated it. Robert Adams is credited with having
been the fi rst to touch on it in print, while the fi rst detailed formulation is
due to Martin Davies.
Such uncertainties about name and origin have possibly to do with the
fact that the maximality paradox is actually a family of closely related, yet
distinct, arguments. For, while each argument relies on a common body of
tenets, namely, well - established facts of standard set theory, it nevertheless
is the case that the salient targets of maximality paradox can, and do, differ.
What is common to each argument, and so what the maximality paradox
essentially consists in, is that a reductio of the hypothesis that a set A exists
of a given sort, namely a totality - set, is arrived at. Different maximality
Just the Arguments: 100 of the Most Important Arguments in Western Philosophy,
First Edition. Edited by Michael Bruce and Steven Barbone.
© 2011 Blackwell Publishing Ltd. Published 2011 by Blackwell Publishing Ltd.
116 Nicola Ciprotti
paradox - style arguments can be wielded, however, against the existence of
distinct set - theoretic (or set - like) totalities, such as, for example, the set of
all possible worlds, the set of all truths, or the set of all states of affairs
(whether or not the maximality paradox also threatens the existence of the
members of such sets, not only the sets themselves, is an issue we shall
briefl y address in closing).
In what follows, we shall focus on Adams ’ original outline of maximality
paradox as subsequently given rigorous shape by John Divers. This version
of the maximality paradox is specifi cally concerned with a particular conception
of possible worlds as world - stories, namely, peculiar sets of propositions.
After due modifi cations, however, the argument can be conferred
wider in scope so as to apply to set - like totalities including elements that
are different from possible worlds.
According to a good deal of philosophers (#99), abstract entities of
various sorts exist. Among them are sets, numbers, states of affairs, propositions,
and properties, to name the ones referred to most often. The majority
of philosophers who believe in abstract objects also include possible worlds
among them. In particular, the suggestion is that possible worlds can be
analyzed as world - stories, that is, sets of propositions that are both (i)
consistent and (ii) maximal collections thereof.
Generally speaking, a set A is consistent if and only if it is possible for
its members to be jointly true (or jointly obtain); a set A is maximal if and
only if, for every proposition p , either A includes p or A includes the contradiction
of p . Such two conditions seem constitutive of the notion of a
possible world: a possible world ought to be possible, that is, a contradiction -
free entity; a possible world ought to be maximal, that is, a complete
alternative way things might be, or have been – one fi lled in up to the
minutest detail.
According to this conception, then, the explicit defi nition of “ possible
world ” is as follows:
(DF) w is a possible world = df w is a set A of propositions such that: (i)
for every proposition p , either p is an element A or p is not an element
A (maximality condition); (ii) the conjunction of the members of A is
consistent (consistency condition).
The main asset of (DF) is that, through it, the existence of possible worlds
is made compatible with an ontology that eschews quantifi cation over
nonactual objects, generally regarded as entia non grata . Qua sets of propositions,
in fact, it is alleged that no more than actually existing abstract
objects – indeed, sets and propositions – is needed for accommodating possible
worlds within a respectable actualist ontology; that is, one free of mere
possibilia . (DF), though, gives rise to the maximality paradox.
The Maximality Paradox 117
Notoriously, the development of a satisfactory logic theory of propositions
[ . . . ] is also beset by formal problems and threats of paradox. One such threat
particularly concerns the [ . . . ] theory [of possible worlds as maximal sets of
propositions]. The theory seems to imply that there are consistent sets composed
of one member of every pair of mutually contradictory propositions.
Furthermore, it follows from the theory, with the assumption that every possible
world is actual in itself, that every world - story, s , has among its members
the proposition that all the members of s are true. Here we are teetering on
the brink of paradox [ . . . ]. This may give rise to a suspicion that the [ . . . ]
theory could not be precisely formulated without engendering some analogue

all cases (logical pluralism, hypothesis to be reduced).

C1. Since they are different collections of valid inferences, there is an

inference X => Y belonging to one of the collections but not to the other

( modus ponens , P4, P5).

C2. If X => Y is a valid inference, then it holds in all cases (equivalence,

simplifi cation, P2).

C3. If X => Y is not a valid inference, then it does not hold in all cases

(equivalence, simplifi cation, P2 ′ ).

C4. X => Y holds in all cases ( modus ponens , C1, C2).

C5. X => Y does not hold in all cases ( modus ponens , C1, C3).

C6. X => Y holds in all cases and X => Y does not hold in all cases (conjunction,

C4, C5).

C7. There are not even two collections of inferences that are different

and hold in all cases ( reductio , P5 – C6).

C8. There is exactly one collection of inferences holding in all cases

(disjunctive syllogism, P3, C7).

29

The Maximality Paradox

Nicola Ciprotti

Adams , Robert . “ Theories of Actuality , ” No û s 8 ( 1974 ): 211 – 31 . Reprinted

in The Possible and the Actual. Readings in the Metaphysics of Modality ,

edited by Michael Loux , 190 – 209 . Ithaca, NY : Cornell University Press ,

1979 . (All subsequent references are to this edition.)

Chihara , Charles . The Worlds of Possibility: Modal Realism and the Semantics

of Modal Logic . Oxford : Clarendon Press , 1998 .

Davies , Martin . Meaning, Quantifi cation, Necessity: Themes in Philosophical

Logic . London : Routledge & Kegan Paul , 1981 .

Divers , John . Possible Worlds . London : Routledge , 2002 .

Grim , Patrick . The Incomplete Universe. Totality, Knowledge, and Truth .

Cambridge, MA : The MIT Press , 1991 .

The suggested label for the argument to follow, the “ maximality paradox, ”

is tentative. As a matter of fact, there currently is no consensus as to what

the most appropriate label might be; what ’ s more, there is not even consensus

as to who fi rst formulated it. Robert Adams is credited with having

been the fi rst to touch on it in print, while the fi rst detailed formulation is

due to Martin Davies.

Such uncertainties about name and origin have possibly to do with the

fact that the maximality paradox is actually a family of closely related, yet

distinct, arguments. For, while each argument relies on a common body of

tenets, namely, well - established facts of standard set theory, it nevertheless

is the case that the salient targets of maximality paradox can, and do, differ.

What is common to each argument, and so what the maximality paradox

essentially consists in, is that a reductio of the hypothesis that a set A exists

of a given sort, namely a totality - set, is arrived at. Different maximality

Just the Arguments: 100 of the Most Important Arguments in Western Philosophy,

First Edition. Edited by Michael Bruce and Steven Barbone.

116 Nicola Ciprotti

paradox - style arguments can be wielded, however, against the existence of

distinct set - theoretic (or set - like) totalities, such as, for example, the set of

all possible worlds, the set of all truths, or the set of all states of affairs

(whether or not the maximality paradox also threatens the existence of the

members of such sets, not only the sets themselves, is an issue we shall

briefl y address in closing).

In what follows, we shall focus on Adams ’ original outline of maximality

paradox as subsequently given rigorous shape by John Divers. This version

of the maximality paradox is specifi cally concerned with a particular conception

of possible worlds as world - stories, namely, peculiar sets of propositions.

After due modifi cations, however, the argument can be conferred

wider in scope so as to apply to set - like totalities including elements that

are different from possible worlds.

According to a good deal of philosophers (#99), abstract entities of

various sorts exist. Among them are sets, numbers, states of affairs, propositions,

and properties, to name the ones referred to most often. The majority

of philosophers who believe in abstract objects also include possible worlds

among them. In particular, the suggestion is that possible worlds can be

analyzed as world - stories, that is, sets of propositions that are both (i)

consistent and (ii) maximal collections thereof.

Generally speaking, a set A is consistent if and only if it is possible for

its members to be jointly true (or jointly obtain); a set A is maximal if and

only if, for every proposition p , either A includes p or A includes the contradiction

of p . Such two conditions seem constitutive of the notion of a

possible world: a possible world ought to be possible, that is, a contradiction -

free entity; a possible world ought to be maximal, that is, a complete

alternative way things might be, or have been – one fi lled in up to the

minutest detail.

According to this conception, then, the explicit defi nition of “ possible

world ” is as follows:

(DF) w is a possible world = df w is a set A of propositions such that: (i)

for every proposition p , either p is an element A or p is not an element

A (maximality condition); (ii) the conjunction of the members of A is

consistent (consistency condition).

The main asset of (DF) is that, through it, the existence of possible worlds

is made compatible with an ontology that eschews quantifi cation over

nonactual objects, generally regarded as entia non grata . Qua sets of propositions,

in fact, it is alleged that no more than actually existing abstract

objects – indeed, sets and propositions – is needed for accommodating possible

worlds within a respectable actualist ontology; that is, one free of mere

possibilia . (DF), though, gives rise to the maximality paradox.

The Maximality Paradox 117

Notoriously, the development of a satisfactory logic theory of propositions

[ . . . ] is also beset by formal problems and threats of paradox. One such threat

particularly concerns the [ . . . ] theory [of possible worlds as maximal sets of

propositions]. The theory seems to imply that there are consistent sets composed

of one member of every pair of mutually contradictory propositions.

Furthermore, it follows from the theory, with the assumption that every possible

world is actual in itself, that every world - story, s , has among its members

the proposition that all the members of s are true. Here we are teetering on

the brink of paradox [ . . . ]. This may give rise to a suspicion that the [ . . . ]

theory could not be precisely formulated without engendering some analogue