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of the semantical paradoxes. (Adams, 207 – 8)
Adams ’ point is as follows. If w is a possible world, namely, a maximal
consistent set of propositions (call it ‘ S ’ ), then qua set S must have a size
– in set - theoretic jargon, a cardinality. What is the cardinality of S? We
know from standard set theory that the power set of S – that is, the set
whose members are all of the subsets of S – has a cardinality that is greater
than S ’ s. It follows that for each member B of the power set, there is the
proposition that B is a set (in fact, it is true that B is a set). Accordingly,
there is a consistent set of propositions that has a cardinality greater than
S, which was supposed to be a maximal consistent set – reductio . Thus, we
have started from the supposition that w was a maximally comprehensive
object, one “ than which nothing greater can be conceived, ” and we have
ended up with an object greater than w. (Taking the union of S and B as
the real maximal consistent set won ’ t do, of course, since by standard set
theory again, there is a set whose cardinality is greater than the union of S
and B.) This is the maximality paradox.
As already hinted at, the maximality paradox has possibly more bite;
while the case below is only concerned with possible worlds as maximal
consistent sets of propositions, structurally identical arguments can be
constructed to question the existence of other set - like totalities. As soon as
some given totality is construed set - theoretically, in fact, there follows by
Cantor ’ s Theorem that such a totality cannot exist. Thus, parallel arguments
have been mounted for proving, for example, that: (i) there is no set
of all truths (Grim); and (ii) there is no set of all possible states of affairs
(Chihara).
Notice that, strictly speaking, maximality paradox - style arguments do not
rule out the (possibility of the) existence of the members involved. As regards,
for example, possible worlds as maximal consistent sets of propositions, their
nonexistence actually follows from the maximality paradox only if the further
premise is taken aboard that, for every possible world, there necessarily is a
corresponding maximal consistent set of propositions. In other words, the
stronger conclusion – that is, the nonexistence of the worlds themselves –
would follow only if the further principle is assumed that, for every domain
118 Nicola Ciprotti
of discourse, the objects in that domain necessarily make up a set or some
set - like object. Unless this is assumed, a possible way out of the maximality
paradox is to treat possible worlds not as sets but proper classes, that is, such
that they cannot in turn be members of a more inclusive collection. Maximality
paradox - style arguments cannot exclude by themselves the (possibility of
the) existence of all - inclusive domains of discourse of a given sort (e.g., the
domain of all possible worlds, the domain of all existent objects, the domain
of all truths, etc.), provided that such domains be (treated as) nonsets. What
maximality paradox - style arguments do rule out is the existence of a set (or
set - like entity) of which all objects of the domain of discourse at stake are
members. Notice fi nally though that regarding possible worlds, the suggested
way out is not trouble free because it seems to undermine a basic tenet of
possible - worlds semantics, that is, that a set W of possible worlds is both
mathematically well defi ned and manageable. This strategy would then
require us to revise robustly our views on what constitutes an acceptable
applied semantical system, like possible - worlds semantics.
The power set of A, symbolized as ℘ (A), is the set of all subsets of a set
A. Thus, ℘ (A) is short for {B \|B ⊆ A}. ℘ (A) has 2 n members if A has n
members.
(Example: suppose that A = {1, 2, 3}. Hence, ℘ (A) = {A, {1, 2}, {1, 3},
{2, 3}, {1}, {2}, {3}, Ø }.)
Theorem (so - called “ Cantor ’ s Theorem, ” CT): For any set A, every
subset of A is smaller than ℘ (A). (Emphasis on “ every ” because every
set A is a subset of itself.)
The Proposition Assumption, PA: For each set A i that is a member of
℘ (A), a proposition p exists that is about that set, namely, the proposition
that A i is a set; if A i ≠ A j , then the proposition that A i is a set and
the proposition that A j is a set are different propositions.
P1. There is a maximal consistent set S of propositions (assumption for
reductio ).
P2. For each set S i that is a member of ℘ (S), there is the proposition p that
S i is a set (Proposition Assumption).
P3. For each such p , either p is an element of S or p is not an element of S
(defi nition of maximality condition).
P4. S includes at least as many propositions as there are elements in ℘ (S)
(P2, P3).
P5. S is a subset of S (standard set theory).
P6. S has a subset that is at least as large as ℘ (S) (P4, P5).
P7. S has no subset as large as ℘ (S) [CT].
C1. There is no maximal consistent set S of propositions ( reductio ,
P1 – P7).
30
An Argument for Free Will
Gerald Harrison
Clarke , Randolph . “ Toward a Credible Agent - Causal Account of Free Will . ”
No û s 27 ( 1993 ): 191 – 203 .
van Inwagen , Peter . An Essay on Free Will . Oxford : Oxford University Press ,
1983 .
___. “ How to Think about the Problem of Free Will . ” Journal of Ethics 12
( 2008 ): 327 – 41 .
Reid , Thomas . Essays on the Active Powers of the Human Mind . Cambridge,
MA : The MIT Press , 1969 .
Strawson , Peter F. “ Freedom and Resentment . ” Proceedings of the British
Academy 48 ( 1962 ): 1 – 25 .
Some philosophers think that our decisions are free only if uncaused, others
that causation is needed to prevent our decisions being uncontrolled; some
think that the causation needs to be indeterministic, others that it needs to
be deterministic, and others that it does not matter either way.
Nevertheless, there is near unanimous agreement that free will is needed
to ground moral responsibility. That is to say, free will is required if we are
to deserve praise, blame, reward, or punishment for our deeds, and if a host
of so - called “ reactive attitudes ” such as resentment, guilt, and forgiveness
are appropriate.
This common ground among disputants provides the basis for a positive
argument for free will. Versions of this argument (which has no specifi c
Just the Arguments: 100 of the Most Important Arguments in Western Philosophy,

of the semantical paradoxes. (Adams, 207 – 8)

Adams ’ point is as follows. If w is a possible world, namely, a maximal

consistent set of propositions (call it ‘ S ’ ), then qua set S must have a size

– in set - theoretic jargon, a cardinality. What is the cardinality of S? We

know from standard set theory that the power set of S – that is, the set

whose members are all of the subsets of S – has a cardinality that is greater

than S ’ s. It follows that for each member B of the power set, there is the

proposition that B is a set (in fact, it is true that B is a set). Accordingly,

there is a consistent set of propositions that has a cardinality greater than

S, which was supposed to be a maximal consistent set – reductio . Thus, we

have started from the supposition that w was a maximally comprehensive

object, one “ than which nothing greater can be conceived, ” and we have

ended up with an object greater than w. (Taking the union of S and B as

the real maximal consistent set won ’ t do, of course, since by standard set

theory again, there is a set whose cardinality is greater than the union of S

and B.) This is the maximality paradox.

As already hinted at, the maximality paradox has possibly more bite;

while the case below is only concerned with possible worlds as maximal

consistent sets of propositions, structurally identical arguments can be

constructed to question the existence of other set - like totalities. As soon as

some given totality is construed set - theoretically, in fact, there follows by

Cantor ’ s Theorem that such a totality cannot exist. Thus, parallel arguments

have been mounted for proving, for example, that: (i) there is no set

of all truths (Grim); and (ii) there is no set of all possible states of affairs

(Chihara).

Notice that, strictly speaking, maximality paradox - style arguments do not

rule out the (possibility of the) existence of the members involved. As regards,

for example, possible worlds as maximal consistent sets of propositions, their

nonexistence actually follows from the maximality paradox only if the further

premise is taken aboard that, for every possible world, there necessarily is a

corresponding maximal consistent set of propositions. In other words, the

stronger conclusion – that is, the nonexistence of the worlds themselves –

would follow only if the further principle is assumed that, for every domain

118 Nicola Ciprotti

of discourse, the objects in that domain necessarily make up a set or some

set - like object. Unless this is assumed, a possible way out of the maximality

paradox is to treat possible worlds not as sets but proper classes, that is, such

that they cannot in turn be members of a more inclusive collection. Maximality

paradox - style arguments cannot exclude by themselves the (possibility of

the) existence of all - inclusive domains of discourse of a given sort (e.g., the

domain of all possible worlds, the domain of all existent objects, the domain

of all truths, etc.), provided that such domains be (treated as) nonsets. What

maximality paradox - style arguments do rule out is the existence of a set (or

set - like entity) of which all objects of the domain of discourse at stake are

members. Notice fi nally though that regarding possible worlds, the suggested

way out is not trouble free because it seems to undermine a basic tenet of

possible - worlds semantics, that is, that a set W of possible worlds is both

mathematically well defi ned and manageable. This strategy would then

require us to revise robustly our views on what constitutes an acceptable

applied semantical system, like possible - worlds semantics.

The power set of A, symbolized as ℘ (A), is the set of all subsets of a set

A. Thus, ℘ (A) is short for {B |B ⊆ A}. ℘ (A) has 2 n members if A has n

members.

(Example: suppose that A = {1, 2, 3}. Hence, ℘ (A) = {A, {1, 2}, {1, 3},

{2, 3}, {1}, {2}, {3}, Ø }.)

Theorem (so - called “ Cantor ’ s Theorem, ” CT): For any set A, every

subset of A is smaller than ℘ (A). (Emphasis on “ every ” because every

set A is a subset of itself.)

The Proposition Assumption, PA: For each set A i that is a member of

℘ (A), a proposition p exists that is about that set, namely, the proposition

that A i is a set; if A i ≠ A j , then the proposition that A i is a set and

the proposition that A j is a set are different propositions.

P1. There is a maximal consistent set S of propositions (assumption for

reductio ).

P2. For each set S i that is a member of ℘ (S), there is the proposition p that

S i is a set (Proposition Assumption).

P3. For each such p , either p is an element of S or p is not an element of S

(defi nition of maximality condition).

P4. S includes at least as many propositions as there are elements in ℘ (S)

(P2, P3).

P5. S is a subset of S (standard set theory).

P6. S has a subset that is at least as large as ℘ (S) (P4, P5).

P7. S has no subset as large as ℘ (S) [CT].

C1. There is no maximal consistent set S of propositions ( reductio ,

P1 – P7).

30

An Argument for Free Will

Gerald Harrison

Clarke , Randolph . “ Toward a Credible Agent - Causal Account of Free Will . ”

No û s 27 ( 1993 ): 191 – 203 .

van Inwagen , Peter . An Essay on Free Will . Oxford : Oxford University Press ,

1983 .

___. “ How to Think about the Problem of Free Will . ” Journal of Ethics 12

( 2008 ): 327 – 41 .

Reid , Thomas . Essays on the Active Powers of the Human Mind . Cambridge,

MA : The MIT Press , 1969 .

Strawson , Peter F. “ Freedom and Resentment . ” Proceedings of the British

Academy 48 ( 1962 ): 1 – 25 .

Some philosophers think that our decisions are free only if uncaused, others

that causation is needed to prevent our decisions being uncontrolled; some

think that the causation needs to be indeterministic, others that it needs to

be deterministic, and others that it does not matter either way.

Nevertheless, there is near unanimous agreement that free will is needed

to ground moral responsibility. That is to say, free will is required if we are

to deserve praise, blame, reward, or punishment for our deeds, and if a host

of so - called “ reactive attitudes ” such as resentment, guilt, and forgiveness

are appropriate.

This common ground among disputants provides the basis for a positive

argument for free will. Versions of this argument (which has no specifi c

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