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Its name is probably derived from the stock example used but connotes also
its sophistication: It was a masterly argument about a master (see Michael
Psellus, Theologica , 3.129 – 35). Together with Aristotle ’ s sea - battle argument
( De Interpretatione 9), it belongs to a series of arguments pertaining
to the discussion of possibility and necessity and their bearing on the determination
of the future. The master argument hinges on the alleged logical
incompatibility of three intuitively valid conceptions:
(1) The necessity of the past: What is past cannot be changed; thus truths
about the past seem to be necessary.
(2) The closure of the possible over entailment: A possible proposition
does not entail any impossible propositions but only possible ones;
this can be used as a test for checking whether something is indeed
possible (cf., Aristotle, Metaphysics IX 3 – 4).
(3) The existence of unrealized possibilities: There seem to be plenty of
unrealized possibilities. For example, it seems both to be possible that
I sit at noon and that I stand at noon, but at most one of these possibilities
will be realized.
Diodorus ’ aim is to disprove (3), that is, to show that it is inconsistent
to assume that a statement such as “ You are a master ” may be possible,
although it neither is nor will be true. On this basis, Diodorus was able to
argue for his characterization of the possible in temporal terms as that
which either is or will be (Cicero, On Fate 13; Boethius, On De
Interpretatione 234.22). But it leads also to a form of “ logical determinism,
” because if there are no unrealized possibilities, everything is necessary.
His fellow Dialectician Panthoides and others, however, used (2) and (3) to
reject (1), and the Stoic Chrysippus used (1) and (3) to reject (2). Anterior
to this debate, Aristotle was able to hold all three ideas by distinguishing
absolute necessity (of, e.g., logical truth) from time - relative necessity. For
it is only now that singular past facts are unchangeable; when they were
still in the future, they were contingent and thus nonnecessary, because they
could have been changed. As we have no ancient sources about the structure
of Diodorus ’ argument, its reconstruction is somewhat speculative, and
several competing reconstructions have been suggested, using different
modern logical systems such as tense logic or quantifi ed temporal logic with
or without indexicals.
These seem to be the sort of starting - points from which the Master
Argument is posed. The following three propositions mutually confl ict: ‘ Every
past truth is necessary ’ ; ‘ Something impossible does not follow from someThe
Master Argument of Diodorus Cronus 75
thing possible ’ ; and ‘ There is something possible which neither is nor will be
true. ’ Diodorus saw this confl ict and exploited the convincingness of the fi rst
two to establish the conclusion that ‘ Nothing which neither is nor will be true
is possible. ’ (Epictetus, 38A)
P1. If α is or has been the case, then it is necessary that α is or has been
the case.
C1. If α is or has at least once not been the case, then it is not possible
that α is and has always been the case (contraposition, P12).
P2. If α necessarily implies β , and α is possible, then β is possible.
C2. If α necessarily implies β , and β is not possible, then α is not possible
(contraposition, P2).
P4. There is a proposition, p , that is possible but neither is nor will be the
case (assumption for reductio ).
C3. p is possible (simplifi cation, P4).
C4. p neither is nor will be the case (simplifi cation, P4).
P5. If p neither is nor will be the case, then it is or has at least once not
been the case that p will be true (tense logic).
C5. It is or has at least once not been the case that p will be true ( modus
ponens , C4, P5).
C6. It is not possible that it is and has always been the case that p will
be true ( modus ponens , C1, C5).
P6. p necessarily implies that it is now and has always been the case that
p will be true (tense logic).
C7. p is not possible ( modus ponens , conjunction, C2, P6, C6).
C8. There is no proposition that is possible but neither is nor will be
true ( reductio , P4 – C7).
19
Lewis ’ Argument for
Possible Worlds
David Vander Laan
Lewis , David . Counterfactuals . Cambridge, MA : Harvard University Press ,
1973 .
___. On the Plurality of Worlds . Malden, MA : Blackwell , 1986 .
van Inwagen , Peter . “ Two Concepts of Possible Worlds , ” in Ontology,
Identity and Modality: Essays in Metaphysics . Cambridge, UK :
Cambridge University Press , 2001 .
Lycan , William . “ The Trouble with Possible Worlds , ” in The Possible and
the Actual: Readings in the Metaphysics of Modality , edited by Michael
J. Loux , 274 – 316 . Ithaca, NY : Cornell University Press , 1979 .
In the middle of the twentieth century, the notion of possible worlds demonstrated
its power by providing a semantics for modal logic, and the idea
has since become standard equipment in the analytic philosopher ’ s toolbox.
Naturally, the notion of possible worlds raises ontological questions. Are
there really such things? If so, what kinds of things are they? David Lewis
was one of the fi rst to take on these questions. In Counterfactuals , Lewis
defended the ontological foundations of his possible worlds analysis of
counterfactual conditionals. Later, in On the Plurality of Worlds , Lewis
made a sustained case for possible worlds and more fully developed his
“ modal realist ” account of what possible worlds are.
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.
Lewis’ Argument for Possible Worlds 77
Lewis ’ earlier argument for possible worlds is characteristically concise.
Lewis notes that we already believe that there are many ways things could
have been, takes this as an affi rmation that certain entities exist, and calls
these entities “ possible worlds. ”
One reason the argument has been controversial is that Lewis took the
actual world to be what we ordinarily call “ the universe ” and took other
possible worlds to differ from the universe “ not in kind but only in what
goes on in them ” (Lewis Counterfactuals , 85). Worlds are thus concrete

Its name is probably derived from the stock example used but connotes also

its sophistication: It was a masterly argument about a master (see Michael

Psellus, Theologica , 3.129 – 35). Together with Aristotle ’ s sea - battle argument

( De Interpretatione 9), it belongs to a series of arguments pertaining

to the discussion of possibility and necessity and their bearing on the determination

of the future. The master argument hinges on the alleged logical

incompatibility of three intuitively valid conceptions:

(1) The necessity of the past: What is past cannot be changed; thus truths

about the past seem to be necessary.

(2) The closure of the possible over entailment: A possible proposition

does not entail any impossible propositions but only possible ones;

this can be used as a test for checking whether something is indeed

possible (cf., Aristotle, Metaphysics IX 3 – 4).

(3) The existence of unrealized possibilities: There seem to be plenty of

unrealized possibilities. For example, it seems both to be possible that

I sit at noon and that I stand at noon, but at most one of these possibilities

will be realized.

Diodorus ’ aim is to disprove (3), that is, to show that it is inconsistent

to assume that a statement such as “ You are a master ” may be possible,

although it neither is nor will be true. On this basis, Diodorus was able to

argue for his characterization of the possible in temporal terms as that

which either is or will be (Cicero, On Fate 13; Boethius, On De

Interpretatione 234.22). But it leads also to a form of “ logical determinism,

” because if there are no unrealized possibilities, everything is necessary.

His fellow Dialectician Panthoides and others, however, used (2) and (3) to

reject (1), and the Stoic Chrysippus used (1) and (3) to reject (2). Anterior

to this debate, Aristotle was able to hold all three ideas by distinguishing

absolute necessity (of, e.g., logical truth) from time - relative necessity. For

it is only now that singular past facts are unchangeable; when they were

still in the future, they were contingent and thus nonnecessary, because they

could have been changed. As we have no ancient sources about the structure

of Diodorus ’ argument, its reconstruction is somewhat speculative, and

several competing reconstructions have been suggested, using different

modern logical systems such as tense logic or quantifi ed temporal logic with

or without indexicals.

These seem to be the sort of starting - points from which the Master

Argument is posed. The following three propositions mutually confl ict: ‘ Every

past truth is necessary ’ ; ‘ Something impossible does not follow from someThe

Master Argument of Diodorus Cronus 75

thing possible ’ ; and ‘ There is something possible which neither is nor will be

true. ’ Diodorus saw this confl ict and exploited the convincingness of the fi rst

two to establish the conclusion that ‘ Nothing which neither is nor will be true

is possible. ’ (Epictetus, 38A)

P1. If α is or has been the case, then it is necessary that α is or has been

the case.

C1. If α is or has at least once not been the case, then it is not possible

that α is and has always been the case (contraposition, P12).

P2. If α necessarily implies β , and α is possible, then β is possible.

C2. If α necessarily implies β , and β is not possible, then α is not possible

(contraposition, P2).

P4. There is a proposition, p , that is possible but neither is nor will be the

case (assumption for reductio ).

C3. p is possible (simplifi cation, P4).

C4. p neither is nor will be the case (simplifi cation, P4).

P5. If p neither is nor will be the case, then it is or has at least once not

been the case that p will be true (tense logic).

C5. It is or has at least once not been the case that p will be true ( modus

ponens , C4, P5).

C6. It is not possible that it is and has always been the case that p will

be true ( modus ponens , C1, C5).

P6. p necessarily implies that it is now and has always been the case that

p will be true (tense logic).

C7. p is not possible ( modus ponens , conjunction, C2, P6, C6).

C8. There is no proposition that is possible but neither is nor will be

true ( reductio , P4 – C7).

19

Lewis ’ Argument for

Possible Worlds

David Vander Laan

Lewis , David . Counterfactuals . Cambridge, MA : Harvard University Press ,

1973 .

___. On the Plurality of Worlds . Malden, MA : Blackwell , 1986 .

van Inwagen , Peter . “ Two Concepts of Possible Worlds , ” in Ontology,

Identity and Modality: Essays in Metaphysics . Cambridge, UK :

Cambridge University Press , 2001 .

Lycan , William . “ The Trouble with Possible Worlds , ” in The Possible and

the Actual: Readings in the Metaphysics of Modality , edited by Michael

J. Loux , 274 – 316 . Ithaca, NY : Cornell University Press , 1979 .

In the middle of the twentieth century, the notion of possible worlds demonstrated

its power by providing a semantics for modal logic, and the idea

has since become standard equipment in the analytic philosopher ’ s toolbox.

Naturally, the notion of possible worlds raises ontological questions. Are

there really such things? If so, what kinds of things are they? David Lewis

was one of the fi rst to take on these questions. In Counterfactuals , Lewis

defended the ontological foundations of his possible worlds analysis of

counterfactual conditionals. Later, in On the Plurality of Worlds , Lewis

made a sustained case for possible worlds and more fully developed his

“ modal realist ” account of what possible worlds are.

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

First Edition. Edited by Michael Bruce and Steven Barbone.

Lewis’ Argument for Possible Worlds 77

Lewis ’ earlier argument for possible worlds is characteristically concise.

Lewis notes that we already believe that there are many ways things could

have been, takes this as an affi rmation that certain entities exist, and calls

these entities “ possible worlds. ”

One reason the argument has been controversial is that Lewis took the

actual world to be what we ordinarily call “ the universe ” and took other

possible worlds to differ from the universe “ not in kind but only in what

goes on in them ” (Lewis Counterfactuals , 85). Worlds are thus concrete