METAPHYSICS
POSSIBLE WORLDS (DRAFT 2)
What is a “Possible World”?
Modality and Possible Worlds
Prequal to Modal Logics
“Normal” Modal Systems and Accessibility
Ontological Status of Possible Worlds
Lewis’s Realism
Fictionalism
Actualism
LECTURE
PROFESSOR JULIE YOO
WHAT IS A “POSSIBLE WORLD”?
People talk a lot about alternative possibilities – you could have gone to a different university,
that global warming could have prevented, George Washington could have remained a farmer
rather than going into politics, or that George Washington could have never been born, and so
on. Sometimes we make these speculations for the fun of it. Other times, we make these kinds
of speculations about alternative ways things could have turned out to give serious explanations
– about why our history led to where we are, why global warming is happening, why planets
don’t collide into each other, and so on.
Equally important, we also talk about things that must be the case or cannot be the case: if you
pour water out of a glass, the water must go down, not up; ice must melt if you plunge it in hot
water; 2 + 2 must be 4 and it can’t be 3, 5, etc.; a triangle must have three sides and can’t ever
have four of five, and so on.
Talk about possibilities, necessities, and impossibilities. This is called modal talk – ‘modal,’
because ‘mode’ connotes ways things could be or ways in which things are true. Philosophers
invoke the notion of possible situations or possible worlds (PW) to accommodate modal talk.
The difference between a possible situation and a possible world is that a possible world is a
complete way the whole world can be, whereas a possible situation a complete way a small
segment of a world can be. Without getting caught up too much at this stage on what a possible
world is, we can look at how philosophers use possible worlds talk to account for modal talk:
•
You could have been born in Canada ⇒ There is a PW where in that world, you exist and
you are born in Canada.
•
It is necessary that 2 + 2 = 4 ⇒ For every PW, it is the case that 2 + 2 = 4.
•
It is impossible that 2 + 2 = 5 ⇒ For no PW, it is the case that 2 + 2 = 5.
Just as propositions can be true or false in the actual world, they can be true or false in other
possible worlds.
MODALITY AND POSSIBLE WORLDS
The various modal claims can be understood in terms of quantification over possible worlds. A
necessary proposition is true in all possible worlds, and an individual’s have a property
essentially amounts to that individual’s having that property in every world in which what
individual exists. A contingent proposition, on the other hand, is true only in some possible
worlds, and an individual’s having a property contingently amounts to that individual’s having
that property only in some of the worlds in which that individual exists.
Necessarily p ↔ For all w, p is true in w.
a necessarily has F ↔ For all w containing a, a has F in w.
Contingently p ↔ There exists a w such that p is true in w.
a contingently has F ↔ There exists a w containing a and a has F in w.
The notion of a possible world can be a very powerful tool to explore, defend, or articulate
philosophical positions on all sorts of things, like creating a perfect society, figuring out
conditions under which we have free will or no free will, figuring out what is ethically important,
and lots of other things.
PREQUAL TO MODAL LOGICS
Just as our talk of what is the case and what is not the case can (partly) be systematized into a
logical system – the system of sentential logic and quantificational logic – that tells us what can
be derived from what, when a logically complex sentence is true, we have a system for dealing
with modalities. Let us refresh our memories of sentential and predicate logic.
The basic units of sentential logic are declarative sentences (represented by upper case letters):
• George Washing was a president of the US: G
• Everyone is miserable at rush hour: M
• Someone stole the last cookie: S
The basic units of predicate (quantificational) logic are certain components of declarative
sentences, namely an individual and a feature that individual has or lacks:
• George Washing: g
being a president: P
Pg
• Everyone: ∀x
miserable at rush hour: M
(∀x) Mx
• Someone: ∃x
stole the last cookie: S
(∃x) Sx
When it comes to modal logic, the things getting systematized are the statement of possibility,
necessity, and impossibility, as we say above, and what philosophers call counterfactual
conditional, or subjunctive conditionals:
•
•
•
If George Washington had remained a farmer, he would not have been a president.
If there weren’t so many cars, we wouldn’t be miserable driving.
If kangaroos had no tails, then they would fall over. (David Lewis’s example)
Counterfactuals are different from regular ‘if – then’ statements/conditionals, because the
antecedent of a counterfactual is false, but that doesn’t automatically make the whole statement
true (which is how regular conditionals are evaluated).
Unlike quantificational logic, where there is only one system, we have four systems from which
to choose in the case of modal logic. Moreover, the each system can be combined with
sentential logic as well as with predicate logic (which gets very complicated). The choice of a
modal logic reflects a philosopher’s philosophical and logical commitments.
“NORMAL” MODAL SYSTEMS AND ACCESSIBILITY
 - Possibility Operator
(∃x)Fx = It is possible that x is/has F: It is possible that there are flying cars.
(∃x)Fx = x is/has F possibly (contingently/accidentally): My car could be free of dents.
 - Necessity Operator:
(∃x)Fx = It is necessary that x is/has F: It is necessary that white bread is white.
(∃x) Fx = x has F necessarily (essentially): A cow is a bovine essentially.
Interdefinability of Possibility and Necessity
[A = ~~ A] ↔ [A = ~~A]
The Accessibility Relation
What is judged necessary or possible, or which statement of necessity or possibility, is
determined true false, is a matter of what is true or false in other words “relative to” the world
w0 in which the statement is made. In other words, there has to be a clear way of restricting the
worlds when judging the truth value of modal claims. Worlds are restricted according to what
logicians call accessibility relations, and following Garson 2006, we can define accessibility
relations (or the Kripke relation, after Saul Kripke, who proposed it):
A relation R holding between worlds w and w* where if A is true in w,
then w* is a world where A must also be true.
In short, worlds, w1 – wn are the worlds we can look at to determine the truth or falsity of a
modal claim in w0 just in case those worlds w1 – wn are accessible to w0. For instance, when
we say that it is possible that you are a resident of Canada, we are saying that there is an
accessible world where in that world, you are a resident of Canada. If so such accessible worlds
exist for this claim, then we would say that it is not possible.
Accesibility Axioms
So the question, then, is what makes certain worlds accessible and not others? The full answer to
this requires a commitment to two separate things: 1) a set of axioms regarding accessibility,
which then shores up a modal system you let yourself use, and 2) thesis about which axioms are
plausible, given your philosophical views about things like logic, laws of nature, and so on.
Let’s just focus on the first thing, the axioms.
Axiom (T): Reflexivity: This is the axiom that says that all worlds are
accessible to themselves.
Axiom (B): Symmetry: This is the axiom that says if w1 is accessible to
w2, then w2 is accessible to w1.
Axiom (4): Transitivity: This is the axiom that says w1 is accessible to
w2, and w2 is accessible to w3, then w1 is accessible to w3.
Axiom (E): Euclidian: This is the axiom that says w1 is accessible to w2,
and w1 is accessible to w3, then w2 is accessible to w3.
With these axioms, construct logical systems that select from among these axioms, starting with
the system K:
K
1) Sentential Logic (SL)
2) If A is a theorem of logic, then so is  K (Necessitation Rule)
3)  (A → B) → (A → B)
Systems
Accessibility Relations
T
1) A → A
K plus Axiom (T): Reflexivity
B
1) System T
2) A →   A
Axioms (T) and (B): Reflexivity and Symmetry
S4
1) System T
2) A →   A
Axioms (T) and (4): Reflexivity and Transitivity
S5
1) System T
2)  A →   A
Axioms (T) and (B) and (4): Reflexivity and
Symmetry and Transitivity
With these systems in place, we can ask: Does physical necessity satisfy these axioms? We can
get even more specific about certain bodies of laws of nature – laws of biology, laws of geology,
laws of psychology, and so on. What about logical necessity – does logical necessity satisfy
these axioms?
Conversely, we can ask: Which system could support the contingency of your existence? Which
axioms support the necessity of your falling under the sortal, human being? What about the
necessity of your origin (coming from that specific egg and sperm that turned into your)? Which
axioms could support the existence of God?1 When dealing with these latter axiom, we expand
1
This excerpt from T. Parent’s Draft of “The Modal Ont Argument ... ” (unpublished), uses quantified modal logic:
When it comes to modal arguments, ‘God’ is given a definition that implies:
our modal systems to include predicate logic. This expansion is called quantified modal logic,
and it gets into a number of very thorny, but interesting, issues.
ONTOLOGICAL STATUS OF POSSIBLE WORLDS
Anti-Realist
-Deflationism
-Modal Fictionalism
Realist
Abstractionism
Concretism
(David Lewis)
Empiricists claim that we do not observe necessities, only what is. Thus, the only necessity is
verbal necessity; it only reflects our decisions to use words a certain way. None of this reflects
anything about the world – only our verbal conventions.
(D1) ‘God’ denotes x only if, necessarily, x exists.
As mentioned, Plantinga in the first instance defines God as “maximally great,” and others similarly define
Him as the “greatest conceivable being,” “the supremely perfect being,” etc. But (D1) isolates a common
element in these definitions; it is also the most important element for argumentative purposes. In fact, if x is
“maximally great” in Plantinga’s terminology, then the existence of x is necessarily necessary. But to
simplify exposition, I here invoke axiom S4 (where adding more boxes on ‘□p’ doesn’t alter its truthcondition). Since Plantinga is already committed to S5, this should be uncontentious in the present debate.
Now unlike the existicorn argument, modal arguments avoid the existential fallacy on first glance. For such
arguments take as an additional premise:
(1) Possibly, God exists.
Prima facie at least, (1) seems entirely reasonable. And (1) with (D1) implies that in some possible world,
God exists in every possible world. Yet given modal axiom B (Adams) or axiom S5 (Plantinga), it follows
that God actually exists. (I henceforth focus on Plantinga’s version, but I believe my remarks carry over to
Adams and the rest as well.) Formally reconstructed, where ‘g’ is a name for God as per (D1), the modal
argument then proceeds as follows:
1.
2.
3.
4.
5.
6.
◇(∃x) x = g
◇□p ⊃ □p
□[(∃x) x = g ⊃ □(∃y) y= g]
◇□(∃y) y = g
□(∃x) x = g
(∃x) x = g
[Assume]
[S5]
[By definition of ‘g’]
[1, 3]
[2, 4]
[5]
Tooley (1981), however, observes that there remains a Gaunilo-type problem. We can alter Salmon’s
“existicorn” example to illustrate:
(D2) ‘Necessicorn’ denotes x only if x is a unicorn and x necessarily exists.
Suppose further that:
(2) Necessicorns are possible.
In a parallel manner, then, it follows that necessicorns are necessary and, hence, that there actually exists a
kind of unicorn.
Early 20th century philosophers worried that modal notions were semantically inferentially illbehaved. Instead of conforming to the extensional systems of propositional logic, predicate
logic, and set theory, modal notions rendered contexts intensional. It was not for lack of modal
logic; indeed, the problem was that there were too many of them, where different systems gave
different answers to the question, “What modal truth follows from what other modal truths?” It
was not until philosophers started to appeal to the Leibnizian notion of possible worlds that the
modal logics became more tractable.
Lewis’s Realism about PW
Lewis – Treat PW as concrete entities, the actual world as only one among the many, and use
PW as a reduction base for a nominalist metaphysics. Contra Plantinga, who treats them as
maximally possible states of affairs, the actual world as the only world (Actualism), and does not
use them as a reduction base for other things, but rather as a part of a network of concepts that
can mutually illuminate each other.
For Lewis, there exist other possible worlds, just as real and concrete as the world in which we
inhabit. Our world is not privileged; “actuality” is an indexical term. Therefore, the use of this
term on a different possible worlds vindicates its real existence as much as our world.
Lewis’s Argument for Realism about Possible Worlds
1. Our talk about alternative possibilities commits us realism about possible worlds.
2. If (1), then possible worlds existence, unless there is an anti-realist paraphrase of
possible worlds.
3. There is no anti-realist paraphrase of possible worlds.
4. Possible worlds exist.