[extropy-chat] Possible Worlds Semantics

[Ian Goddard]

'Possible worlds' is a concept underlying Saul
Kripke's modal logic. Many express bewilderment with
the concept, assuming it refers to spooky
quasi-existent worlds. In Kate Kearns' text
"Semantics" [*] I just ran into the clearest
presentation of possible worlds I've seen that also
makes clear that they exist as (imaginary) points of
reference in everyday natural language. As Kearns
says: "the actual world is not enough to pin down a
sentence meaning [...] other worlds are also needed."

Kearns gives the example of a brown dog named Midge.
(I'll elaborate further by using standard modal
notation she does not.) Now, in the actual world where
Midge is brown (let's call the actual world w1) these
three statements have the same truth value:

a) Midge isn't purple.
b) Midge isn't white.
c) Midge is brown.

Where V is a valuation function mapping sentences a,
b, and c at w1 to truth values (1 = true; 0 = false):

V(w1,a) = 1
V(w1,b) = 1
V(w1,c) = 1

So the *extensional* meaning of all the sentences is
the same (the extensional meaning of a sentence is its
truth value wrt the actual world). But we know that
the meanings of the sentences are not the same. Their
meanings reside in their *intensional* meaning, which
is found in the unique Truth Set of each sentence. The
truth set of a sentence is the set of its truth values
in all possible worlds.

In the possible world w2 where Midge is purple, we
have it that of the three sentences above:

V(w2,a) = 0
V(w2,b) = 1
V(w2,c) = 0

In the possible world w3 where Midge is white, we have
it that of the three sentences above:

V(w3,a) = 1
V(w3,b) = 0
V(w3,c) = 0

Here then are the Truth Sets for the three sample
sentences (represented by their respective capital
letters) over the three sample worlds:

A = { (w1,1), (w2,0), (w3,1) }
B = { (w1,1), (w2,1), (w3,0) }
C = { (w1,1), (w2,0), (w3,0) }

So while the extensional meaning confined to the
actual world (w1) is the same for the three sentences,
the unique intensional meaning of each sentence exists
in its unique truth set.

I've read a number of texts on modal logic and from
them have gained a good understanding. But the short
description in Kearns "Semantics" [*] outlined above
just struck me as the clearest I've read. Moreover,
her example seems to make it obvious that possible
worlds is not some aloof Ivory Tower pipedream, but
seems inherent in natural language. When I say "My car
isn't orange," my statement refers to a possible, or
conceivable, world where it IS orange and denotes that
the actual world differs from that world, which is to
say the valuation of my statement in this world
differs from its valuation in that conceivable world.
Ie, this world is not a possible world where my car is
orange. ~Ian 

_____________________________________________________
[*] Kearns, K. (2000). "Semantics." New York: St
Martin's Press.
http://www.amazon.com/gp/product/0312231830/

http://iangoddard.net

"All inferences from experience suppose, as their
foundation, that the future will resemble the past,
and that similar powers will be conjoined with similar
sensible qualities. If there be any suspicion, that
the course of nature may change, and that the past may
be no rule for the future, all experience becomes
useless, and can give rise to no inference or
conclusion." - David Hume 
 



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