Propositions

Recap

Common Three-Way Equivalence:

• Sentence meanings• The objects of the attitudes• The referents of ‘that’-clauses

We can call whatever is all of these things a “proposition.” Now we have the question: what are propositions?

Facts

Facts are complex entities composed of objects, properties, and relations “going together” in the world– e.g. objects instantiating properties and multiple objects instantiating relations.

Facts as Propositions?

There aren’t any “false facts.” But there are:

• Sentences that are false.• Beliefs that are false.• Attitude ascriptions that ascribe false attitudes (e.g. beleifs).

States of Affairs

States of affairs are like facts (composed of objects, properties, and relations “going together”), but they can be merely possible.

The state of affairs Michael is not wearing pants exists, even though Michael is not not wearing pants. It exists but it fails to obtain. It is merely possible.

States of Affairs as Propositions?

1. There are no impossible states of affairs, but we can believe or mean impossible things.

2. The state of affairs Superman can fly is the same state of affairs Clark Kent can fly, but I can believe that Superman can fly without believing Clark Kent can fly.

3. States of affairs are not truth-evaluable, but we might think propositions are: the things we believe are true or false.

4. Compositionality

Compositionality Worry

TRUE: Lois Lane believes Superman can fly.

FALSE: Lois Lane believe Clark Kent can fly.

What a Theory of Propositions NeedsPropositions should be:

• Fine-grained• Truth-evaluable• Sometimes necessarily false (impossible)• Compositionally determined

Propositions as Sets of Possible Worlds

Possible Worlds

Some things are not true, but they could have been true.

It could have been true that there were talking donkeys, even though there aren’t actually any talking donkeys.

Note on ‘Could Have’English modal auxiliaries have lots of meanings:

Personal possibility: I could not have killed him; he was my brother.Physical possibility: I could have killed him; I had a gun.Physical possibility: I could not have caught the plane; I arrived too late.Metaphysical possibility: I could have killed him, I could have caught the plane.Metaphysical possibility: I could not have been red and green all over.

Possible WorldsSome philosophers have tried to analyze possibility in terms of possible worlds:

It could have been true that P.=

In some possible world, it is true that P.

The MultiverseImportant: some physicists believe our universe is part of a multiverse of universes. This is different from the philosophers notion of possible world.

In particular, physicists are not committed to the claim that in some other universe, donkeys talk.

Possible WorldsPhilosophers who believe in possible worlds disagree about what they are. According to Lewis:• Possible worlds are just as real,

and made out of the same sorts of things as the world we live in.• They are universes that are not

spatially connected to ours, so we cannot go there or change what happens there.

Possible WorldsRobert Stalnaker, however, thinks that possible worlds are not concrete universes. They are instead a type of uninstantiated property:

Possible worlds are maximally specific ways that our world could have been.

Possible Worlds

Possible worlds are maximally specific ways that our world could have been.

This doesn’t particularly help with analyzing modality.

Truth-Conditions

The verificationists thought that the meaning of a sentence was the set of all experiences, which if you had them you would know the sentence was true.

Another, currently more popular view, is that the meaning of a sentence is the set of all conditions under which it is true (whether we would know it or not).

Possible Worlds Semantics

One way of understanding truth conditions is with possible worlds:

The meaning of a sentence S is the set of all possible worlds where that sentence is true, {w: S is true in w}.

Common Ground

Following Stalnaker, let’s define the common ground as the set of all possible worlds that are live options in our conversation: things that might be true, for all we are presupposing.

Common Ground

Assertion

Now suppose someone asserts something new, something that we did not presuppose to be true already: P.

According to possible worlds semantics the meaning of P is the set of worlds {w: P is true in w}

P worldsNot-P worlds

Update of Common Ground

If the assertion is accepted, then we remove all the not-P worlds from the common ground, arriving at a new common ground where P is presupposed.

P worldsNot-P worlds

BeliefRemember that propositions are not just the meanings of sentences, they’re supposed to be the objects of the attitudes.We can define your belief state similar to the common ground. Your belief state is the set of all possible worlds that your evidence hasn’t eliminated as actual.

Belief

A believes P = P is true in every world of A’s belief state.

BeliefThe goal of conversation is to narrow down the common ground.

The goal of inquiry is to narrow down the belief state. If we got it down to just one possible world, then we would know everything: we’d have ruled out every other possibility.

ImpossibilityWe can believe impossible things. Can the possible worlds account of propositions explain this?

Yes! Impossible propositions are all identical with the null set (the set that has no members).

This does mean that if you believe one impossible thing, you believe them all (and also everything else).

Problems for Propositions = Sets of Possible Worlds

The Deduction ProblemThe axioms of set theory are easy to understand, and once they are explained to someone, in general that person will believe them. They contain things like:

• Two sets are equal (are the same set) if they have the same elements.• If x and y are sets, then there exists a set which contains x and y as

elements.• For any set x, there is a set y that contains every subset of x.

The Deduction ProblemBut the basic, seemingly obvious axioms of set theory allow us to prove very strange things, like the Banach-Tarski paradox:

The Deduction Problem

On the possible worlds conception of propositions, you believe every deductive consequence of the things you believe.

Why? Because if P is true in every world in your belief state, AND Q is a logical consequence of P, then Q will be true in every world in your belief state.

No Such Thing as Logic PuzzlesSuppose there is this little town with a finite numer of people: (1) No two inhabitants have exactly the same number of hairs. (2) No inhabitant has exactly 409 hairs. (3) There are more inhabitants than there are hairs on the head of any

inhabitant.

According to the possible worlds conception of propositions, you know the largest possible number of people who live in the town.

The Aboutness ProblemIt seems like a semantic feature of a sentence that it is about certain things and not about others. Similarly, we like to think that our beliefs are about certain things and not about others.

There isn’t really any sense to be made of “aboutness” on the propositions-are-sets-of-possible-worlds view.

Suppose I believe: Angelina Jolie is the highest paid actor in Hollywood.

Jolie highest paid actor.

The Aboutness ProblemEvery world where it is true that Angleina Jolie is the highest paid actor in Hollywood is a world where Angelina Jolie exists.

But we can’t analyze “Sentence S is about person P” as “P is in every world in which S is true.”

Why not? Well many philosophers think that your parents are essential to your existence. You could not have been born to different parents. If you exist, your parents exist.

Jolie highest paid actor.

Lewis’s Two Gods“Consider the case of the two gods. They inhabit a certain possible world, and they know exactly which world it is. Therefore they know every proposition that is true at their world. Insofar as knowledge is a propositional attitude, they are omniscient…”

Lewis’s Two Gods

“…Still I can imagine them to suffer ignorance: neither one knows which of the two he is. They are not exactly alike. One lives on top of the tallest mountain and throws down manna; the other lives on top of the coldest mountain and throws down thunderbolts. Neither one knows whether he lives on the tallest mountain or the coldest mountain; nor whether he throws manna or thunderbolts.”

De Se Exceptionalism

1. The manna god knows exactly which world she inhabits.2. She does not know that *I am the manna god.*3. Therefore, *I am the manna god* is not solely about which world

she inhabits.4. Therefore, the de se is special and subject to special semantic

treatment.

Fine-GrainednessOne of the problems with treating states of affairs as meanings was that the state of affairs wherein Clark Kent flies is the same state of affairs wherein Superman flies.

Sets of possible worlds have the same problem: the set of worlds where Clark Kent flies is the set of worlds where Superman flies.

Thus, on this account, if you believe the one proposition, you believe the other as well.

Structured Propositions

Syntactic Structure

Michael likes Paisley

LexiconThe lexicon is a pairing of words with their meanings.

“Michael” →

“Likes” →

“Paisley” →

Structured Proposition

Structural IsomorphismIn general, there is a structural isomorphism (“sameness of form”) between sentences and structured propositions.

To get a structured proposition, we replace the words in the syntactic structure with their meanings.

Some philosophers have thought the isomorphism is violated in cases involving unarticulated constituents.

Unarticulated Constituents

Michael didn’t drink wine

Meaning?

That’s Better

InstantiationLast time we talked about the fundamental way in which objects in the world “go together” with properties and relations: instantiation.

If Michael instantiates baldness, then he’s bald. And if he instantiates having hair, he has hair.

New Way of “Going Together”

Structured propositions introduce a new way for objects and properties (and relations) to go together in the world.

Michael can “go together” with having hair in the proposition that Michael has hair, even if Michael does not have hair.

Proposition: Reality:

Systematicity

One positive aspect of structured propositions is that if they are the meanings of sentences, then they explain systematicity.

When This Is True:

• E1 can combine with E2 to form a grammatical sentence [E1 E2].• E3 can combine with E4 to form a grammatical sentence [E3 E4].• E1 is of the same grammatical category as E3• E2 is of the same grammatical category as E4

When This Is True: Example: ‘Dogs’ can combine with ‘chase cars’ to form the sentence ‘Dogs chase cars.’‘Cats’ can combine with ‘eat mice’ to form the sentence ‘Cats eat mice.’‘Dogs’ is of the same grammatical category as ‘cats.’ (Plural Noun Phrases)‘Chase cars’ is of the same grammatical category as ‘eat mice.’ (Verb Phrases)

Then This Is True

The meanings of [E1 E2] and [E3 E4] are predictably related to the meanings of [E1 E4] and [E3 E2], when the latter are well-formed.

Example: ‘dogs chase cars’ has a meaning that is predictably related to both ‘dogs eat mice’ and ‘cats chase cars.’

Meaning of “Cats Eat Mice”

Meaning of “Dogs Eat Mice”

Compositionality

“The meaning of the whole depends on (and only on) the meanings of the parts and the way they are combined.”

Structured propositions are obviously compositional: they are made out of the meanings of the sentence’s parts and the way it those parts are combined!

Reverse Compositionality

Some philosophers have argued also that “reverse compositionality” is true: if you know the meaning of a complex expression E, then you know the meaning of each of E’s simple parts.

Structured propositions also have an easy time explaining this fact (if it is a fact).

Other Positive FeaturesNo logical omniscience problem: the structured propositions theorist does not have to say that everyone knows all the logical truths.

No deduction problem: she does not have to say that everyone knows the logical consequences of what they know: logic puzzles exist!

No aboutness problem: the proposition that Michael is not wearing pants is about Michael in the following sense– Michael is literally a part of that proposition.

Problems for Structured Propositions

Fineness of GrainProblem for propositions = sets of possible worlds: the set of worlds where

“2 + 2 = 4” is true

is the same as the set of worlds where

“eiπ + 1 = 0” is true

Fineness of Grain

But many people believe that “2 + 2 = 4” is true without believing that “eiπ + 1 = 0” is true.

The structured propositions theorist has no such problem: she can say there are two propositions. On proposition, for example, contains the number 2, the other does not.

Grain Too Fine?However, the structured propositions theorist will also be forced to admit that these are different propositions:

A & B A&B

Meaning of “Superman Flies”

Meaning of “Clark Kent Flies”

Why Are They Truth-Evaluable?

According to the structured propositions theorist, propositions are abstract structures with objects and properties occupying certain places in those structures.

Most abstract structures are not true or false. Why are these ones?

Language-Like StructureStructured propositions obviously have a structure.

Where do they get it from? Why do they have the structure they do?

The isomorphism with language suggests the structure comes from language.

Young Propositions

But if structured propositions get their structure from language, then they could not exist before language.

Thus the proposition that dinosaurs exist did not itself exist until dinosaurs were extinct!

Animal Thought

Also, if the structure of propositions comes from language, and propositions are the objects of thought, doesn’t that mean non-linguistic animals cannot think? (Cf. LOT)

Interpreted Logical Forms

Fineness of Grain

Returning to the Superman/ Clark Kent problem, perhaps we can increase the fineness of grain of structured propositions by not replacing the words with their meanings, but instead adding the meanings to the words.

Interpreted Logical Form

Michael, likes, Paisley,

Meaning of “Superman Flies”

Superman,

flies,

Meaning of “Clark Kent Flies”

Clark Kent,

flies,

Language-Bound Beliefs

The unfortunate thing about ILFs is that now it seems that French people don’t believe the world is round!

Le monde est rond.