CAS LX 502CAS LX 502SemanticsSemantics

1b. The Truth1b. The Truth

Ch. 1Ch. 1

Specifics of meaningSpecifics of meaning

Basic goals initially:Basic goals initially: What kind of thing is the meaning What kind of thing is the meaning of a sentence?of a sentence?

What kind of thing is the meaning What kind of thing is the meaning of a word (or word part)?of a word (or word part)?

How do they relate?How do they relate?

These initial goals will take us These initial goals will take us through much of the semester, in some through much of the semester, in some form or anotherform or another

Meaning?Meaning?

It’s difficult to pin down It’s difficult to pin down anything we could confidently call anything we could confidently call meaningmeaning..

A sentence might “mean something,” A sentence might “mean something,” but what does that tell us?but what does that tell us?

Where do we start? Images? Where do we start? Images? Memories? Associations?Memories? Associations?

The meaning of a The meaning of a (simple) sentence(simple) sentence

Starting point: Starting point: Homer is Homer is standingstanding..

Whatever the meaning is, if we Whatever the meaning is, if we know it, there’s a trick we can know it, there’s a trick we can do.do.

Separating situationsSeparating situations

Reduce the problem to an easier one:Reduce the problem to an easier one: Forget trying to decide what the Forget trying to decide what the sentence sentence Homer is standingHomer is standing means.means.

Concentrate instead on what someone who Concentrate instead on what someone who knows what knows what Homer is standingHomer is standing means can means can do.do.

If we know the meaning of If we know the meaning of Homer is Homer is standingstanding, we can separate situations , we can separate situations into those in which it is true and into those in which it is true and those in which it is false. We know the those in which it is false. We know the truth conditionstruth conditions..

Truth conditionsTruth conditions

So, whatever meaning of a (simple) So, whatever meaning of a (simple) sentence is, it includes the truth sentence is, it includes the truth conditions of that sentence.conditions of that sentence. Perhaps that’s even Perhaps that’s even allall meaning is. meaning is.

Inside every hydrogen atom, there Inside every hydrogen atom, there is a small blue-and-white orb is a small blue-and-white orb which continually displays the which continually displays the original episode of original episode of The Twilight The Twilight ZoneZone..

Sentences vs. Sentences vs. propositionspropositions

If we’re basing our semantics on a notion of If we’re basing our semantics on a notion of true, false, and possible worlds, the true, false, and possible worlds, the objects we are investigating are things that objects we are investigating are things that can be true or false in a possible world.can be true or false in a possible world.

John realized that Mary closed the John realized that Mary closed the door and began to dance.door and began to dance.

Mary closed the door.Mary closed the door. John realized (something).John realized (something). John is dancing.John is dancing. John was not dancing before.John was not dancing before.

Truth conditions = Truth conditions = meaning?meaning?

Intuitions about logic.Intuitions about logic. Mary closed the door.Mary closed the door. John danced.John danced. Mary closed the door and John Mary closed the door and John danced.danced.

What is the effect of What is the effect of andand??

Truth conditions = Truth conditions = meaning?meaning?

Intuitions about logic.Intuitions about logic. Mary closed the door or John Mary closed the door or John danced.danced.

Mary closed the door and John did Mary closed the door and John did not dance.not dance.

Aside on set theoryAside on set theory

A set is a collection of things:A set is a collection of things: {a, 1, Bill, tomorrow} = A{a, 1, Bill, tomorrow} = A

Here, Here, aa is a member of the set A, is a member of the set A, and and improbableimprobable is not. is not. a a {a, 1, Bill, tomorrow} {a, 1, Bill, tomorrow} improbable improbable {a, 1, Bill, tomorrow} {a, 1, Bill, tomorrow}

Things are either members of a set Things are either members of a set or not.or not. {a, a} = {a}.{a, a} = {a}.

Aside on set theoryAside on set theory There are relations we can describe There are relations we can describe between sets:between sets: Elements that are in both sets are said to be Elements that are in both sets are said to be in the in the intersectionintersection of the sets. Elements that of the sets. Elements that are in one or the other are said to be in the are in one or the other are said to be in the unionunion of the sets. of the sets.

{A, B, D, E} {A, B, D, E} {B, D, F} = {B, D} {B, D, F} = {B, D} {A, B, D} {A, B, D} {B, F, G} = {A, B, D, F, G} {B, F, G} = {A, B, D, F, G}

A set A is a A set A is a subsetsubset of a set B if every element of a set B if every element in A is also in B.in A is also in B.

{A, B} {A, B} {A, B, C}; {A, B, C} {A, B, C}; {A, B, C} {A, B, C} {A, B, C} A is a A is a proper subsetproper subset of B if (1) A is a subset of B if (1) A is a subset of B, and (2) there is some element in B that of B, and (2) there is some element in B that is not in A.is not in A.

{A, B} {A, B} {A, B, C}; {A, B, C} {A, B, C}; {A, B, C} {A, B, C} {A, B, C}

Aside on set theoryAside on set theory If we designate a If we designate a domaindomain, we can also , we can also talk about the complement of a set.talk about the complement of a set. Domain: {A, B, C, D}Domain: {A, B, C, D} A = {A, C}A = {A, C} AA = {B, D} = {B, D}

An element x is in AAn element x is in A if (1) it is not in A, if (1) it is not in A, (2) under no other conditions.(2) under no other conditions.

An element x is in AAn element x is in A if and only if x is not if and only if x is not in A.in A.

An element x is in AAn element x is in A iff x is not in A. iff x is not in A. x x A A iff x iff x A A x x A A x x A A x x A A x x A A

Special setsSpecial sets

We sometimes have cause to talk of We sometimes have cause to talk of a set that has no members: The a set that has no members: The empty setempty set.. {} or Ø{} or Ø

Or a set that has exactly one Or a set that has exactly one member, a member, a singleton setsingleton set.. {Ralph}{Ralph}

Set notationSet notation Set notation is a convenient (and Set notation is a convenient (and potentially quite precise) terminology potentially quite precise) terminology in which to discuss meanings in terms in which to discuss meanings in terms of possible worlds.of possible worlds. P = the set of worlds in which P = the set of worlds in which pp is true. is true. Q = the set of worlds in which Q = the set of worlds in which qq is true. is true. P P Q ? Q ? P P Q ? Q ? PP??

We can define “logical connectives” We can define “logical connectives” quite simply, in a way which matches quite simply, in a way which matches our intuitions about that they mean.our intuitions about that they mean.

More on intuitionsMore on intuitions When are two propositions synonymous?When are two propositions synonymous?

The switch is on. The switch is not off.The switch is on. The switch is not off. Or contrary/opposite?Or contrary/opposite?

The switch is on. The switch is off.The switch is on. The switch is off. When is a proposition tautological?When is a proposition tautological?

The switch is on or the switch is off.The switch is on or the switch is off. When is a proposition contradictory?When is a proposition contradictory?

The switch is on and the switch is off.The switch is on and the switch is off.

EntailmentEntailment This is a red dot.This is a red dot. This is red.This is red. This is a dot.This is a dot.

We believe (intuitively) that if the We believe (intuitively) that if the first one is true, the second and first one is true, the second and third are also third are also necessarilynecessarily true. true.

The first one The first one entailsentails the second and the second and third.third.

And the relation of entailment is easy to And the relation of entailment is easy to state and understand in the set/world terms state and understand in the set/world terms we are developing. This is promising…we are developing. This is promising…

Thoughts and beliefsThoughts and beliefs Sets of possible worlds can also model Sets of possible worlds can also model quite nicely quite nicely information statesinformation states..

Suppose I believe that Mary opened the Suppose I believe that Mary opened the door.door.

There are many, many possible worlds.There are many, many possible worlds. With respect to With respect to Mary opened the doorMary opened the door, , they can be divided into two sets.they can be divided into two sets.

The actual world is one of the possible The actual world is one of the possible worlds.worlds.

My belief is essentially one about which My belief is essentially one about which set of possible worlds contains the set of possible worlds contains the actual world.actual world.

Thoughts and beliefsThoughts and beliefs ww00 = {w: Mary opened the door in w} = {w: Mary opened the door in w}

We can divide the worlds in which Mary We can divide the worlds in which Mary opened the door into two subsets with opened the door into two subsets with respect to the proposition respect to the proposition John dropped John dropped his pencilhis pencil..

If all I know is that Mary opened the If all I know is that Mary opened the door, and someone tells me that John door, and someone tells me that John dropped his pencil (and I believe it), dropped his pencil (and I believe it), then I have narrowed the possibilities then I have narrowed the possibilities for where the actual world is in the for where the actual world is in the space of all possible worlds.space of all possible worlds.

The games we playThe games we play

At least a certain type of At least a certain type of information interchange can be information interchange can be thought of as a game in which thought of as a game in which the goal is to “find the actual the goal is to “find the actual world” by narrowing down the world” by narrowing down the places it might be.places it might be.

The more information you have, The more information you have, the closer you are to finding the closer you are to finding the actual world—the more the actual world—the more places you know it isn’t.places you know it isn’t.

Did Mary open the door?Did Mary open the door?

We can actually get pretty far We can actually get pretty far working with the machinery so working with the machinery so far developed, but it is worth far developed, but it is worth considering that there are other considering that there are other kinds of sentences that don’t kinds of sentences that don’t immediately lend themselves to immediately lend themselves to an interpretation directly in an interpretation directly in terms of truth conditions.terms of truth conditions.

Is “Did Mary open the door?” Is “Did Mary open the door?” true or false?true or false?

Declaratives/Declaratives/interrogativesinterrogatives

We’ve been discussing We’ve been discussing declarativesdeclaratives, , statements.statements.

An An interrogativeinterrogative seems to do something seems to do something else, but we can still understand it by else, but we can still understand it by applying the same kinds of analysis.applying the same kinds of analysis. A declarative divides possible worlds into A declarative divides possible worlds into two (disjoint) sets. And so does a yes/no two (disjoint) sets. And so does a yes/no question. The difference seems to be that a question. The difference seems to be that a declarative is positioning wdeclarative is positioning w00 in one of in one of them, while an interrogative leaves the them, while an interrogative leaves the position of wposition of w00 open—interpreted as a open—interpreted as a request.request.

ImperativesImperatives

(You) close the door(!).(You) close the door(!). Also not true and not false. Also not true and not false. But the fact that it is But the fact that it is imperative seems to be imperative seems to be communicating something like: communicating something like: Make it come about that wMake it come about that w00 is is in the true set of in the true set of the door is the door is closedclosed..

ConversationConversation Actual language use can be quite a bit Actual language use can be quite a bit more complicated, however.more complicated, however.

By leveraging the literal meaning of By leveraging the literal meaning of sentences and using them in certain ways, sentences and using them in certain ways, we can communicate something different. we can communicate something different. In part, this often relies on a mutual In part, this often relies on a mutual “agreement” between speakers that they “agreement” between speakers that they will converse according to certain will converse according to certain conventions, such as:conventions, such as: Do not assert something you know to be false.Do not assert something you know to be false. Be as informative as possible (within Be as informative as possible (within reason).reason).

A distant previewA distant preview How many computers do you own?How many computers do you own?

I own two computers.I own two computers. I own n computers entails that I own n-1 computers.I own n computers entails that I own n-1 computers. I have an actual number k of computers. To say I own I have an actual number k of computers. To say I own k+1 computers would be false.k+1 computers would be false.

It is true that I have 2 computers whether I have 2 It is true that I have 2 computers whether I have 2 or more computers. It’s true that I have 3 computers or more computers. It’s true that I have 3 computers in fewer situations. It’s more informative (stronger) in fewer situations. It’s more informative (stronger) to say that I have 3 computers.to say that I have 3 computers.

Conclusion: I do not own more than two Conclusion: I do not own more than two computers.computers. But notice this is not part of the literal But notice this is not part of the literal meaning, it comes from working out meaning, it comes from working out whywhy this was this was said and not something else, according to the said and not something else, according to the “rules of the game.”“rules of the game.”