CAS LX 502 Semantics

CAS LX 502CAS LX 502SemanticsSemantics

2a. Reference, 2a. Reference, Compositionality, LogicCompositionality, Logic

2.1-2.32.1-2.3

Meaning as truth Meaning as truth conditionsconditions

We know the meaning of We know the meaning of pp if we know if we know the conditions under which the conditions under which pp is is true.true. conditions under which conditions under which pp is true = is true = which are the possible worlds in which which are the possible worlds in which pp holds holds

possible worlds = ways things might bepossible worlds = ways things might be The meaning of The meaning of pp: A specification : A specification of possible worlds.of possible worlds.

Recall the trick we can Recall the trick we can dodo

Homer standsHomer stands.. True iff Homer stands.True iff Homer stands.

How do we arrive at How do we arrive at truth conditions?truth conditions?

Homer standsHomer stands. . Marge standsMarge stands.. True iff Homer stands.True iff Homer stands. True iff Marge stands.True iff Marge stands.

Two parts of Two parts of Homer Homer standsstands

Homer standsHomer stands. . Marge standsMarge stands.. Homer stands. Bart’s father Homer stands. Bart’s father stands.stands.

The The HomerHomer part and the part and the standsstands part part

Homer stands.Homer stands. We use the name We use the name HomerHomer to to refer to that guy.refer to that guy.

Homer stands is true Homer stands is true when that guy has the when that guy has the property that he property that he standsstands (being upright on his (being upright on his feet).feet).

Other things/people can Other things/people can standstand, and we feel that , and we feel that standingstanding should be should be basically the same basically the same regardless of who we say regardless of who we say holds that property.holds that property.

Unsaturated Unsaturated propositionspropositions

A proposition with a “hole” in it A proposition with a “hole” in it is called an is called an unsaturated unsaturated propositionproposition..

It’s something that, once we fill It’s something that, once we fill in the hole, will be true or in the hole, will be true or false (of a given possible false (of a given possible world).world).

Portner drawsPortner drawsthese like so:these like so:

true

false

Unsaturated Unsaturated propositionspropositions

A proposition with a “hole” in A proposition with a “hole” in it is called an it is called an unsaturated unsaturated propositionproposition..

Thus:Thus: true

false

Unsaturated Unsaturated propositionspropositions

Although perhaps we could come up Although perhaps we could come up with a better picture, the idea is with a better picture, the idea is that:that: Homer standsHomer stands: the possible worlds in : the possible worlds in which Homer standswhich Homer stands

StandsStands: (unsaturated) Given a referent : (unsaturated) Given a referent x, the possible worlds in which x, the possible worlds in which xx stands.stands.

HomerHomer::

true

false

Meaning is Meaning is compositionalcompositional

It seems that there is something common It seems that there is something common across all the propositions we might across all the propositions we might express using express using HomerHomer..

And something in common across all the And something in common across all the propositions we might express using propositions we might express using standsstands.. Homer stands. Homer snores. Marge standsHomer stands. Homer snores. Marge stands..

Given that each word seems to have a Given that each word seems to have a consistent contribution to the meaning, consistent contribution to the meaning, (to some extent) regardless of the (to some extent) regardless of the sentence in which it appears…sentence in which it appears…

Meaning is Meaning is compositionalcompositional

We hypothesize this:We hypothesize this:

Meaning is compositionalMeaning is compositionalThe meaning of a sentence is formed The meaning of a sentence is formed from the meanings of its parts, and from the meanings of its parts, and the way in which they are arranged.the way in which they are arranged.

Homer strangles Bart. Bart strangles Homer strangles Bart. Bart strangles Homer.Homer.

Meaning is Meaning is compositionalcompositional

And it really And it really hashas to be compositional. to be compositional. We after all know what the world has to We after all know what the world has to look like in order for a sentence to be look like in order for a sentence to be true, even if we haven’t heard the true, even if we haven’t heard the sentence before—and have to compute the sentence before—and have to compute the meaning.meaning.

So the project here is really:So the project here is really: Understand the pieces of meaningUnderstand the pieces of meaning Understand how they combine to form larger Understand how they combine to form larger units of meaningunits of meaning

Where are we so far?Where are we so far? In the set of things that we’ve been In the set of things that we’ve been considering as part of meaning:considering as part of meaning:

Possible worldPossible world: A state of affairs.: A state of affairs. One special possible world is the actual One special possible world is the actual world, wworld, w00..

IndividualsIndividuals: Referents, like Homer.: Referents, like Homer. PropositionsPropositions: Sets of possible worlds: Sets of possible worlds

In which the proposition is “true.”In which the proposition is “true.”

Unsaturated Unsaturated propositionspropositions

We’ve added the idea of an We’ve added the idea of an unsaturated proposition, which unsaturated proposition, which would be a proposition, but for would be a proposition, but for the lack of an individual.the lack of an individual.

Given an individual, it would be a Given an individual, it would be a set of possible worlds.set of possible worlds.

It’s “waiting for an individual.”It’s “waiting for an individual.” It, in a sense, turns individuals It, in a sense, turns individuals into sets of possible worlds.into sets of possible worlds.

Limiting our attention Limiting our attention to wto wkk

For simplicity in presentation, let’s For simplicity in presentation, let’s stop thinking about sets of possible stop thinking about sets of possible worlds briefly, and limit out focus to worlds briefly, and limit out focus to specific possible worlds.specific possible worlds.

One good candidate would be wOne good candidate would be w00, but it , but it doesn’t have to be that one necessarily.doesn’t have to be that one necessarily.

If we do this, we can consider a If we do this, we can consider a proposition to be either true or false.proposition to be either true or false. Though in the back of our minds, we know that Though in the back of our minds, we know that this is in a particular possible world.this is in a particular possible world.

Semantic typeSemantic type The entire semantics that we are The entire semantics that we are creating here depends on two types of creating here depends on two types of things, things, individuals individuals and and truth valuestruth values..

We can label individuals as being of We can label individuals as being of type “type “ee” (traditional, think ” (traditional, think “entity”), and truth values as being “entity”), and truth values as being of type “of type “tt”.”.

In these terms, names like In these terms, names like Homer Homer are are of type of type <e><e>, and sentences like , and sentences like Homer Homer stands stands are of type are of type <t><t>..

A formal systemA formal systemof semantic typesof semantic types

<e><e> is a basic type. is a basic type. <t><t> is a basic type. is a basic type. If If and and are types, are types, <<,,>> is a type. is a type.

<<,,> > is a function that takes something is a function that takes something of type of type and returns something of type and returns something of type ..

<e,t><e,t> is a type. is a type. <<e,t>,<e,<e,t>>><<e,t>,<e,<e,t>>> is is a type.a type.

<e,t,e><e,t,e> is not, nor is is not, nor is <<e,t>><<e,t>>..

FunctionsFunctions A function transforms one thing into A function transforms one thing into another.another.

We can define the We can define the squaring functionsquaring function as a function that takes a number as a function that takes a number and gives back that number and gives back that number multiplied with itselfmultiplied with itself

Square(Square(nn) = ) = nn nn This is a function from numbers to This is a function from numbers to numbers. It takes a number, it gives numbers. It takes a number, it gives back a number.back a number.

FunctionsFunctions A function doesn’t need to give back A function doesn’t need to give back the same kind of thing it gets. the same kind of thing it gets. Usually, the thing it gives back Usually, the thing it gives back depends on the thing it gets, but it depends on the thing it gets, but it doesn’t need to be of the same doesn’t need to be of the same typetype..

Change-machine($Change-machine($nn-bill) = 4 -bill) = 4 nn quarters.quarters.

This is a function from bills to This is a function from bills to quarters.quarters.

<e,t> functions<e,t> functions An intransitive verb like An intransitive verb like standsstands can be can be viewed as a function from individuals to viewed as a function from individuals to truth values. Given an individual truth values. Given an individual xx, it , it will return will return truetrue if if xx is boring, or is boring, or falsefalse if if xx is not boring. is not boring.

Stands(Stands(xx) = true if ) = true if xx stands; false stands; false otherwise.otherwise.

This is a function from individuals This is a function from individuals (type <e>) to truth values (type <t>). (type <e>) to truth values (type <t>). That is, it has type <e,t>.That is, it has type <e,t>.

Enter the Enter the

There is a way to write functions that we will get There is a way to write functions that we will get some experience with as the semester progresses, some experience with as the semester progresses, using using lambda notationlambda notation. Here’s a first introduction. Here’s a first introduction The structure of a function written in lambda notation The structure of a function written in lambda notation is:is: argumentargument [ [ return value return value ]]

So, for the meaning of So, for the meaning of standsstands, we might write , we might write this:this:

xx [ [ x x stands (in wstands (in wkk) ]) ] Type <e,t>Type <e,t>

argumentargument [ [ return return value value ]]

Change-machine($Change-machine($nn-bill) = 4 -bill) = 4 nn quarters.quarters.

Change-machine = Change-machine = $ $nn-bill [ 4 -bill [ 4 nn quarters ]quarters ]

Square = Square = n n [ [ nn n n ]]

Not very complicated, just a short way Not very complicated, just a short way to write “that function to write “that function ff such that, such that, given given argumentargument, returns , returns return valuereturn value.”.”

valuevalue argumentargument [ [ return value return value ]]

Square = Square = n n [ [ nn n n ]] Square(3) = 3 Square(3) = 3 3 = 9 3 = 9 Square(4) = 4 Square(4) = 4 4 = 16. 4 = 16.

To evaluate a function, we take the To evaluate a function, we take the valuevalue and substitute it in for the and substitute it in for the argumentargument within the within the return valuereturn value. If . If we give it a 3, and the argument is we give it a 3, and the argument is nn, then we replace all of the , then we replace all of the nns s with 3s and evaluate the return with 3s and evaluate the return value.value.

valuevalue argumentargument [ [ return value return value ]]

Strictly speaking, there’s an intermediate Strictly speaking, there’s an intermediate step, which is written like so:step, which is written like so:

Square = Square = n n [ [ nn n n ]] Square(3) = 3 Square(3) = 3 n n [ [ nn n n ] = 3 ] = 3 3 = 9 3 = 9 Square(4) = 4 Square(4) = 4 n n [ [ nn n n ] = 4 ] = 4 4 = 16. 4 = 16.

What What value value argument argument [[ return value return value ] ] means is: Replace every instance of means is: Replace every instance of argumentargument within within return valuereturn value with with valuevalue, , then evaluate then evaluate return valuereturn value.. This operation goes by the name This operation goes by the name lambda conversionlambda conversion..

valuevalue argumentargument [ [ return value return value ]]

One last piece of terminology:One last piece of terminology: Instances Instances of of argumentargument within within return valuereturn value are said are said to be to be variablesvariables that that boundbound by the lambda by the lambda operator.operator.

Triple = Triple = n n [ 3 [ 3 n n ]]

Bound variableLambda operator

Desiderata for a theory Desiderata for a theory of meaningof meaning

A is A is synonymoussynonymous with B with B A has the same meaning as BA has the same meaning as B

A A entailsentails B B If A holds then B automatically holdsIf A holds then B automatically holds

A A contradictscontradicts B B A is inconsistent with BA is inconsistent with B

A A presupposespresupposes B B B is part of the assumed background against which A B is part of the assumed background against which A is said.is said.

A is a A is a tautologytautology A is automatically true, regardless of the factsA is automatically true, regardless of the facts

A is a A is a contradictioncontradiction A is automatically false, regardless of the factsA is automatically false, regardless of the facts

Intuitions about logicIntuitions about logic If it’s Thursday, If it’s Thursday, ERER will be on at 10. will be on at 10.It’s Thursday.It’s Thursday.ERER will be on at 10. will be on at 10.

Modus PonensModus Ponens

Logic is essentially the study of valid Logic is essentially the study of valid argumentation and inferences.argumentation and inferences. If the If the premisespremises are true, the are true, the conclusionconclusion will will be true.be true.

Truth out there in the Truth out there in the worldworld

A statement like A statement like It’s ThursdayIt’s Thursday is either is either truetrue (corresponding to the facts of the (corresponding to the facts of the world) or it is world) or it is falsefalse (not corresponding (not corresponding to the facts of the world).to the facts of the world).

Same for the statement Same for the statement ER is on at 10ER is on at 10.. It turns out that It turns out that modus ponensmodus ponens is a valid is a valid form of argument, no matter what form of argument, no matter what statements we use. Let’s just say we have statements we use. Let’s just say we have a statement—we’ll call it a statement—we’ll call it pp. The statement . The statement (proposition) (proposition) pp can be either true or can be either true or false. And another one, we’ll call it false. And another one, we’ll call it qq..

Modus ponensModus ponens So, whatever So, whatever pp and and qq are: are:

If If pp then then q.q.p.p.qq..

Granting the premises Granting the premises If p then If p then qq and and pp, we can conclude , we can conclude qq..

An invalid argumentAn invalid argument Incidentally, some things are not Incidentally, some things are not valid arguments. Modus ponens and valid arguments. Modus ponens and modus tollens are. This is not:modus tollens are. This is not:

If it is Thursday, then If it is Thursday, then ERER is on is on at 10.at 10.It is not ThursdayIt is not Thursday**ERER is not on at 10. is not on at 10.

Other forms of valid Other forms of valid argumentargument

If it is Thursday, then If it is Thursday, then ERER is on. is on.If If ERER is on, Pat will watch TV. is on, Pat will watch TV.If it is Thursday, the Pat will If it is Thursday, the Pat will watch TV.watch TV.

Hypothetical syllogismHypothetical syllogism

If If pp then then qq..If If qq then then rr..If If pp then then rr..

Other forms of valid Other forms of valid argumentargument

Pat is watching TV or Pat is Pat is watching TV or Pat is asleep.asleep.Pat is not asleep.Pat is not asleep.Pat is watching TV.Pat is watching TV.

Disjunctive syllogismDisjunctive syllogism

pp or or qq..qq..pp..

Logical syntaxLogical syntax A proposition, say A proposition, say pp, has a , has a truth valuetruth value. . In light of the facts of the world, it is In light of the facts of the world, it is either either truetrue or or falsefalse. The conditions under . The conditions under which which pp is is truetrue is are called its is are called its truth truth conditionsconditions..

We can also create complex expressions by We can also create complex expressions by combining propositions. For example, combining propositions. For example, qq. . That’s true whenever That’s true whenever qq is false. is false. is the is the negation operatornegation operator (“not”).(“not”).

Logical connectivesLogical connectives We can combine propositions with We can combine propositions with connectives like connectives like andand, , oror. In . In logical notation, “logical notation, “p p and and qq” is ” is written with the logical connective written with the logical connective (“and”): (“and”): pp qq; “; “pp or or qq” is ” is written with written with (“or”): (“or”): pp qq..

pp qq is true whenever is true whenever pp is true is true and and qq is true. Whenever either is true. Whenever either pp or or qq is false, is false, pp qq is false.is false.

Truth tablesTruth tables We can show the effect of logical We can show the effect of logical operators and connectives in operators and connectives in truth truth tablestables..

pp ppTT FFFF TT

pp qq ppqq

TT TT TTTT FF FFFF TT FFFF FF FF

pp qq ppqq

TT TT TTTT FF TTFF TT TTFF FF FF

OrOr v. v. v. v. ee The meaning we give to The meaning we give to oror in English (or in English (or any other natural language) is not quite any other natural language) is not quite the same as the meaning that of the the same as the meaning that of the logical connective logical connective .. We’re going to South Carolina or Oklahoma.We’re going to South Carolina or Oklahoma.

Seems odd to say this if we’re going to both South Seems odd to say this if we’re going to both South Carolina and Oklahoma.Carolina and Oklahoma.

You will pay the fine or you will go to jail.You will pay the fine or you will go to jail. Seems a bit unfair if you get put in jail even Seems a bit unfair if you get put in jail even after paying the fine.after paying the fine.

We will preboard anyone who has small children We will preboard anyone who has small children or needs special assistance.or needs special assistance.

Doesn’t seem to exclude people who both need Doesn’t seem to exclude people who both need special assistance and have small children.special assistance and have small children.

OrOr v. v. v. v. ee There are two interpretations of There are two interpretations of oror, differing in , differing in their interpretation with respect to what happens if their interpretation with respect to what happens if both connected propositions are true.both connected propositions are true.

Exclusive or Exclusive or ((ee) is “either…or…but not both.”) is “either…or…but not both.” Inclusive or Inclusive or ((disjunctiondisjunction; ; ) is “either…or…or both.”) is “either…or…or both.”

pp qq ppqq

TT TT TTTT FF TTFF TT TTFF FF FF

pp qq ppeeqq

TT TT FFTT FF TTFF TT TTFF FF FF

Material implicationMaterial implication The logic of The logic of if…thenif…then statements is statements is covered by the connective covered by the connective .. If it rains, you’ll get wet.If it rains, you’ll get wet.((ppqq, where , where pp==it rainsit rains, , qq==you’ll get you’ll get wetwet))

pp qq ppqq

TT TT TTTT FF FFFF TT TTFF FF TT

What is the truth value of What is the truth value of If it rains, you’ll get wetIf it rains, you’ll get wet?? Well, it’s true if it rains Well, it’s true if it rains and you get wet, it’s false if and you get wet, it’s false if it rains and you don’t get it rains and you don’t get wet. But what if it doesn’t wet. But what if it doesn’t rain?rain?

Truth and the worldTruth and the world

In most cases, the truth or falsity of a In most cases, the truth or falsity of a statement has to do with the facts of the statement has to do with the facts of the world. We cannot know without checking. It is world. We cannot know without checking. It is contingent contingent on the facts of the world on the facts of the world ((syntheticsynthetic).). John Wilkes Booth acted alone.John Wilkes Booth acted alone.

Sometimes, though, the very form of the Sometimes, though, the very form of the statement guarantees that it is true no matter statement guarantees that it is true no matter what the world is like (what the world is like (analyticanalytic).). Either John Wilkes Booth acted alone or he didn’t.Either John Wilkes Booth acted alone or he didn’t. John Wilkes Booth acted alone and he didn’t.John Wilkes Booth acted alone and he didn’t.

The first is The first is necessarily truenecessarily true, a , a tautologytautology, the , the second is second is necessarily falsenecessarily false, a , a contradictioncontradiction..

Limits of propositional Limits of propositional logiclogic

There are some kinds of logical intuitions There are some kinds of logical intuitions that are not captured by propositional that are not captured by propositional logic. For example:logic. For example: All men are mortal.All men are mortal.Socrates is a man.Socrates is a man.Socrates is mortal.Socrates is mortal.

Try as we might, we can’t prove this Try as we might, we can’t prove this logically with only logically with only pp, , qq, and , and rr to work to work with, but it nevertheless seems to have with, but it nevertheless seems to have the same deductive quality as other the same deductive quality as other syllogisms (like modus ponens).syllogisms (like modus ponens).

Predicate logicPredicate logic Propositional logic is about predicting Propositional logic is about predicting the truth and falsity of propositions when the truth and falsity of propositions when combined with one another and subjected to combined with one another and subjected to operators like negation.operators like negation.

What we need for the What we need for the All men are mortalAll men are mortal case is something like:case is something like: For any individual For any individual xx, if , if xx is a man, then is a man, then xx is is mortal.mortal.

That is, we need to be able to look inside That is, we need to be able to look inside the sentence, to refer to predicates the sentence, to refer to predicates (properties) not just to truth and (properties) not just to truth and falsities of entire propositions.falsities of entire propositions.

Predicate logicPredicate logic Predicate logic is an extension of Predicate logic is an extension of propositional logic that allows us propositional logic that allows us to do this.to do this. Mortal(Socrates)Mortal(Socrates)True if the predicate Mortal holds of True if the predicate Mortal holds of the individual Socrates.the individual Socrates.

Individuals have properties, and Individuals have properties, and just like we labeled our just like we labeled our propositions propositions pp, , qq, , rr, we can label , we can label properties abstractly like properties abstractly like AA, , BB, , CC..

Predicate logicPredicate logic Thus:Thus:

Man(x) Man(x) Mortal(x) Mortal(x) AA((xx) ) BB((xx))Man(Socrates)Man(Socrates) AA(S)(S)Mortal(Socrates)Mortal(Socrates) BB(S)(S)

Note: This is not exactly in the right form Note: This is not exactly in the right form yet, but it’s close. The right form of the yet, but it’s close. The right form of the first premise is actually first premise is actually xx[Man([Man(xx))Mortal(Mortal(xx)]. More on that later.)]. More on that later.

EntailmentEntailment From the standpoint of linguistic From the standpoint of linguistic knowledge of meaning (intuition), there knowledge of meaning (intuition), there are sentences that stand in a are sentences that stand in a implicational relation, where the truth of implicational relation, where the truth of the first guarantees the truth of the the first guarantees the truth of the second.second. The anarchist assassinated the emperor.The anarchist assassinated the emperor. The emperor died.The emperor died.

It is part of the meaning of It is part of the meaning of assassinateassassinate that the unlucky recipient dies. So, the that the unlucky recipient dies. So, the first sentence first sentence entails entails the second. the second.

EntailmentEntailment

This is the same relationship as This is the same relationship as ppqq from from before. If we know before. If we know pp is true, we know is true, we know qq is is true—and if we know true—and if we know qq is false, we know is false, we know pp is is false.false. The anarchist assassinated the emperor.The anarchist assassinated the emperor. The emperor died.The emperor died.

At the same time, knowing At the same time, knowing qq is true doesn’t is true doesn’t tell us one way or the other about whether tell us one way or the other about whether pp is true—and knowing is true—and knowing pp is false doesn’t tell us is false doesn’t tell us one way or the other about whether one way or the other about whether qq is false. is false. We take entailment relations to be those that We take entailment relations to be those that specifically arise from specifically arise from linguistic structure linguistic structure (synonymy, hyponymy, etc.).(synonymy, hyponymy, etc.).

SynonymySynonymy For a paraphrase to be a good one, and For a paraphrase to be a good one, and accurate rendering of the meaning, the accurate rendering of the meaning, the sentence should entail its paraphrase and sentence should entail its paraphrase and the paraphrase should entail the sentence.the paraphrase should entail the sentence. The dog ate my homework.The dog ate my homework. My homework was eaten by the dog.My homework was eaten by the dog.

This kind of This kind of mutual entailmentmutual entailment (like (like from earlier) is a requirement for from earlier) is a requirement for synonymy.synonymy.

Truth and meaningTruth and meaning A young boy named Rickie burned down the A young boy named Rickie burned down the library at Alexandria in 639 AD by library at Alexandria in 639 AD by accidentally failing to extinguish his accidentally failing to extinguish his cigarette properly.cigarette properly.

True? Well, we’ll pretty much never True? Well, we’ll pretty much never know (though perhaps we can rate its know (though perhaps we can rate its likelihood). But knowing whether it is likelihood). But knowing whether it is true or not is not a prerequisite for true or not is not a prerequisite for knowing its meaning.knowing its meaning.

Rather, what’s important is that we Rather, what’s important is that we know its know its truth conditionstruth conditions—we know what —we know what the world must be like if it is true.the world must be like if it is true.

Truth and meaningTruth and meaning If we know what a sentence means If we know what a sentence means we know (at least) the conditions we know (at least) the conditions under which it is true.under which it is true.

On that assumption, we proceed in On that assumption, we proceed in our quest to understand meaning our quest to understand meaning in terms of truth conditions. in terms of truth conditions. Understanding how the words and Understanding how the words and structures combine to predict the structures combine to predict the truth conditions of sentences.truth conditions of sentences.