Linguistics Semantics paper about 2 page

Lecture notes

Linguistics 53

February 19, 2019

1 The syntax of predicate calculus (cont’d)

1.1 Some remarks about linking adjectives

Question: How can we translate an adjective inside a noun phrase in a copula sentence?

(1) Suzie is a tired student. (tired′(s1)∧ student ′(s1))

Here, we have translated both the noun and adjective as separate predicates that apply to a single term, which are combined by conjucntion. This makes intuitive sense: Suzie is both tired and a student.

But when we see adjectives, we have have to be careful! Not all adjectives have this kind of inter- sective meaning.

Question: Can all of the adjectives below be translated with conjunction?

(2) a. Max is a French student. b. Josie is an excellent doctor. c. Ethel is a former performer. d. Lucy is a fake dancer.

1.2 Adding n-ary predicates

So far, we only have 1-place predicates. But there are also (di)transitive verbs…

(3) Josie likes Suzie.

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The verb like has two arguments. So, we need to add to both the lexicon and the syntax.

Adding to the former is easy. We just need to recognize a class of 2-place predicates, alongside 1-place predicates:

◦ predicates:

– 1-place predicates: disappear′( ), run′( ), student ′( ), tired′( ), . . . – 2-place predicates: like′( , ), see′( , ), . . .

But, how do we now add to the syntax to allow for 2-place predicates to combine with the right number of terms? Here is one possible way of doing this:

- If P is a 1-place predicate and t1 is a term, then P(t1) is a formula.
- If P is a 2-place predicate and t1 and t2 are terms, then P(t1, t2) is a formula.
- If A is a formula, so is ¬A.
- If A and B are formulas, so are (A∧B), (A∨B), (A→ B), and (A B).

In other words, we could just add another clause for 2-place predicates. This allows us to translate a sentence with a 2-place predicate in the following way:

(4) Josie likes Suzie. like′( j3,s2)

Alternately, it is possible generalize the original rule for 1-place predicates to allow for a predicate with any number of argument slots:.

- If P is an n-ary predicate and t1, . . . , tn are terms, then P(t1, . . . , tn) is a formula.
- If A is a formula, so is ¬A.
- If A and B are formulas, so are (A∧B), (A∨B), (A→ B), and (A B).

Some terminology: An n-ary predicate is an “n-place” predicate or a predicate with n argument slots (where n is an integer).

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2 Pronouns

We have translated proper names as individual constants:

(5) a. Ethel e1 or e9 b. Josie j4 or j6

These are composed of two parts: a letter and an index. Different letters correspond to different linguistic expressions, while different indices correspond to different individuals in the context.

Since the same name can be used to refer to more than one person, the choice of index in predicate calculus disambiguates who the referent for some use of a name is.

Question: Should pronouns be translated as constants, too?

No, a pronoun does not always pick out the same referent:

(6) A: What do you think of Josie? B: She is smart.

(7) A: What do you think of Suzie? B: She is smart.

Both of these discourses can be uttered in the same possible world where the same facts hold. Yet, the referent of she in each discourse is different, Josie in one case and Suize in the other.

In other words, she needs to be translated as something whose value can change depending on the context, including the surrounding discourse (or conversation).

2.1 Introducing individual variables

Pronouns will be translated as individual variables. So, the vocabulary of predicate calculus we have so far is:

- predicates:

◦ 1-place predicates: disappear′( ), run′( ), student ′( ), tired′( ), . . . ◦ 2-place predicates: like′( , ), see′( , ), . . .

- terms

◦ individual constants: j1, s2, p3, . . . ◦ individual variables: x1, x2, x3, . . .

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- connectives: ¬, ∧, ∨,→,

Individual variables are terms, so they go in all the same positions as individual constants:

(8) She is smart. smart ′(x3)

Variables always use the letter x, indicating that they are variables and not constants. But like constants, they have an index that serves the same function: it picks out the referent for the variable.

There is one number used per individual in the context, not per pronoun or proper name. This allows us to represent whether two arguments within a single sentence are co-referent (they pick out the same referent) or not:

(9) a. She saw her. see′(x3,x4) b. She saw herself. see′(x3,x3)

Question: Can either of these sentences be translated in any other way?

The two pronouns (she, her, he, him, etc.) in (9a) cannot be co-referent, and hence these variables must have different indices.

By contrast, the reflexive (herself, himself, etc.) in (9b) must be co-referent with the subject pro- noun. Hence, these are both translated as variables with the same index.

2.2 Names, pronouns, and reflexives

In many languages, different classes of arguments show systematic patterns in how they pick out a referent. We have seen three classes so far for English: names, pronouns, and reflexives.

Question: How do these three classes of arguments systematically differ in their co-reference possibilities?

(10) Josie saw her.

(11) She saw Josie.

(12) Josie saw herself.

(13) * Herself saw Josie.

(14) Josie thought that she would win.

(15) * Josie thought that herself would win.

(16) She thought that Josie would win.

Here is a first stab at a generalization about the reference possibilities for names, pronouns, and reflexives:

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◦ reflexives: must be co-referent with the subject of the same predicate

◦ pronoun: cannot be co-referent with the subject of same predicate

◦ names: cannot be co-referent with anything that precedes them

In a syntax class, you might discover that these generalizations are somewhat more complex and depend on the abstract structure of a sentence rather than linear order. But for now, this is enough.

2.3 Pronoun shape features

The shapes of pronouns differ based on certain properties of the referent:

◦ person: first, second, third (i.e., I, you, her)

◦ number: singular, plural (i.e., her, them)

◦ gender: feminine, masculine, neuter (i.e., her, him, it)

We will only be translating third person pronouns as variables. (More on first and second person pronouns later.) But for these, we need some way to represent the semantic effects of these different shapes.

We can do this using superscripted features indicating what properties the referent must have for the pronoun to be used felicitiously:

(17) she x1 f ,sg

(18) him x4m,sg

So, the individual variable can be annotated with features according to the following rule:

(19) An individual variable xi (where i is some index) can optionally bear the following features:

i) m, f , or n ii) sg or pl

2.4 Practice with translation

Question: Can you translate the following into predicate calculus?

(20) She is talking to herself.

(21) If it falls on him, he will know.

(22) They introduced Bill to her.

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3 Summary: Predicate calculus so far

Lexicon:

- predicates:

◦ 1-place predicates: disappear′( ), run′( ), student ′( ), tired′( ), . . . ◦ 2-place predicates: like′( , ), see′( , ), . . .

- terms

◦ individual constants: j1, s2, p3, . . . ◦ individual variables: x1, x2, x3, . . . (with features: m, f , n and sg, pl)

- connectives: ¬, ∧, ∨,→,

Syntax:

- If P is an n-ary predicate and t1, . . . , tn are terms, then P(t1, . . . , tn) is a formula.
- If A is a formula, so is ¬A.
- If A and B are formulas, so are (A∧B), (A∨B), (A→ B), and (A B).

We covered some informal linking rules for translating sentences of English into predicate calculus:

◦ atomic expressions

– proper names individual constants – nouns, adjectives, verbs predicates

◦ formulas

– NP V V′(NP′) – NPS V NPO V′(NPS′,NPO′) – NPS V NPO NPIO V′(NPS′,NPO′,NPIO′) – NP be A A′(NP′) – NP be a N N′(NP′)

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4 Summary: Predicate calculus so far

Lexicon:

- predicates:

◦ 1-place predicates: disappear′( ), run′( ), student ′( ), tired′( ), . . . ◦ 2-place predicates: like′( , ), see′( , ), . . .

- terms

◦ individual constants: j1, s2, p3, . . . ◦ individual variables: x1, x2, x3, . . . (with features: m, f , n and sg, pl)

- connectives: ¬, ∧, ∨,→,

Syntax:

- If P is an n-ary predicate and t1, . . . , tn are terms, then P(t1, . . . , tn) is a formula.
- If A is a formula, so is ¬A.
- If A and B are formulas, so are (A∧B), (A∨B), (A→ B), and (A B).

We covered some informal linking rules for translating sentences of English into predicate calculus:

◦ atomic expressions

– proper names individual constants – pronouns, reflexives individual variables – nouns, adjectives, verbs predicates

◦ formulas

– NP V V′(NP′) – NPS V NPO V′(NPS′,NPO′) – NPS V NPO NPIO V′(NPS′,NPO′,NPIO′) – NP be A A′(NP′) – NP be a N N′(NP′)

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5 Practice with translation

Question: Can you translate the following into predicate calculus?

(23) She is talking to herself.

(24) If it falls on him, he will know.

(25) They introduced Bill to her.

6 The semantics of predicate calculus

We have covered part of the lexicon and syntax of predicate calculus. Now we need to do the semantics. This will map expressions of predicate calculus (terms, predicates, formulas) to their semantic values.

We use the interpretation function — notated with brackets: J .K — to represent what the semantics is doing. It takes an expression of predicate calculus as its input and outputs its semantic value:

(26) Jα K = β ‘The semantic value of α is β.’

What is the semantic value of an expression of predicate calculus? For individual constants, the semantic value is a single individual in the world:

(27) J l3 K = a specific individual

So, what is the semantic value of a predicate? Starting with 1-place predicates, we might think this is a type of situation. But what it means to be situation or even a type of situation is hard to grasp.

An easier way of thinking about the semantic value of a predicate is as the set of participants involved in a situation described by the predicate:

(28) Jdisappear′( ) K = the set of individuals who have disappeared

The clearest way to see how this semantics for predicate calculus works is to look at one concrete instance of how the world could be — a single model — where the possible referents for a proper name or the individuals in the semantic value of a predicate are defined.

6.1 Defining a model

For predicate calculus, a model has three parts: (i) the domain, a set of individuals, (ii) the semantic value for each individual constant, (iii) the semantic value for each predicate.

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The “facts” we considered a couple classes ago were, in essence, such a model, which we can call M1:

(29) M1 (i) D = { , , }

(ii) J j KM1 = J s KM1 =

(iii) J tall′( ) KM1 = { , } J short ′( ) KM1 = { } J intelligent ′( ) KM1 = { , , } Jhappy′( ) KM1 = { } Junhappy′( ) KM1 = { }

You can think of this model as a possible world. M1 specifies which individuals exist. It further specifies what the referent of individual constants are and what the semantic value for predicates are. (Right now, we are looking solely at 1-place predicates.)

Since we are talking about the semantic value of these expressions in a specific model, we notate this by superscripting the name of the model to the interpretation brackets. J .KM1 gives the semantic value for an expression in M1.

6.2 Toward defining semantic rules

The interpretation function will also give a semantic value for an entire formula in a given model. For instance, for all the following simple and complex formulas, it will yield T (true) or F (false), depending on what the facts in the model are.

(30) a. J tall′(s1) KM? = ? b. J (tall′( j2)∧ intelligent ′(s1)) KM? = ? c. J¬tall′(s1) KM? = ?

How does the interpretation function do this? We will state semantic rules defining what the inter- pretation function spits out.

To see how this might work, let’s consider another model, M2, a different set of facts corresponding to a different possible world:

(31) M2 (i) D = { , , }

(ii) J j KM2 = J s KM2 =

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(iii) J tall′( ) KM2 = { , } J intelligent ′( ) KM2 = { , , } Jdoctor′( ) KM2 = { } J plumber′( ) KM2 = { }

We can think about these facts informally using a Venn diagram:

(32) !”#$%&'(#)*+$,-./”#01$20/0#)-&-&3$4#”/5

!!!”osie &uzie

!!!!! “!!!!!!#

!”## $%!&##$’&%! ()*!)+ ,#-./&+

)”#$%!*+!*%,-..*/-%,)!!*+!,01-!2-3’1+-!,$-!(-‘%*%/!#4!”#$%!*+!*%!

,$-!(-‘%*%/!#4!*%,-..*/-%,

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Question: Are the following sentences true or false in this model?

(33) Josie is intelligent.

(34) Suzie is a doctor.

(35) Josie is tall.

Now, here is a more formal representation of the same Venn diagram above, replacing English words with expressions of predicate calculus:

(36) !”#$%&'(‘)*+,-#+.)/”#$%&+0’12-#3

!!! !!” !#!

!!! !”##$ %&!’##%(‘&! $ “!!!!!!!!# )*+!*,$ -#./0′,$

! $%&'(!$)*+,-.

” 1′!12*32*04’+!1/!+!01&!#23#’4#!%5!+

” %&1!,.+!%*&123*,25*62!5*1’21’!12+*,,’1-*&)2!*2-,’)%+”!’1/!

-6&%74%89:!)!4;’!38%<1!#’4

” %&1!,.+!%*&123*,265%+52*04’+!12&”/’12-%+72*.!/!-6″:)!”

=>

Question: Are the following formulas true or false in this model (M2)? (What is their semantic value?)

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(37) intelligent ′( j2)

(38) doctor′(s3)

(39) tall′( j2)

Intuitively, (40) has the semantic value of T if the semantic value of j2 is in the semantic value of tall′( ). (The expression iff stands for if and only if.)

(40) J tall′( j2) KM2 = T iff J j2 KM2 is in J tall′( ) KM2 = T iff is in { , }

These are the truth conditions for this formula of predicate calculus. Given these truth conditions, is this formula true or false in M2? True!

Similarly, for (41), this formula has the semantic value of T if and only if the semantic value of s3 is in the semantic value of doctor′( ):

(41) Jdoctor′(s3) KM2 = T iff J s3 KM2 is in Jdoctor′( ) KM2 = T iff is in { }

This formula is false in M2.

6.3 The semantic rules of predicate calculus

To determine whether a sentence was true or false, we needed:

◦ a set of individuals (the domain)

◦ for proper names, instructions for picking out a referent

◦ for predicates, instructions for picking out a subset of the domain corresponding to that predicate

The model provides all three of these ingredients. The semantic rules for predicate calculus will give us the rest, providing instructions for determining the semantic value of a formula based on the semantic values of its component parts.

These instructions allow, in other words, the calculation of the truth conditions for any formula of predicate calculus, thereby allowing for that formula to be associated with its semantic value (T or F).

(42) Predicate Calculus Semantics For a model M,

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- if P(t) is a formula with a 1-place predicate P and term t, JP(t) KM = T iff J t KM is in JP KM
- if t i is an individual constant, J t i KM = J t KM 3. for formulas containing a connective:

(i) J¬p KM = T iff J p KM = F (ii) J (p∧q) KM = T iff J p KM = T and Jq KM = T

(iii) J(p∨q)KM = T in only these cases: (i) J pKM = T , (ii) JqKM = T , (iii) J pKM = T and Jq KM = T

(iv) J (p→ q) KM = T unless J p KM = T and Jq KM = F (v) J (p q) KM = T whenever J p KM = Jq KM

This semantics gives truth conditions with respect to a model. Only by knowing what the model actually assigns to each atomic element can we determine whether a formula something is actuall true.

6.4 Calculating truth conditions

We can now calculate the truth conditions for one of the formulas above — tall′( j2) — and hence determine its truth value in a model.

We start with the largest formulas we can find. Since this formula is simple — it does not contain a connection —- all we have to do is apply rule 1:

(43) J tall′( j2) KM2 = T iff J j2 KM2 is in J tall′( ) KM2

On the right hand side of the equal sign, there is nothing more to break down. Both j2 and tall′( ) are atomic expressions of predicate calculus. So, we can just calculate the formula’s truth value in M2.

(44) J tall′( j2) KM2 = T iff is in { , }

Let’s try a more complicated example:

(45) J¬tall′( j2) KM2 = T iff J tall′( j2) KM2 = F = T iff J j2 KM2 is not in J tall′( ) KM2 = T iff is not in { , }

Unlike (44), (45) is false in M2! (This, of course, makes sense, since (44) is true.)

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