Scales: A simulation semantics accountsnarayan/Scales.pdf•Scales are linear Paths (Lakoff 1994) –Changing value on a scale is like moving along the path –Scalar inferences correspond

Scales: A simulation semantics account

Srini Narayanan

Talk Outline

• Introduction and Basic Result

• Implementation Basics

• Polarity and Scales

• Coordinated Scales and Actions

• Scales and Decision Making

• Questions

• Negation and Polarity

• Speaker dependent information and Scalar reasoning

Basic Aspects of Scales

• Scales are orderings – Usually have a bottom and top – Orderings go from low to high (vertically)

• Scales can be reversed – Hot scale versus cold scale

• Warm is lower than hot in the hot scale • Cool is lower than cold in the cold scale

• Scalar values are properties or propositions • The extent to which a property P has value y • The likelihood of a proposition being true

• Scalar items can be compared to each other

Scales, Polarity, Pragmatics • Contrast

– Both contrasting properties cannot hold • Hot-cold • Tall-short

– but the conjunction of the negation can hold • Not(hot) and not(cold)

• Reversal – Negation reverses scales (Israel 2001, 2004)

• John can solve a hard problem => John can solve simple problems • John can’t solve a simple problem => John can’t solve a hard

problem

• Emphasis (Horn 2001, Ladusaw 2009, Israel 2001, 2004) – Negative and Positive Polarity items (NPI, PPI)

• Present in most languages • Truth conditional semantics problematic

Scale

Difficulty of puzzle scale

BOTTOM

TOP

Least difficult is the Bottom of the scale

Most difficult is the top of the scale

John can solve difficult problems

John can solve easy problems

Propositional Function

Basic Modeling Results 1. Scalar reasoning is conceptualized as simulated action, often as

motion along a path 1. Scales are linear paths (Lakoff 94) 2. The logic of paths and motion along paths is projected to structure

scalar inferences and entailments. 3. Projections from regions in the path form ranges of the scale

1. Projections can incrementally create subscales through zoom in

2. The structure of the action coordinates two different kinds of scales 1. Expenditures, resource consumption, and impediments to action 2. Enablers, stimuli, and reward scales.

3. Scales provide information for decision making 1. Comparison often uses the More IS Up metaphor. 2. Information in questions can be quantified as entropy reduction 3. Scalar use by speakers trades off between the goals of

1. Maximally informative action selection and 2. Maintaining a social image or reputation

Phenomena Modeled

• Scalar reasoning in canonical and reverse scales • Polarity sensitive items (emphatic and attenuative)

– Negative Polarity Items (NPI) – Positive Polarity Items (PPI)

• Coordinated scales – “Let alone” construction

• Questions, scale reversal and information • Negation, Scales, and Polarity

– Contradictory as contrary phenomena – Neg raising – Speaker dependence and scales.

Talk Outline






• Questions



Implementation Assumption

• Scales are linear Paths (Lakoff 1994)

– Changing value on a scale is like moving along the path

– Scalar inferences correspond to regions visited on the path

– Canonical scales are conceptualized as motion from the bottom to the top of the scale • Places visited go from bottom to top

– Reverse scales are conceptualized as motion from top to bottom of the scale • Places visited go from top to bottom of the canonical scale

Projections and Scale Density

BOTTOM

Bottom

R2 Top R1

R Bottom

Top

PATH LOCATIONS SOURCE GOAL

TOP

FINE

COARSE

Projection

Projection

Projection

Projection

SCALE POSITIONS

Projection and Subscale Selection

BOTTOM

R Bottom

Top Bottom

Bottom


TOP

Projection

Subscale Selection

Projection

Subscale

Zoom in and Subscale Selection

BOTTOM

Bottom

Top

R Bottom

Top


TOP

FINE

COARSE

Projection Projection

Top R2 Bottom

R1

SCALE POSITIONS

Subscale ZOOM IN

Possible Projection Strategies

• Point – Location to Scale position projection

• dense to dense

• Interval – Spatial region (1-D bounded path) to scale position

• Discrete • Parametric (Gaussian) • Probabilistic (with or without threshold) • Fuzzy

• Scale spaces – Edge preserving mappings

• Linear (ex. convolution with a gaussian kernel) • Non-linear (used in image processing)

One Dimensional Scale Space • For continuous scalars (ex. Location.heights)

– Convolve the discrete space with a sampled (or discrete) Gaussian kernel

– Which is truncated at the end to give a filter with a finite impulse response

Circuit Implementation

Normal Level

Basic Insight

• Scales are orderings that reflect the extent to which a property (or propositional function) is realized. – Height(x), intensity(x), height(John, tall).

• The realization of a property is like a movement action from a low to a high realization of that property. – Move from low values (short, dim) to high values (tall,

intense).

• Scalar values and position indicators differ based on how they impact the realization of the proposition. – Causal forces, stimuli, rewards accelerate the realization

– Obtacles, resources required impede the realization.

Talk Outline






• Questions



20

The structure of Actions

• Key elements – preconditions, resources, effects, sub-events – evoked by frames (alternatively: predicates, words)

21

Simulation of Actions using X-nets

• Actions & States

• Sequentiality

• Concurrency & Synchronization

a

b

• Alternative

• Stochasticity

a

b

<rate 1>

<rate 2>

• Asynchronous Control

Action 1 Transition

State / Resource Place

Action 2 Transition

Consume res Produce res

Scalar Inferences

• Realizing propositions is like moving on a path from the low to high levels of the scale. – Places visited are available as inferences

• Negation corresponds to motion along segments of the path not visited. – This corresponds to the reverse scale with simulated

motion from the high to low location.

• Inferences on locations visited from the simulated motion carry over to inferences on scalar values. – Entailments (downward and upward) fall out of the

simulation on forward and reverse simulations.

John can solve the hard puzzle

1. Select the Hardness of Puzzle Scale

2. Simulate John solving puzzles upto high point in the hardness scale.

Normal Level

John can not solve a simple puzzle

1. Select the Hardness of Puzzle Reverse Scanning

2. Simulate John solving puzzles upto simple point in the hardness scale.

Normal Level

A Basic action coordinates multiple possible scalar entities

Scale of cost R1 with minimum to maximum (unit is W1)

Scale of cost R2 with minimum to maximum (unit is W1)

R O

R1

R2

W1

W2

Cost or resources consumed :W1 and W2 are the minimum amounts of Resources required for Action, Benefit or Resources Produced, W3 is the amount of resource produced

Reward Function Scale for reaching the new state

Action rate (canonical value T). Rate may be a scalar depending on resources available

W3

Action Requirements and Enablers

• An action coordinates multiple scales – Some of the scalar entities are requirements

• Costs, obstacles, impedences

– Some of the scalar entities are enablers • External stimuli, rewards, resources produced, removal

of obstacles

– From the viewpoint of the action these scales are treated as reverse scales • Lower costs and higher benefits make the action more

likely

Action Needs and Stimuli

• The action needs to: – Overcome impedences, obstacles, and meet requirements

• Competence • Ability • Energy • Resistance/Obstacles • Resource Production scalar examples

• The action is helped by – External enabling forces, values produced, rewards

• External Stimuli (provides resource) • Rewards

• An agent tries to minimize resources consumed and maximize resources produced by acting.

Coordinated scales

T R

M

O

W1

W2

W3

C

C T R

M

O

W1

W2

W3

Coordinating Scales

• Minimum unit of cost/requirement (M, O, W, C) and maximum unit of Reward (R).

• Action A is strictly more likely than Action B – If all the costs for A are less than B AND – If all the rewards for A are greater than B

• Can we collapse the cost and reward scales into one scale for comparison? – A > B if

• i Cost/Requirement {Cia < Cib} and

• i Benefits {Bia > Bib}

Coordinated Contrast

T R

M

O

W1

W2

W3

C C T R

M

O

W1

W2

W3

Placing two actions together can create a coherent contrast case: Let

1. M be individuals ranked in a scale of need 2. C be circumstances ranked in a scale of difficulty 3. O be an object ranked in a scale of size 4. W2 be an effort (cost) resource ranked in a scale from low to high effort 5. R be a scale of reward for an action

The let alone construction Fillmore, Kay, and O’Connor (1988)

T R

M

O

W1

W2

W3

C C T R

M

O

W1

W2

W3

1. M be individuals ranked in a scale of need 2. C be circumstances ranked in a scale of difficulty 3. O be an object ranked in a scale of size 4. W2 be an effort (cost) resource ranked in a scale from low to high effort 5. R be a scale of reward for an action

You can’t get a poor man in bad times to wash a car for $2 let alone a rich man to wash a truck for $1 in prosperous times.

Some (non) puzzles

• Conjunction analysis (Fillmore,Kay,O’Connor) – John barely reached Denver, let alone Chicago

• Downward Entailment requirement – He has climbed Mt. Everest, let alone the Berkeley

hills (Tooshirvadani)

– Is he pleased? He is delighted, let alone pleased.

Reach_Denver(John) Not(Reach_Chicago(John)

Different Polar Operators Michael Israel (2001, 2004)

NPI

• Attenuative NPI: much, long, any too, all that

• Emphatic NPI: lift a finger, sleep a wink, at all, the least bit

PPI

• Emphatic PPI: always, tons, utterly, absolutely, a heap, insanely

• Attenuative PPI: a little bit, sorta, rather, somewhat

Bottom of scale

Top of Scale

NPI’s correspond the reverse scale

NPI map to the reverse scale

Normal Level

Emphatic NPI’s maximize the inference by including the canonical case


Emphatic NPIs specify scalar Simulation to include the canonical instance

Normal Level

Attenuative NPI’s minimize inference by excluding the canonical case


Attenuative NPIs specify scalar Simulation to exclude the canonical instance

Normal Level

PPI’s correspond the canonical scale

PPI map to the canonical scale

Normal Level

Emphatic PPI’s maximize the inference by including the canonical case


Emphatic PPIs specify scalar simulation to include the canonical instance

Normal Level

Attenuative PPI’s minimize inference by excluding the canonical case


Attenuative PPIs specify scalar simulation to exclude the canonical instance

Normal Level

Inverted Polarity Items (Israel 2001)

• ‘maximizing’ NPIs—forms which emphatically strengthen negation precisely by virtue of their high quantitative values. – Wild horse s could *(-n’t) keep me away. – I would *(-n’t) do it for all the tea in China . – I wouldn’t touch it with a ten-foot pole .

• ‘minimizing’ emphatic PPI –which clearly designate low scalar values and yet produce emphatic propositions in affirmative contexts. – Godfrey is (*not) scared of his own shadow . – She would (*not) betray us at the drop of a hat . – You could have knocked me over with a feather .

Canonical and Inverted Emphatic Items (Michael Israel 2001)

Influence of Thematic Role

• Canonical polarity items tend to refer to entities affected by the action – a patient (crack a book, hurt a fly), a – theme (lift a finger, move a muscle, bat an eye), or

more generally some sort of – increment (sleep a wink, drink a drop, budge an inch,

breathe a word). All these forms involve entities which are somehow affected by the action of the verb: they are

• low in the thematic hierarchy, or, in other terms, near the bottom of the action chain (Langacker 1987).

Influence of Thematic Role (2)

• Inverted polarity items, on the other hand, tend to involve participants at the top of the thematic hierarchy—entities which somehow play a causal, or at least a facilitating role in the realization of an eventuality. (Michael Israel, 2001).

– Wild Horses

– All the tea in China

If Property Realization is a Movement Action

• For resources, impedences, and obstacles – Emphasize the MAXIMAL amount that of resource

expenditure or cost for to realize the proposition (Emphatic PPI)

– Emphasize the MINIMAL amount that of resource expenditure or cost to realize the proposition (Emphatic NPI)

• For causal forces, enabling stimuli, rewards – Emphasize the MAXIMAL amount that will still not

realize the proposition (Emphatic NPI) – Emphasize the MINIMAL amount that will make the

proposition true (Emphatic PPI)

Emphasis and Actions

• When dealing with requirements and obstacles: – Emphatic NPI deny even spending the least amount

• Lift a finger, sleep a wink

– Emphatic PPI emphasize high cost or resource spent • Way to hard, utterly disappointed, an arm and a leg

• When dealing with enablers (produced resources and rewards) – Emphatic NPI deny the action even with the greatest

amount produced • Wild horses, all the tea in China

– Emphatic PPI emphasize the action even with the least amount produced • At the drop of a hat, for a pittance, in a new york minute

NPIs depend on the action context

• Yoshimura (1994) shows that the use of NPIs in before clauses may depend on pragmatic assumptions about canonical action schemas. – Miss Prism {spilled/??poured} her wine before she

had drunk a drop. – The alarm clock was {ringing/??plugged in} before I

could sleep a wink.

• Emphatic NPI’s need to produce inferences that include the canonical instance – Cecily didn’t eat a bite of her food. – ??Cecily didn’t stare at a bite of her food

Time can make actions unlikely

• They have been married for {ages/ever/eternity} is an emphatic PPI that

– maintains the state even in the face of resistance (temptation)

• I won’t talk to her even for a minute is an emphatic NPI that

– Denies spending even the minimal time resource for the action

Time can make actions likely

• I will marry her in a new york minute is an emphatic PPI that – Performs the action with minimal (less than

canonical) deliberation

• He hasn’t taken an exam in ages is an emphatic NPI that – Denies the action despite a lot of time (chance to

perform action).

Different uses of Perception

• Fauconnier (1975,1980)

• Both maximal and minimal elements can be paired with verbs of perception and paraphrased with any

– She couldn’t hear even the faintest noise.

– She couldn’t hear the loudest noise.

– She couldn’t hear any noise.

Two uses of perception

Threshold

Effect

Stimulus

Ability 𝑺𝒕𝒊𝒎𝒖𝒍𝒖𝒔 ∗ 𝑨𝒃𝒊𝒍𝒊𝒕𝒚 > 𝑻𝒉𝒓𝒆𝒔𝒉𝒐𝒍𝒅

High Ability (detector sensitivity) with Low Stimulus could create Effect Low Ability (detector sensitivity) with High Stimulus could create Effect

Different uses of Perception

• NPI’s could communicate either the requirement (ability) or the external produced force (stimulus) – In one case, the NPI denies even the smallest

helpful stimulus (didn’t help a finger to help) • She couldn’t hear even the faintest noise.

– In the other case, the NPI denies the action even with the maximal stimulus (the wild horses couldn’t.. example) • She couldn’t hear the loudest noise.

Talk Outline






• Questions



Scales are related to decisions

• Scalar assertions often provide an argument for some conclusion (Israel 2001): – Hank can run a five minute mile indicates that Hank is

athletic, or that he has a chance of winning some race.

• Given such argumentative goals, the ordering in a scalar model effectively determines what counts as a strong or a weak argument for a given conclusion.

• The strength of an argument is related to the informativeness of the scalar argument

Scalar inference helps decision making

• Scalar inference can

– Help decision making through informative questions

– Serve the twin communicative goals of being informative while maximizing social image.

• Question: How do we quantify informative?

Gricean Implicature and speaker goals (Horn 1984;1989:194).

• Two opposing principles

– the Q-Principle, that a speaker should say enough to achieve her communicative goals

• Upper bound reasoning.

– the R-Principle, that a speaker should say no more than is necessary to achieve her goals

• Lower bound reasoning

Decisions with multiple goals

• In decision making there are often at least two possibly competing goals – Select the optimal action (Maximum expected utility) – Maintain/enhance social image

• Evidence from sociology, social psych/neuroscience

• For instance, one’s maintaining a social image could include – Minimizing negative evaluation of others – Distancing by committing to weaker proposition (not

happy versus sad)

• Different individuals could weigh these two goals differently

Multiple goals and scalar reasoning

• Trade-off between speaker goals of providing information while keeping self image explains the use of not happy for sad.

• The hearer can infer the stronger version which is among the entailed possibilities.

• Can this explain some scalar properties of negation – Contradicton as contraries (Horn 1989, 2000)

– Affixal negation

– Certain types of neg raising

Speaker characteristics

• In being maximally informative, speakers may bias the optimal decision versus self image/reputation differently

– May involve high bias on social image

• Understatement as being more rational

• Exaggeration/story telling as interesting/passionate?

– May involve high bias on action selection

• Exaggeration as advertising/marketing?

Understatement

• Often, understatements are litotes.

• Litote: An understatement where the affirmative is expressed as the negative of the contrary.

– He’s not a bad ball player.

• Here the speaker doesn’t commit but the hearer uses the sharper definition (good ball player)

• Intonation can give further constructional clues

– Not bad versus Not bad

Changing the range of a scale

Changing the range of the scale to select a subscale corresponds to binding a new bottom and top of the scale. The rest of the scalar simulation remains unchanged. Notice that the levels in the original scale that lie outside the range of the subscale do not activate either the bottom or top of the subscale. In parts of the original scale with is in the range of the subscale, the simulation of movement on the original scale is reflected in the subscale.

Subscale selection can occur with both canonical and reverse scales.

Assertions, Communication and Information

• Speakers often use the negation of an evaluatively positive (e-positive) predicates the denial of which may indirectly express an evaluatively negative (e-negative) judgment.

• Hearers often use the most informative reading of the evaluation to reach a negative judgment.

Hearers may choose restrictive subscales

• Hearers may change the range of the scale to the relevant subscale to interpret the negation of e-positive predicates as e-negative predicates.

Contradictories as contraries

• Contradictions are interpreted as contraries (Horn 2000) – John is not happy (contradiction) is often interpreted

as – John is sad (contrary)

• The model of information with multiple goals is compatible with the speaker’s use of the contradiction.

• Our model predicts that the contrary interpretation is more informative for the hearer (to make a decision).

Other examples

• He’s not nice. (= ‘he’s mean’)

• She’s not happy. (= ‘she’s sad’)

• He’s not mean. (≠ ‘he’s nice’)

• She’s not sad. (≠ ‘she’s happy’)

• She’s not ecstatic. (≠ ‘she’s miserable’)

Changing the range on the scale

• Metalinguistic negation operators often change the range on the scale from weaker to stronger ranges – I don’t like it, I love it.

– It is not cold, its freezing.

– I am not happy, I am elated.

• In all these cases, negation is not truth conditional, it operates to tighten the range of the inferred property

Neg raising

• Speakers may say – I don’t suppose you’d like to dance.

– I don’t think John is tall.

• Hearers may interpret as – I suppose you would’nt like to dance

– I think John is not tall.

• Some of the reasons for speakers using neg-raising is related to the goal of maintaining a social image.

• Other aspects come from semantic considerations (Horn 1989, 2000) Heim and students 1983,95,2005).

Two uses of but

• Metalinguistic – Tighten the range to result in stronger claim.

• It is not hot, but scalding

• It is not hot, it is scalding

• *It is not hot, but it is scalding

• Description/Concessive – Deny an entailed reading to a lower level on the

scale • It is not scalding, but it is hot

• She doesn’t love him, but she does like him.

Questions and Polarity (Fauconnier 1980, van Rooy 2003, Israel 2004)

• Scale reversal with NPI • Did Jane lift a finger to help?

• Did she talk to him even once?

– In both these cases the expected answer is No.

• Scale reversal with PPI • Wouldn’t you rather stay here?

• Aren’t you pretty tired?

– In both these cases the expected answer is Yes

Questions and information

• NPI and PPI in questions can be viewed as being maximally informative in the presence of bias (where the expected answers are not all (equally)) likely. (van Rooy 2003)

Basic Intuition Krifka(1995), van Rooy 2003)

• With Questions the reduction in entropy is

– E(Q) = - Sk e Q P(k) log P(k)

– The highest entropy reduction is when the n answers are equally likely (log2(n))

– Lowest entropy reduction is when the answer is known (log2(1)) = 0

– Any bias makes the entropy reduction lie in between 0 and (log2(n))

Questions and Polarity

• Minimal NPI questions tighten the range to get an informative answer – Did she (even) lift a finger to help?

– Did she help?

• PPI with explicit negation – Wouldn’t you rather stay here?

– Wouldn’t you stay here?

• In both cases the use of polarity items makes the question more informative.

Questions, Polarity, and Action Scales

• With stimulus or rewards, the questions keep the information content – NPI for stimuli/rewards

• Would she solve it if she had infinite time?

• Would she marry him for all the money in the world?

• Expected answer : No in both cases.

– PPI for stimuli/rewards • Wouldn’t they marry in a jiffy?

• Wouldn’t she join the peace corps for a pittance?

• Expected answer: Yes in both cases.

Scalars and Neg raising domains (Horn 1989)

Weaker

Medium

Strong

Be able May, Might, Can

Allow, Permit, Let

Be legal, Be ethical Possible

Know, Realize

Clear, Evident, Be sure

Must, Necessary

Require, Order, Demand

Certain Obligatory, Mandatory

Believe, Suppose, Think

Seem Appear , Look like

Should, Ought

Want Intend

Likely, Probable

Advisable, Desirable

Negation of Domain Scales

• With the logic of reverse scales

– Strong scalars (know, certain) when negated will result in the weak value (be able, possible)

– Intermediate scalars (likely, believe) when negated will result in an intermediate scalar value (not likely, not believe)

– Weak scalars (possible, allow) when negated will result in a strong scalar (impossible, forbid)

Neg raising and inference

• Neg raising works with intermediate scalars – I don’t suppose that X = (I suppose that (not X)) – I don’t believe that X = (I believe that (not X))

• Neg raising does not work with weak scalars – I am not able to X =! I am able to not X – It is not possible to X =! It is possible to not X

• Typically Neg raising does not work with strong scalars** – I am not certain that X =! I am certain that (not X) – I am not obliged to do X =! I am obliged to (do not X)

Non-cumulative projections

• Special class of projections with partial transfer of path logic.

• Projections are from key way points to points on a scale. – Scale values are mutually exclusive.

• Examples include ranking scales of different types. – It is a felony does not entail it is a misdemeanor. – It is a felony implies that it is at least a misdemeanor. – It is a misdemeanor implies it is not (yet) a felony. – It stops short of being a felony.

Scales and Properties of Scales SCALES AND PROPERTIES OF SCALES

(scale s): s is a scale.

(partialOrdering e x y s) e is a partial ordering on the components

of s, where x is less than y.

(inScale y s): y is a component of the scale s.

(lts x y s): x is less than y in the partial ordering

for scale s.

(gts x y s): x is greater than y in the partial ordering

for scale s.

(leqs x y s): x is less than or equal to y in the partial

ordering for scale s.

(geqs x y s): x is greater than or equal to y in the

partial ordering for scale s.

(top x s): x is the highest element in the scale s.

(bottom x s): x is the lowest element in the scale s.

(subscale s1 s): s1 is a subscale of scale s.

(reverse s1 s): s1 is the reverse of scale s.

(disjoint s1 s2): The component sets of scales s1 and s2 are

disjoint.

(totalOrdering e x y s): e, the partial ordering on the components

of s, where x is less than y, is in fact

total.

(function f s1 s2): f is a function from a set or scale s1 onto

a set or scale s2.

(monotoneIncreasing f): Function f is monotone-increasing

scale-to-scale function preserving the

scales' "less than" ordering.

(functionInto f s1 s2): f is a function from a set or scale s1 into

a set or scale s2.

(scaleDefinedBy s s1 e): s is the scale with components s1 and

partial ordering defined by relation e.

(subsetConsistent s e): s is a scale whose ordering is consistent

with the subset ordering among sets

associated by the relation e with

entities placed at points in s.

(compositeScale s s1 s2): s is a composite scale with the same

components as scales s1 and s2 and a

partial ordering consistent with the

partial orderings of s1 and s2.

(Hi s1 s): s1 is the high region of scale s.

(Md s1 s): s1 is the middle region of scale s.

(Lo s1 s): s1 is the low region of scale s.

(scaleFor s e): The property e corresponds to being in the

Hi region of scale s.

Jen’s example

• She's already too small. – Background

• 32 weeks gestation • at this early point in my pregnancy, before slow growth is

typically calculated and/or an issue at all, this fetus is smaller than the normal size of a fetus at this gestational age.)

• Previous pregnancy measured small fetus.

– Scales involved • prototypical growth of fetus • prototypical time in pregnancy when fetal growth rate can

potentially become an issue • the implicit comparison to previous pregnancy -where slow

growth rate was a problem.

Grow Size

Documents

Scales: A simulation semantics accountsnarayan/Scales.pdf•Scales are linear Paths (Lakoff 1994) –Changing value on a scale is like moving along the path –Scalar inferences correspond