Formal Semantic of Natural LanguagesFocus: Vagueness

Introduction

for TU-Wien / SS 2012 / 185.A53 / Fermüller

Hilbert Heikenwälder / Werner Doubek

Content• Introducing reasoning about vagueness• Phenomena of Vagueness

– Borderline cases– Sorites paradoxa– Multi-dimensional vagueness– Higher-order vagueness

• Further notes to Vagueness– Vagueness is helpful– Not expected vagueness

• Theories of Vagueness– Three-value logic– Infinite-value logic– Supervaluation– The epistemic/pragmatic approach

• Persistency and elasticity of vagueness• Summary

Introducing reasoning about vagueness• Precision is not our nature. We are children of

vagueness.

• Yet we learned to create islands of precision within the ocean of vagueness. This way we conquer mathematics, logic, „hard“ science. But now we depend on precision.

• When trying to understand our origins in vagueness, we require precision.

• So now, minds of vagueness who learned to create precision go back to vagueness using precision.

• Warning: Philosophy may harm your mental stability!

Phenomena of Vagueness (1)What is constituting vagueness, what is not ? What it isn’t:• Ambiguity:

Example “bank”.• Undecidability:

mathematical undecidability.• Uncertainity of a statement:

Example “Next Monday it will rain in Vienna”.• (excessive) Generality:

Example “coloured”.

Generality is not always vague: Example ‘prime number’.

Phenomena of Vagueness (2)

What it is:• Borderline cases• Sorites paradoxa• Multi-dimensional vagueness• Higher-order vagueness

Phenomena of Vagueness (3) Borderline cases

• Questions:– What is the maximum height of a short man? – When develops a child to an adult?– Is this colour reddish or not?

• Measurable Items:– Tallness: pygmy vs. Masai– Although borderline case: being tall or not being it for a Masai.

• No simple measurement:– Hichcock’s: ‘All actors are children’ is not a borderline case.– A mothers statement: ‘My son is yet a child’, is a borderline-case.

Phenomena of Vagueness (4) Sorites paradoxa

• Eubulides of Milet (4. century b.C.) riddle of the heap of seeds (greek: soros) of a fern: – One single seed is not a heap. – If some seeds do not make a heap then one seed more does also not

make a heap. – If you follow this, even 1 million of seeds build not a heap.

• Reverse: – If there is a heap of sand-corns, – then it rests also a heap, if there is removed one single sand-corn. – So it follows that also as few as possible sand-corns are a heap.

• Hints:– Sorites problems exist in great variety.– The introduction of borderlines does not always eliminate the problem

(fuzzy boundaries).

Phenomena of Vagueness (5)Multi-dimensional vagueness:

• Questions: What makes a man interesting for a woman? What makes a women (or a scientific article) nice?

• Properties can be: not single, not continuous, not measurable.

• Bundle of parameters:The bundle can be varying in different directions.Two given ‘complex cases’ can be both ‘true’ (or both ‘false’),although the underlying parameters are quite different.

• Vague predicates: Common used items (‘nice’ or ‘wonderful’) are often vague.In meaning: ‘the wonderful Greta Garbo’ vs. ‘the wonderful Kurt Gödel’.In quality: ‘the wonderful Greta Garbo’ vs. ‘the wonderful wife’ and ‘the wonderful Kurt Gödel’ vs. ‘the wonderful Hilbert Heikenwälder’.

Phenomena of Vagueness (6) Higher-order vagueness:• Back to the borderline-case ‘Tallness’:

– To decide, ‘this special case is a borderline case’ is a vague decision itself.

– ‘The vagueness of the vagueness of . . . of vague’ is a given phenomenon.

– A theory should be capable to model this.

What we addressed in this chapter:

• Ambiguity, uncertainity, undecidability and generality are not seen as phenomena of vagueness.

• Borderline cases, sorites paradoxa, multi-dimensionality of vagueness and higher-order vagueness are phenomena of vagueness, which should be ‘managed’ by the theories of vagueness.

Further Notes about Vagueness (1)Not expected vagueness

The biologists definition of species:• First: ‘The individuals of a species are looking similarly’. • Second: ‘The individuals of a species are looking similarly

and can interbreed and their offspring are fertile.’

Esatina Salamanders in California• Geographical: E1 w E2 w E3

w central valley E4 w E6 w E5

• Interbreeding: Each group also with its neighbours, but no ‘transitivity’!

Result• The definition of the species seems (today) not to be sufficient.• Thinking ahead (to the year 3000): Probably time (by survival of few species)

reduces the problem (no ‘transitivity-rule’ is hurt nor needed).

Further Notes about Vagueness (2) Vagueness is helpful:• Example:

The forgotten ‘blue’ book (out of 1000).

• Seeking without color-check needs more time.• Seeking with color-check is more efficient. What we addressed in this chapter:• Sometimes definitions seem to be precisely,

although they are not. • Vagueness helps for efficiency.

Vagueness is succeeding.

Theories of Vagueness (1)Many-valued logics• Borderline questions show certainly true or certainly false cases,

but there are a lot of cases which are neither true nor false. • So we need some intermediate (non-classical) truth-value(s).

Questions: • How many of them? • Are the sentential connectives truth-functional? • What is the semantics of connectives and quantifiers? • What is their validity and how is the classical notion to be

generalized? Approaches to many-valued logics:• Three-value logic• Infinite-value logic

Theories of Vagueness (2) Three-value logic: • Michael Tye used three truth-values: T=true, F=false and I=indefinite.

Truth-tables for the connectives Ø, Ù, Ú, Þ, Û are defined.

• Example:Prop. Truth-values Prop. Truth-values .P T I F p T T T I I I F F FØ p F I T q T I F T I F T I F

p Ú q T T T T I I T I F

• No tautologies, (none of the basic connections are always true). The ‘tercium non datur’: p Ú Ø p is not true if p is indefinite.

• The third truth-value defines the existence of a gap between true and false (and not a ‘reality-based’ truth-value near to continuous measurements).

• Multi-valued (finite) logic systems are not mightier than a ‘simple’ three-value logic.

Theories of Vagueness (3)Infinite-value logic: • Truth values: real numbers in the interval [0, 1] (K.F.Machina).

The value of the connectives Ø, Ù, Ú, Þ, Û are defined.

• Example:|Ø p| = 1 - |p| (where |p| denotes the truth-value of p) |p Ú q| = max (|p|, |q|) (i.e. the math. maximum of the conjunct-values).

• Validity as preservation of truth degrees: A conclusion is at least as true as the least true premise.

• Motivation: allow a continuous range of truth-values according to the continuity of given facts (see the given sorites paradoxa).

• Yes-No-Situations: Checking the statement ‘The angle is acute’The continuous range of degrees of the angle should always lead to the ‘pure’ answer Yes or No.

Theories of Vagueness (4)Supervaluation (1):• The truth of a statement is defined by checking all possible situations for the

statement in the following way (see R.Keefe):

• Definition: Be v(p) the truth-value of p in classical logic and let be given an index i to all precifications of a clause, then is defined as true, iff vi(p) is true for all i and V(p) is defined as false, iff vi(p) is false for all i; is not defined for the other cases (i.e. when there are existing some vi which are true

and some other vi which are false). • The ‘undefined’-status:

A ‘hidden’ third truth-value or the ‘gap-indicator’ of a clause.

• D- and I-Operator:K.Fine defined the D- and I-Operator for a general clause A:– DA is true (‘definitely’ A) if A is true in the sense of supervaluation,

t.m. at (all) the base points of the viewed space;– IA (’indefinitely’ A) is (in the sense of supervaluation) undefined.

Theories of Vagueness (5)The epistemic/pragmatic approach. • Back to communication. We communicate what we think we

perceive, and we think we perceive roughly what is relevant to us.

• Imagine giving this introduction in front of crocodiles. According to which criteria will they classify what they see?

• Simple judgements are usually shared by almost all individuals.

• Complex judgement is shared with less probability!• Therefore, growing complexity places judgement increasingly

into the freedom and responsibility of the listener.

Theories of Vagueness (6)• The epistemic/pragmatic approach. • Until now, we have encountered methods of reasoning

about vague statements and their truth values with respect to real world situations.

• STATEMENT <=> REAL WORLD

• Now we introduce another actor into the scene:

• STATEMENT <=> PERCEPTION/ <=> REAL WORLD JUDGEMENT

• Let's view “supervaluation” in this context.

Theories of Vagueness (7)• Supervaluation (2): • Supervaluation can be interpreted as a modal logic

approach. E.g. in „V(P) is true if P is definitely true”, „definitely” is turned into a modal operator. But definitely is also a judgement about one's judgement.

• Individual valuations can be identified with individual judgement, or with specific systems of precification. Example: “approximately 300/345.4 million years“ – approximately 345.4 million years.” (which impose classes of precision).

• By Supervaluation, „approximately“ is not defined within the real world (supertrue). It is defined within the judgement system (true).

Theories of Vagueness (8)Supervaluation implicitly creates a kind of three valued logic

– and a problem with it:

Is (A is undefined) supertrue?

Is there a border case between a border case and a none border case?

Is there a border case of ((A is undefined) is undefined) and ((A is undefined) is defined)?

Probably supervaluation works best based on vagueness.

But it can work: If supervaluation relates to real world facts, and valuation relates to judgement, and we only remain in that view!

Theories of Vagueness (9)What we addressed in this chapter:• Three-value logic

is a ‘gap-theory’ for the indefinite truth-value.• Infinite-value logic

is an approach to continuous parameters.• Supervaluation

is a system of defining truth-values by checking all the truth-possibilities of a statement and concluding (depending on them) the super-truth or super-falseness of a statement.

• The epistemic/pragmatic approach introduces the perception of the individuals be sited between the statements and the real world and showed an interpretation of supervaluation.

Persistency and elasticityA: “Will it work well?”

B: “It will work!”A: “But will it work well?”

B: “I guess ...”A: “What you guess? Will it work well or not?”

B: “Yes, it will work well or not!”A: “You know what I mean! I need you to tell me that it will definitely work well!”

B: “I think it is possible that it will definitely work well.”A: “I must go and tell C that it will work well!!!”

B: “You can tell C that it will work well.”A: “So that is what you say?”

B: “No, that is what you say!”A: “I need you to tell me that it will work well!”

B: “Well – yes, it will work well!”A: “Thank God!”

B: “.. unless it won't”

Summary• Introduction: Vagueness is an efficient and succeeding element in natural

languages, but it brings also some difficulties in communication.

• The most significant phenomena of vagueness were noted:borderline-cases, sorites problems, multi-dimension and higher-order vagueness. Points of hidden vagueness: An example showed where an (un-) precise definition creates a transitivity-problem.

• The main theories of vagueness were introduced:three-value logic, infinite-value logic, supervaluation and the epistemic/pragmatic approach.

• Special questions: persistence and elasticity.

So we hope to have given an interesting overview over the theme as short as an introduction has to be and as long as it is needed to awake some further interest for it.

Formal Semantic of Natural LanguagesFocus: Vagueness

End of Introduction

for TU-Wien / SS 2012 / 185.A53 / Fermüller

Thanks from Hilbert Heikenwälder and Werner Doubek