24QL Trans

8/3/2019 24QL Trans

http://slidepdf.com/reader/full/24ql-trans 1/27

More QL Translations

() More QL Translations 1 / 27

http://find/

http://goback/

8/3/2019 24QL Trans


1 Introduction

2 Translating English into QL

3 Translating from QL

4Moving Quantiers


http://find/

http://goback/

8/3/2019 24QL Trans


Familiarizing with the Language of QL

Our goal here is to get familiarized with the the language of QL

The best way to do so would be to tackle some translationexercises.


http://find/

http://goback/

8/3/2019 24QL Trans


1 Introduction



4

Moving Quantiers


http://find/

http://goback/

8/3/2019 24QL Trans


Individual Variables and Predicate Letters

Interpretation of individual variable’m’ means Maldwyn’n’ means Nerys’o’ means Owen

Predicate Letters’F’ means is a man’G’ means is a woman’L’ means ...loves ...

’M’ means ...is married to ...’R’ means ...prefers ... to ..

The domain of discourse : All people.


http://find/

http://goback/

8/3/2019 24QL Trans


Two stages of Translation I

Stage One

Re-express the proposition in English itself usingprexed-quantiers plus variable terminology.The quantiers

used will be restricted quantiers.The the restricted quantiers would be either simple ones like’some man x is such that’, ’Every woman y is such that,’ etc orelse ones involving relative clauses like ’Every one x who is a ma

is such that’ ’Some woman y who loves Owen is such that,’ etc.


http://find/

http://goback/

8/3/2019 24QL Trans


Stage TwoTranslate the simple restricted Quantiers into QL as follows:

(Every man is such that) [...x ...x ...]→ ∀ x(Fx ⊃ [...x...x...])

(Some woman y is such that) [...y ...y ...]→ ∃ y(Gy ∧ [...y...y...])(No man z is such that) [...z ...z ...]→ ∀ z(Fz ⊃¬ [...z...z...])

The arrow ’→ ’ Can be read as ’translated as’


http://find/

http://goback/

8/3/2019 24QL Trans


Translating quantiers with relative clauses:’Every one x who is man is such that’ is equivalent to ’everyman is such that.’ And so it gets translated the same way.’Some woman y who loves Own’ is equivalent to ’someone ywho-is-a woman-and-loves-Own.’

Thus we can translate this more complex quantication by usingconjunctive restrictive clauses.


http://find/

http://goback/

8/3/2019 24QL Trans


Examples of Translations into QL I

1 Whoever is loved by Owen is loved by Maldwyn too→ ∀ x(Lox⊃ Lmx)

2 Every woman who loves Maldwyn is loved by Owen→ ∀ x((Gx ∧ Lxm)⊃ Lox)3 Maldwyn loves some woman who loves owen

→ ∃ x((Gx ∧ Lxo) ∧ Lmx)


http://find/

http://goback/

8/3/2019 24QL Trans


Translations ...

4 No man who loves Nerys loves own or Maldwyn→ ¬∃ x((Fx ∧ Lxn) ∧ (Lxo ∨ Lxm))Or alternatively→ ∀ x((Fx ∧ Lxn) ⊃ ¬ (Lxo ∨ Lxm))

5 Every man loves someone→ ∀ x(Fx⊃∃ yLxy)This can also be translated as∀x∃y(Fx ⊃ Lxy)The other very much less natural, reading of the English can berendered∃y∀x(Fx ⊃ Lxy)


http://find/

http://goback/

8/3/2019 24QL Trans


Examples ... I

6 Every one Owen loves is loved by someone Nerys Loves→ ∀ x(Lox⊃ ∃ y(Lny ∧ Lyx))

7 No woman loves every man→ ¬∃ x(Gx ∧ ∀y(Fy ⊃ Lxy))0r ∀x(Gx ⊃ ¬ ∀ y(Fy ⊃ Lxy))

8 No woman loves any man→ ¬∃ x (Gx∧ ∃ y(Fy ∧ Lxy))or ∀x(Gx ⊃ ¬∃ y(Fy ∧ Lxy))

9 If everyone loves Nerys then Owen does.→ (∀xLxn⊃ Lon)


http://find/

http://goback/

8/3/2019 24QL Trans


Examples ... II

(∀xLxn⊃ Lon)is quite different from∀x(Lxn⊃ Lon).

This wff is true in different circumstances.Suppose that Maldwyn loves Nerys and Owen does not.Then’∀xLxn’and ’Lon’are both false. So the conditional ’(∀xLxn⊃ Lon)’istrue . However, (Lmn⊃ Lon) is false. So its universal quantication∀x (Lxn⊃ Lon) is false too.


http://find/

http://goback/

8/3/2019 24QL Trans


Examples,...9 In most contexts, ’If anyone loves Nerys, then Owen does’. This

is the same as ’If theres some one or other who loves Nerys,then own loves her.’ If any one loves Nerys then Owen does.→ (∃x Lxn⊃ Lon)

10 Someone who is married loves NerysWe have in our QL vocabulary thetwo-place predicate ’M’meaning ’... is married to...’But how do we translate the one-place predicate ’...is married’?To be married is to be married to someone.So ’x is married’ can be translated∃yMxy. so 10 can betranslated as→ ∃ x(∃yMxy∧ Lxn)

11 Anyone who is married loves someone they arent married to..→ ∀ x(∃zMxz ⊃ ∃ y(¬ Mxy ∧ Lxy))


http://find/

http://goback/

8/3/2019 24QL Trans


Examples...It is a very good policy when introducing new variables into a wff,always use letters that do not already appear in the wff. But in thelast step, it would have been permissible to use ’y’ again and write→ ∀ x(∃yMxy⊃ ∃ y(¬ Mxy ∧ Lxy))We can do this because these two existential quantication areisolated from each other, their scope do not overlap, and the localcross linking of variables and quaners is quite unambiguous.

12 A married man only loves women→ ∀ x((Fx ∧ ∃ zMxz)⊃ ∀y(Lxy⊃ Gy))

13 Not every married man loves any woman who loves him.→ ¬ ∀ x((Fx ∧ ∃ zMxz)⊃ ∀y((Gy ∧ Lxy)⊃ Lxy))

14 Nerys loves any married men who prefer her to whomever theyare married to.→ ∀ x([Fx ∧ ∃ zMxz]∧ ∀y[Mxy⊃ Rxny]⊃ lnx).


http://find/

http://goback/

8/3/2019 24QL Trans


1Introduction



4 Moving Quantiers


http://find/

http://goback/

8/3/2019 24QL Trans


Once we are familiar with the kind of devices that QL uses forexpressing restricted quantication, it is easy to decode the

message.Consider:∀x(∃y(Mxy∧ Lxy)⊃ ¬ ∃ y(Lxy∧ ∀ z(Mxz ⊃ Rxyz)))It is of the form∀x(A(...x...)⊃ ¬ B(...x...)), So says’ No x which

is A is such that x is B ’As a rst stage, then we have,No x who is such that∃y(Mxy∧ Lxy) is such that∃ y (Lxy∧∀z (Mxz⊃ Rxyz))

This is interpreted asNo x who is married to someone who loves them is such that∃y(Lxy∧ ∀z(Mxz ⊃ Rxyz))


http://find/

http://goback/

8/3/2019 24QL Trans


This in turn decodes asNo one x , who is married to someone who loves them, is suchthat theres some one y they love who is such that∀y(Mxz⊃Rxyz)That saysNo one x , who is married to someone who loves them, is suchthat theres someone y they love who is such thatx prefers y towhoever x is married to.Or in augmented (though not quite unambiguous) English:No one who is married to some one who loves them loves

someone they prefer to whoever they are married to.


http://find/

http://goback/

8/3/2019 24QL Trans


1 Introduction



4 Moving Quantiers


http://find/

http://goback/

8/3/2019 24QL Trans


Alternative ways of translating IThere are number of cases where there are distinct but equally goodtranslations:

For translating ’no’ sentence we have two good options. Forexample, the sentenceNo humans are perfectcan be translated as either∀x(Hx ⊃ ¬ Px)or as¬∃ x(Hx ∧ Px).Alphabetical choice of variables is arbitrary. So everyone is wisefor example can be translated as either ’∀xFx’ or ’∀yFy’While translating ’Someone loves someone,’ there is nothing tochoose between’∃x ∃yLxy’ and ’∃y ∃xLxy’


http://find/

http://goback/

8/3/2019 24QL Trans


Alternative ways of translating II

Similarly there is nothing to choose between’∀x∀yLxy’ and ’∀y ∀xLxy’We can unrestrictedly swap quantiers around. Indeed the wholerationale of our notation is that it allows us unambiguously markthe difference between messages expressed by eg.’∀x∃yLxy’ and’∃y∀xLxy’.However, immediately adjacent quantiers of the same can beinterchanged.


http://find/

http://goback/

8/3/2019 24QL Trans


Neighbouring Quantiers I

Consider the case where we have neighboring existential quantiers:

’∃yLnxy’ attributes Nerys the property of loving someone.When we quantify in we get’∃x∃yLxy,which says that someone has the property that ∃yLny attributes

Nerys, .i.e., the property of loving someone.So ’∃x∃yLxy’ holds just if there is a pair of people in the domain(not necessarily distinct) such that the rst loves the second.Like wise ∃xLxn attributes to Nerys the property of being loved

by someone.Quantifying in we get ∃y∃xLxywhich says that someone has the property that ’∃xLxn attributesto Nerys, i.e. the property of being loved by someone.


http://find/

http://goback/

8/3/2019 24QL Trans


Neighbouring Quantiers II

So ∃y∃xLxy holds if there is a pair of people in the domain (notnecessarily distinct.) such that the rst loves the second.That is the same truth-condition for ∃x∃yLxy

And the point generalizes to any pair of wffs of the form ’∃v ∃w C(...v ...W ...)’ and ’∃w ∃vC(...v ...w ...)’.Similarly for pairs of universal quantiers.


h f l l

http://find/

http://goback/

8/3/2019 24QL Trans


Another sort of alternative translation ISuppose we want to translate ’Nerys is a woman everyone loves’There are alternative translations:

1 (Gn ∧ ∀xLxn)2 ∀x(Gn ∧ Lxn)

The second holds just in case every one x is such as to make it true

that (Nerys is a woman and x loves her).This holds in case Nerys is a woman loved by every one which is whathe rst says.More generally speaking following equivalences hold:

1 (A ∧ ∀vB (...v ...)) ≡ ∀ v (A ∧ B(...v ...))if the variablev doesn’t occur in A. The order of conjuncts doesnot matter.

2 (A ∧ ∃ v B(...v ...)) ≡ ∃ v (A ∧ B(...v ...))if the variablev does not occur in A.


A h f l i l i II

http://find/

http://goback/

8/3/2019 24QL Trans


Another sort of alternative translation II’Someone who is married loves Nerys’ can be translated as∃x(∃yMxy∧ Lxn)

as we have done earlier.It would have been equally legitimate to write∃x∃y(Mxy∧ Lxn)given the variable y does not occur in the second conjunct.

We can do similar manipulations with disjunctions:Suppose we want to translate ’Either Owen loves Nerys or nobodydoes.’We could equally well write (Lon∨ ∀x¬ Lxn)Or ∀x(Lon ∨ ¬ Lxn)More generally we have:

1 (A ∨ ∀v B(...v )) ≡ ∀ v (A ∨ B (...v ...))where v does not occur in A and order of disjuncts does notmatter.


A h f l i l i III

http://find/

http://goback/

8/3/2019 24QL Trans


Another sort of alternative translation III

2 (A ∨ ∃ v B(...v ...))


C diti l d M i f Q ti I

http://find/

http://goback/

8/3/2019 24QL Trans


Conditionals and Moving of Quantiers IThe case where we really have to be careful involves conditionalsWe have as you might expect,

1 (A ⊃ ∀ v B(...v ...)) ≡ ∀ v (A ⊃ B(... v ...))2 (A ⊃ ∃ v B(...v ...)) ≡ ∃ v (A ⊃ B(...v ...)) ≡ ∃ v (A ⊃ B (...v ...))

However, note very carefully the following:1 (∀v (B ... v ...)⊃ A) ≡ ∃ v (...v ...)⊃ A)2 (∃v B(...v ...)⊃ A) ≡ ∀ v (B(... v ...) ⊃ A)

Extracting universal quantication from antecedent of aconditional (when the variable does not occur in theconsequent)turns it into an existential quantication.This is so because antecedent of conditionals are like negateddisjuncts - remember (A⊃ B) is equivalent to (¬ A ∨ B)


C diti l d M i g f Q ti II

http://find/

http://goback/

8/3/2019 24QL Trans


Conditionals and Moving of Quantiers II

Remeber also that when quantiers tangle with negation they’ip’ into the other quantier.Consider for example, the following chain of equivalences.

(∀x Fx ⊃ Fn) ≡ (¬∀ xFx ∨ Fn) ≡ (∃x¬ Fx ∨ Fn) ≡ ∃ x(¬ Fx ∨Fn) ≡∃ x (Fx ⊃ Fn)


http://find/

http://goback/

24QL Trans

Documents

Transcript of 24QL Trans