logic an axiomatisation

Introduction

To my teachers

1.1.1

All knowledge will be expressed in a language .

1.1.2 A language will be dened with reference to a set of propositionsand a set of relations .

The set of a language 's propositions will also be called thelanguage 's vocabulary .

1.1.3

The subset of a language 's vocabulary the members of which arethose and only those members of the set that are true will be called theset of the language 's elements , or the language 's narrative .1.1.4

A language 's vocabulary will circumscribe the totality of thatwhich is expressible in terms of the language .

1.1.5

The members of a language 's vocabulary will also be called wellformed formulae , or candidate states of aairs .1.1.6

The members of a language 's narrative will also be called theorems, or states of aairs , or events .1.1.7

The members of the set of a language 's relations will also becalled functions , or predicates , or attributes , or Universals , or qualities.

1.1.8

1.2.1

A suitably irreducable subset of the set of a language 's relations1

will be called an atomic set of relations , its members , atomic relationsor atoms with respect to the subset .For any language there may be dierent irreducible subsets of theset of the language 's relations , that is, for any language there may bedierent sets of atoms .1.2.2

All members of the set of a language 's relations that are notatoms with respect to an irreducible subset of rls will be called divisible, or composed relations with respect to the subset .1.2.3

1.2.4 A language 's vocabulary will be dened with reference to a setof the language 's atoms .

1.3.1

A language will dene a set of elements .

A world will be a set the members of which are the world 'selements .1.3.2

There will be a subset of the set of elements dened by a languagethe members of which are all those and only those members of the setof elements dened by the language that are characterised by the factthat they contain elements , or that they constitute the empty world .1.3.3

1.3.4

A language will dene a set of worlds .

1.3.5

A language will constitute a world .

There will be a member of the set of worlds dened by a languageconstituting a world of candidate elements .1.3.6

The member of the set of worlds dened by a language constitutinga world of candidate elements will contain all elements dened by thelanguage . All members of all worlds dened by a language will be1.3.7

2

members of the world of candidate elements dened by the language .The member of the set of worlds dened by a language constitutinga world of candidate elements will also be called a world of conceivableelements .1.3.8

Any member of the set of worlds dened by a language willbe characterised in its entirety by the membership of the members of theworld of candidate elements dened by the language that are its members.1.3.9

Any member of the world of candidate elements dened by alanguage - and any member of any world dened by the language - willbe characterised in its entirety by the subset of the set of the language's attributes the members of which are those members of the set thatpertain to it.

1.4.1

1.4.2

There will not be a thing in itself .

Attributes dening an element 's material composition will not befundamentally dierent from the element 's other attributes .1.4.3

There will not be a fundamental distinction of the positions known, respectively , as 'materialism' and 'idealism' .1.4.4

Any two members of the world of candidate elements dened bya language will be related by a relation of identity in the case and onlyin the case that they agree in all of their attributes . Any two members ofthe world of candidate elements dened by a language will be relatedby a relation of dierence in the case and only in the case that there isat least one attribute with respect to which they disagree .1.4.5

The members of pairs of members of the world of candidateelements dened by a language related by a relation of identity will becalled the same element , the members of pairs of members of the world1.4.6

3

of candidate elements dened by the language related by a relation ofdierence , dierent elements .Relations of identity and dierence of members of the world ofcandidate elements dened by a language will be established withreference to relations of identity and dierence of sets of attributes .1.4.7

Any member of the world of candidate elements dened by alanguage and any member of the set constituting an epiphenomenon withrespect to the former will constitute the same element .1.4.8

The subset of the set of a language 's attributes the members ofwhich pertain to a given member of the world of candidate elementsdened by the language will be called the element 's form , or the element's context .1.4.9

A member of the set of a language 's Universals will be calleda contextual element .1.4.10

Any member of the world of candidate elements dened by alanguage will be characterised in its entirety by its form , or context .1.4.11

Any member of the world of candidate elements dened by alanguage will obtain meaning with reference to its form .1.4.12

1.4.13 Establishment of the meaning of a member of the world of candidate elements dened by a language will entail distinguishing it fromall other elements dened by the language .

Establishment of attributes pertaining to a member of the worldof candidate elements dened by a language will progressively narrowthe element 's meaning , thereby dening it.

1.4.14

1.5.1

A world will be dened by a world underlying the world .4

1.5.2

A world underlying a world will be called a language .

1.5.3

A world will be dened by a language .

1.5.4

A world will be dened in a hirarchy of worlds .

1.5.5

A language will be dened in a hirarchy of languages .

Plato 's cave will be a member of a hirarchy of caves , the outsideof any given cave being the inside of an encompassing cave .1.5.6

1.5.7

.

The world underlying a world will be called the world 's grammar

1.5.8 The language underlying a language will be called the language's grammar .

The language underlying a world will also be called the world 'sinterpretation , or the world 's justication .1.5.9

The language underlying a given world will also be called aphysics with respect to the world , the language underlying the languagea metaphysics , or a metaphysics of order ' one ' with respect to the world.1.5.10

The members of the set of elements of a language dening agiven world will also be called laws of physics , the members of the set ofelements of the language underlying the language , laws of metaphysicswith respect to the world .1.5.11

The language constituting an interpretation of the languageconstituting a given world 's physics will be the world 's metaphysics .1.5.12

5

A language dening a world will be called a logic of order 'zero ' with respect to the world , or a metaphysics of order ' zero ' withrespect to the world .1.5.13

A language constituting a world 's physics will constitute ametaphysics of order ' zero ' with respect to the world .1.5.14

A language dening a language constituting a logic of order 'n ' with respect to a world will be called a logic of order ' n+1 ' withrespect to the world , or a metaphysics of order ' n+1 ' with respect tothe world .

1.5.15

The sets of rules governing the scientic method and the methodof deduction with respect to a world dened by a language will be subsetsof the set of elements of the language dening the language , that is,they will be part of the world 's metaphysics . This will answer the setof beliefs dening the position known as logical positivism .

1.5.16

1.6.1

A world cannot estblsh its own interpretation .

A world cannot arm or deny its own existence . Descartes 'scogito fails .1.6.2

Members of the set of a language 's Universals constituting attributes regarding elements ' existence with respect to a given worldwill not constitute a priori attributes unless the latter is the world ofcandidate elements . This will answer Alm's argument .1.6.3

1.6.4

A world will obtain meaning from without itself .

Although , according to the terminology advanced here , neithersynthetic , nor a priori , those members of the vocabulary of the languageunderlng a language that are true with respect to the latter may beidentied as Kant 's synthetic , a priori propositions with respect to theformer language .1.6.5

6

There will not be two members of the set of worlds dened by alanguage such that there is a member of the second world constitutinga member of the rst world in the case and only in the case that it isnot a member of the rst world .1.7.1

In the case that a member of the set of worlds dened by alanguage contains all those , and only those , members of a world denedby the language that are not members of the themselves the former willnot be a member of the latter .1.7.2

1.7.3 There will not be a member of the set of worlds dened by alanguage constituting a Russell set , that is, a set the members of whichare all those and only those members of the world of candidate elementsdened by the language that are not members of themselves . A Russellset will be inconceivable .

Members of the set of a language 's unary relations will pertainto individual members of the set of candidate elements dened by thelanguage . Members of the set of a language 's binary relations willjointly pertain to the members of ordered pairs of members of the set .Members of the set of a language 's n-ary relations will jointly pertainto the members of ordered n- tuples of members of the set of candidateelements dened by the language .

1.8.1

The members of the set of a language 's relations will occur inpairs the members of which constitute complements with respect to eachother .1.8.2

To each member of the set of elements dened by a language one,and only one, member of each complementary pair of unary attributeswill pertain .

1.8.3

1.8.4 Any member of the set of candidate elements dened by a languagemay be identied positively , with reference to the set of those unaryattributes that pertain to it, or negatively , with reference to the set of

7

those unary attributes that do not pertain to it: Any member of theset of candidate elements dened by a language may be dened by thatwhich it is, or alternatively , by that which it is not .In practise , a world 's denition will begin with an originallanguage the interpretation of which reveals itself . A thinking mind 'soriginal language will be its language of experience .1.9.1

1.9.2

A language of experience will constitute a conscious mind .

Members of the set of elements of a language constituting a conscious mind 's language of experience will be called beliefs , the set , 'knowledge(0) ', or ' knowledge by acquaintance ' with respect to the mind.1.9.3

The subset of the vocabulary of a language constituting a consciousmind 's language of experience the members of which are those membersof the set that are true with respect to that language and with respect to asecond language characterised by the same vocabulary as the former willbe called ' knowledge(1) ', or ' knowledge by description ' wrt the consciousmind , and the second language .

1.9.4

knowledge(0) may be considered a special case of knowledge(1) ,the two languages being the same language .

1.9.5

A conscious mind cannot be mistaken about its own beliefs , thatis, it cannot be mistaken about its knowledge(0) . However , any givenmind will reach its own consclusions as to which of another mind 'sbeliefs it will consider knowledge(1) with respect to itself .1.9.6

Members of the vocabulary of a language constituting a consciousmind 's interpretation that are true with respect to the former will notconstitute beliefs with respect to the mind as they are not part of itsvocabulary . To a conscious mind the mind 's interpretation will revealitself .1.9.7

8

All members of all worlds dened by a language will be characterised by their attributes . As attributes confer meaning any world denedby a language will be meaningful . Chaos , that is, the meaningless world, will be inconceivable .1.10.1

1.10.2

Nothingness , that is, the empty world , will be conceivable .

Any language , insofar as it denes any element characterisedby the pertinence of attributes , will dene at least two elements , theelement , and an element that is related to the former by dierence .1.10.3

It may be argued that an empty language , that is, a languagethe set of attributes of which , and the vocabulary of which are emptywill dene a single unspeakable - and unspoken - entity characterised bythe absence of attributes .1.10.4

9

1.1.1

An unary natural logic w.ll be an ordered 10 - tuple

L : fL Le Lp LD LI S0 R S0 p S0 D S0 I S0 I g;

;

;

;

;

;

;

;

;

w

of sets , the members of which are, respectively , a s.t L the members ofwhich are the logic 's propositions , a set Lp the members of which arethe logic 's primary propositions , a set Le the members of which are thelogic 's elemental propositions , a set LD the members of which are thelogic 's Dirac propositions , a set LI the members of which are the logic's reduced Dirac propositions , a set S0 R the members of which are thelogic 's unary relations , a set S0 p the members of which are the logic 'sprimary unary relations , a set S0 D the members of which are the logic's Dirac unary relations , a set S0 I the members of which are the logic 'sparticular element identifying unary relations and a set S0 Iw the membersof which are the logic 's particular world identifying unary relations .Set S0 I of a natural logic 's particular element identifying unaryrelations will be a subset of set S0 R of the logic 's unary relations , thatis, there will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative rst law of the composition ofthe set of particular element identifying unary relations , given as

1.2.1

(L R(nt:l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )0

0

(S I R S R )

!(

):

Members of set S0 I of a natural logic 's particular element identifyingunary relations w.ll also be called , the logic 's characteristic unary names, or the logic 's characteristic names , or the logic 's unary names .1.2.2

Set S0 I of a natural logic 's particular element identifying unaryrelations will constitute an atomic set of unary relations with respectto set S0 p of the logic 's primary unary relations .

1.2.3

10

Each member Ix of set S0 I of a natural logic 's particular elementidentifying unary relations will pertain to one, and only one, member xof the set S(cd te) of elements dened by the logic . For each member xof the set S(cd te) of elements dened by a natural logic there will bea member Ix of set S0 I of the logic 's particular element identifying unaryrelations that pertains to it.1.2.4

:

:

The members of subset S0 p of set S0 R of a natural logic 'sunary relations constituting primary unary relations will also be calledgeneralised element identifying unary relations , or generalised elementidentifying relations .1.2.5

Member Q^T of set S0 R of a natural logic 's unary relationsconstituting a basic conjunctively necessary unary relation , or a basicdisjunctively forbidden unary relation , w.ll also be called , a general elementidentifying unary relation , or a general element identifying relation .1.2.6

Set S0 Iw of a natural logic 's particular world identifying unaryrelations will be a subset of set S0 R of the logic 's unary relations , thatis, there will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative rst law of the composition ofthe set of particular world identifying unary relations , given as1.3.1

(L R(nt:l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )0

0

(S Iw R S R )

!(

):

Set S0 Iw of a natural logic 's particular world identifying unaryrelations will be a subset of set S0 I of the logic 's particular elementidentifying unary relations , that is, there will be a member of the set ofelements of the logic underlying a natural logic constituting a derivative1.3.2

11

rst law of the composition of the set of particular world identifying unaryrelations , given as(L R(nt:l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )0

0

(S Iw R S I )

!(

):

The members of set S0 Iw of a natural logic 's particular worldidentifying unary relations will also be called , the logic 's characteristicunary world names , the logic 's characteristic world names , or the logic's unary world names .

1.3.3

Each member Iw of set S0 Iw of a natural logic 's particular worldidentifying unary relations will pertain to one, and only one, member w ofthe set of worlds dened by the logic . For each member w of the set ofworlds dened by a natural logic there will be a member Iw of the setof the logic 's particular world identifying unary relations that pertains toit.1.3.4

For any ordered triple fIw ; Ix ; Qg of members of set S0 R of a naturallogic 's unary relations the rst two members of which are members of setS0 I of the logic 's particular element identifying unary relations there will bea member p of set L of the logic 's propositions constituting a particularworld universal , particular element existential basic proposition withrespect to the triple , that is, there will be a member of the set ofelements of the logic underlying a natural logic constituting a derivativelaw of the existence of the particular world universal , particular elementexistential basic propositions , given as2.1.1

(L R(nt:l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

!(

12

8Iw 8Ix8Q9p(Iw 2 S I ) ! (Ix 2 S I ) !(Q 2 S R ) !(p 2 L) ^;

;

;

;

0

0

0

(p R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw Ix Q LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )):

Ix will also be called the proposition 's denite noun phrase Universal ,or the proposition 's noun phrase Universal , Q the proposition 's verbphrase Universal . g of members of set S0 R of a naturalFor any ordered triple fIw ; Ix ; Qlogic 's unary relations the rst two members of which are members ofset S0 I of the logic 's particular element identifying unary relations therewill be a member p of set L of the logic 's propositions constituting aparticular world universal , particular element universal basic propositionwith respect to the triple , that is, there will be a member of the set ofelements of the logic underlying a natural logic constituting a derivativelaw of the existence of the particular world universal , particular elementuniversal basic propositions , given as

2.1.2


8Iw 8Ix8Q9p(Iw 2 S I ) ! (Ix 2 S I ) !(p 2 L) ^ 2 S R) !(Q;

!(

;

;

;

0

0

0

LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(p R[p:`8w;8x;(w2w(cd:te) )!(x2w)!Ìw 0 w!(Ìx 0 x!`Q0 x)0 ] Iw Ix Q

13

):

Ix will also be called the proposition 's denite noun phrase Universal , the proposition 's verbor the proposition 's noun phrase Universal , Qphrase Universal .With respect to any member p of set L of a natural logic 'spropositions any member Q of set S0 R of the logic 's unary relations mayoccur as a verb phrase Universal , or as a noun phrase Universal .2.1.3

For any ordered pair fIw ; Qg of members of set S0 R of a naturallogic 's unary relations the rst member of which is a member of set S0 Iof the logic 's particular element identifying unary relations there will bea member p of set L of the logic 's propositions constituting a particularworld universal , general element existential basic proposition withrespect to the pair , that is, there will be a member of the set ofelements of the logic underlying a natural logic constituting a derivativelaw of the existence of the particular world universal , general elementexistential basic propositions , given as

2.2.1


8Iw8Q 9p(Iw 2 S I ) !(Q 2 S R ) ! (p 2 L) ^

!(

;

;

;

0

0

(p R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!`Q0 x0 ] Iw Q LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )):

14

For any ordered triple fIw ; Q0 x ; Q^T g of members of set S0 R of anatural logic 's unary relations the rst member of which is a memberof set S0 I of the logic 's particular element identifying unary relations , thethird member , a member of set S0 R constituting a basic conjunctivelynecessary unary relation , there will be a member p of set L of the logic 'spropositions constituting a particular world universal , generalised elementexistential basic proposition with respect to the triple , that is, therewill be a member of the set of elements of the logic underlying a naturallogic constituting a derivative law of the existence of the particular worlduniversal , generalised element existential basic propositions , given as2.2.2


8Iw8Qx 8QT9p(Iw 2 S I ) !(Qx 2 S I ) ! (QT 2 S R ) !(p 2 L) ^

!(

;

^

;

;

;

0

0

^

0

00 00 0(Q^T R[Q: b:c c:j:v:ly nc:ss:ry u:ry r:l:n] LLe Lp LD LI S R S p S D S I S Iw )^

!

(p R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw Qx QT LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

):

p will also be called an open verb phrase existential proposition ( ` Thereis a cat. ' ), Qx the proposition 's noun phrase Universal . g of members of set S0 R of a naturalFor any ordered pair fIw ; Qlogic 's unary relations the rst member of which is a member of set S0 Iof the logic 's particular element identifying unary relations there will bea member p of set L of the logic 's propositions constituting a particularworld universal , general element universal basic proposition with respectto the pair , that is, there will be a member of the set of elements of

2.2.3

15

the logic underlying a natural logic constituting a derivative law of theexistence of the particular world universal , general element universalbasic propositions , given as

!(

(L R(nt:l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw );Iw ; Q

8 89p 2 S R) !(Iw 2 S I ) ! (Q(p 2 L) ^;

0

0

LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(p R[p:`8w;8x;(w2w(cd:te) )!(x2w)!Ìw 0 w!`Q0 x0 ] Iw Q

):

p will also be called an open noun phrase universal proposition ( ` the proposition 's verb phrase Universal .Everything is beautiful. ' ), QThere will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the composition of theparticular world universal, general element existential basic propositions ,given as2.3.1


8Iw8QT 8Q8p(Iw 2 S I ) !(QT 2 S R ) ! (Q 2 S R ) !(p 2 L) !

!(

;

^

;

;

;

0

^

0

0

00 00 0(Q^T R[Q: b:c c:j:v:ly nc:ss:ry u:ry r:l:n] LLe Lp LD LI S R S p S D S I S Iw )

(

!

(p R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!`Q0 x0 ] Iw Q LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

16

$

00 00 0(p R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw Q^T Q LLe Lp LD LI S R S p S D S I S Iw )

)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the basic propositions'conjunctive attribute transfer , given as

2.3.2


8Iw8QT 8Qx8p(Iw 2 S I ) !(Qx 2 S I ) ! (QT 2 S R ) !(p 2 L) ^

!(

;

^

;

;

;

0

0

^

0


(

!

00 00 0(p R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw Qx Q^T LLe Lp LD LI S R S p S D S I S Iw )

$

00 00 0(p R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw Q^T Qx LLe Lp LD LI S R S p S D S I S Iw )

)):

There will be a member of the set of elements of the logic underlyingnatural logic constituting a derivative law of the composition of

2.3.3

a

17

the particular world universal, general element universal basic propositions ,given as(L R(nt:l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

8Iw 8QT8Q 8p(Iw 2 S I ) ! (QT 2 S R ) ! 2 S R) !(Q(p 2 L) !^

;

;

!(

;

;

0

^

0

0


(

!

LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(p R[p:`8w;8x;(w2w(cd:te) )!(x2w)!Ìw 0 w!`Q0 x0 ] Iw Q

$

00 00 0

(p R[p:`8w;8x;(w2w(cd:te) )!(x2w)!Ìw 0 w!(Ìx 0 x!`Q0 x)0 ] Iw Q^T Q LLe Lp LD LI S R S p S D S I S Iw )

)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the composition of theparticular world universal, general element existential basic propositions ,given as

2.4.1


8L(pwr) 8L8Iw8Q 8p(L(pwr) 2 S) ! (L 2 S) !(Iw 2 S I ) ! R ! (QSp 2 L) !;

_

!(

;

;

;

;

_

0

0

18

(L(pwr) R[S0 : pwr st;S] L)(L(

2

_

8p

L(pwr) )

!

!

2 L) !(p 2L )$(9Ix (Ix 2 S I) ^_

;

(p_

_

_

0

;

LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(p_ R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw Ix Q

)

)(

! LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(p R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!`Q0 x0 ] Iw Q

$

(p S[p: d:sjnt prp:s:n; L_ ] L_ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ))):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the composition ofthe particular world universal, general element universal basic propositions ,given as

2.4.2


8L(pwr) 8L8Iw8Q 8p(L(pwr) 2 S) ! (L 2 S) !(Iw 2 S I ) !(Q 2 S R ) ! (p 2 L) !(L(pwr) R[S : pwr st S] L) !;

^

!(

;

;

;

;

^

0

0

0

;

19

2 L(pwr)) !

(L^(

8p

2 L) !(p 2 L ) $ (9Ix (Ix 2 S I) ^^

;

(p^

^

^

0

;

(p^ R[p:`8w;8x;(w2w(cd:te) )!(x2w)!Ìw 0 w!(Ìx 0 x!`Q0 x)0 ] Iw Ix Q LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

)

)(

!(p R[p:`8w;8x;(w2w(cd:te) )!(x2w)!Ìw 0 w!`Q0 x0 ] Iw Q LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

$

(p S[p: jnt prp:s:n; L^ ] L^ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ))):

There will be a member of the set of elements of the logic underlying a natural logic constituting a derivative law of the resolutionof the particular world universal, particular element existential basic normalpropositions , given as2.5.1


8Iw8Q8pT

(cd:te) ;

8Ix

!(

;

;

_

;

2

(Iw(cd:te) S0 I ) S0 R )(Q

2 !(pT 2 L) !

! (Ix 2 S I) !0

_

0

0

0

0

0

!

] LLe Lp LD LI S R S p S D S I S Iw )00 00 0

(p_T R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw(cd:te) Ix Q LLe Lp LD LI S R S p S D S I S Iw )(Iw(cd:te) R[Iw : I

w(cd:te)

20

!

(

R[Q0 : c:j:v:ly c:t:t frb:dd:n u:ry r:l:n; Q] Ix LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(Q00 00 0(p_T T_ LLe Lp LD LI S R S p S D S I S Iw )

!

)):

There will be a member of the set of elements of the logic underlying a natural logic constituting a derivative law of the resolutionof the particular world universal, particular element existential basic normalpropositions , given as2.5.2


8Iw8Q8pT

(cd:te) ;

8Ix

!(

;

;

^

;

2 S I) ! (Ix 2 S I) !(Q 2 S R ) !(pT 2 L) !

(Iw(cd:te)

0

0

0

^

(Iw(cd:te) R[Iw : I

0

0

0

0

0

!

] LLe Lp LD LI S R S p S D S I S Iw )00 00 0(p^T R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw(cd:te) Ix Q LLe Lp LD LI S R S p S D S I S Iw )w(cd:te)

((Q R[Q0 : c:j:v:ly c:t:t nc:ss:ry u:ry r:l:n; Q] Ix LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )00 00 0(p^T T^ LLe Lp LD LI S R S p S D S I S Iw ))):

21

!

!

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the resolution of theparticular world universal, particular element existential basic propositions ,given as

2.6.1


8Iw 8Ix8Q8pT(Iw 2 S I ) ! (Ix 2 S I ) ! 2 S R) !(Q(pT 2 L) !;

!(

;

;

_

;

0

0

0

_

00 00 0

(p_T R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw Ix Q LLe Lp LD LI S R S p S D S I S Iw )

(

R[Q0 : c:j:v:ly c:t:t frb:dd:n u:ry r:l:n; Q] Ix LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(Q00 00 0(p_T T_ LLe Lp LD LI S R S p S D S I S Iw )

!

)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the resolution of theparticular world universal, particular element universal basic propositions ,given as

2.6.2


8Iw 8Ix8Q8pT(Iw 2 S I ) ! (Ix 2 S I ) !;

!(

;

;

^

;

0

0

22

!

2 S R) !(pT 2 L) !(Q

0

^

00 00 0(p^T R[p:`8w;8x;(w2w(cd:te) )!(x2w)!Ìw 0 w!(Ìx 0 x!`Q0 x)0 ] Iw Ix Q LLe Lp LD LI S R S p S D S I S Iw )

((Q R[Q0 : c:j:v:ly c:t:t nc:ss:ry u:ry r:l:n; Q] Ix LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )00 00 0(p^T T^ LLe Lp LD LI S R S p S D S I S Iw )

!

)):

For any member p of set L of a natural logic 's propositions therewill be a member p of the set constituting a complementary propositionwith respect to p, that is, there will be a member of the set of elements ofthe logic underlying a natural logic constituting a derivative law ofthe existence of the propositions' complements , given as

3.1.1


8p9p

!(

;;

2 L) !(p 2 L) ^(p

(p Rp p LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )):

The second member of any ordered pair fp; pg of members of setL of a natural logic 's propositions will constitute a complementaryproposition with respect to the pair 's rst member in the case and only3.1.2

23

!

in the case that the pair 's rst member constitutes a complementaryproposition with respect to the pair 's second member , that is, therewill be a member of the set of elements of the logic underlying a naturallogic constituting a derivative law of the reciprocity of the propositions'complementarity , given as(L R(nt:l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

8p 8p(p 2 L) ! (p 2 L) !;

!(

;

(

(p Rp p LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

$

(p Rp p LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ))):

For any member Q of set S0 R of a natural logic 's unary relations of the set constituting a complementary unarythere will be a member Qrelation with respect to Q, that is, there will be a member of the set ofelements of the logic underlying a natural logic constituting a derivativelaw of the existence of the unary relations' complements , given as3.2.1


8Q9Q

!(

;;

(Q

(Q

2 S R) !2 S R) ^00

R Q LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(QQ):

24

; Qg of members of theThe second member of any ordered pair fQset of a natural logic 's unary relations will constitute a complementaryunary relation with respect to the pair 's rst mmb in the case and onlyin the case that the pair 's rst member constitutes a complementaryunary relation with respect to the pair 's second member , that is, therewill be a member of the set of elements of the logic underlying a naturallogic constituting a derivative law of the reciprocity of the unary relations'complementarity , given as3.2.2

(L R(nt:l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ) ; Q;Q

8 8 2 S R ) ! (Q 2 S R ) !(Q0

(

!(

0

R Q LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(QQ

$

)

LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(Q RQ Q

):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the complements ofthe particular world universal, particular element existential basic normalpropositions , given as

3.3.1


8Iw

(cd:te) ;

8Ix

!(

;

25

8Q 8Q8p 8p;

;

;

;

2

! 2 !2 ! 2 !(p 2 L) ! (p 2 L) !

(Iw(cd:te) S0 I )(Ix S0 I ) S0 R )(Q(Q S0 R )(Iw(cd:te) R[Iw : I

0

0

0

0

0

!

] LLe Lp LD LI S R S p S D S I S Iw ) LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(p R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw(cd:te) Ix Qw(cd:te)

(p R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw(cd:te)(

R Q LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(QQ0

0

0

0

0

(p Rp p LLe Lp LD LI S R S p S D S I S Iw )

!Ix Q LLe Lp LD LI S R S p S D S I S I ) !0

0

0

0

!

)):

There will be a member of the set of elements of the logic underlying a natural logic constituting a derivative law of the complementsof the particular world universal, general element existential basic normalpropositions , given as3.3.2


8Iw8Q 8Q8p 8p

!(

(cd:te) ;

;

;

;

;

2

!2 ! 2 S R) !(p 2 L) ! (p 2 L) !(Iw(cd:te) S0 I ) S0 R )(Q(Q

0

!

00 00 0] LLe Lp LD LI S R S p S D S I S Iw ) LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(p R[p:`8w;8x;(w2w(cd:te) )!(x2w)!Ìw 0 w!`Q0 x0 ] Iw(cd:te) Q

(Iw(cd:te) R[Iw : I

w(cd:te)

(p R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!`Q0 x0 ] Iw(cd:te)

26

!Q LLe Lp LD LI S R S p S D S I S I ) !0

0

0

0

0

w

0

w

(

R Q LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(QQ0

0

0

0

0

(p Rp p LLe Lp LD LI S R S p S D S I S Iw )

!

)):

For any ordered pair fp00 ; p0 g of members of set L of elementsof a natural logic 's propositions there will be a member p of the setconstituting a negatively conjunctive proposition with respect to the pair, titwbam law of the existence of the negatively conjunctive propositions ,given as

4.1.1


8p 8p9p(p 2 L) ! (p 2 L) !(p 2 L) ^00

;

0

!(

;

;

00

0

(p S[p: p00 !^p0 ] p00 p0 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

):

For any ordered pair fQ00 ; Q0 g of members of set S0 R a natural of the set constitutinglogic 's unary relations there will be a member Qa negatively conjunctive unary relation with respect to the pair ,titwbam law of the existence of the negatively conjunctive unary relations ,given as4.2.1


27

!(

8Q 8Q9Q(Q 2 S R ) ! (Q 2 S R ) ! 2 S R) ^(Q00

0

;

;

;

00

0

0

0

0

S[Qx: Q00 x!^Q0 x] Q00 Q0 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(Q):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the resolution of thenegatively conjunctive propositions , given as4.3.1


8p 8p8p(p 2 L) ! (p 2 L) !(p 2 L) !00

;

0

!(

;

;

00

0

(p S[p: p00 !^p0 ] p00 p0 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(((p00 T^ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

^

(p0 T^ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

)

$

(p0 T_ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

)):

28

!

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the resolution of thenegatively conjunctive propositions , given as

4.3.2


8p 8p8p(p 2 L) ! (p 2 L) !(p 2 L) !00

;

0

!(

;

;

00

0

(p S[p: p00 !^p0 ] p00 p0 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(

!

((p00 T_ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

_

(p0 T_ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

)

!

(p0 T^ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

)):

A truth table of logical connectives will be dened in a hirarchy of truthtables of logical connectives .There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the resolution of thenegatively conjunctive unary relations , given as4.4.1


8Iw

(cd:te) ;

8Ix

!(

;

29

8Q 8Q8Q8p 8p8p00

0

;

;

;

00

;

0

;

;

2 S I) ! (Ix 2 S I) !(Q 2 S R ) ! (Q 2 S R ) ! 2 S R) !(Q(p 2 L) ! (p 2 L) !(p 2 L) !(Iw(cd:te)00

0

0

0

0

0

0

00

0

(Iw(cd:te) R[Iw : I

0

0

0

0

0

!

] LLe Lp LD LI S R S p S D S I S Iw )(p00 R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw(cd:te) Ix Q00 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )w(cd:te)

!Ix Q LLe Lp LD LI S R S p S D S I S I ) ! LLe Lp LD LI S R S p S D S I S I ) !Ix Q

(p0 R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw(cd:te)(p R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw(cd:te)(

S[Qx: Q00 x!^Q0 x] Q00 Q0 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(Q00 0

0

0

0

0

0

(p S[p: p00 !^p0 ] p p LLe Lp LD LI S R S p S D S I S Iw )

0

0

0

0

0

!

)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the composition of thepropositions' complements , given as

5.1.1


8p 8p(p 2 L) ! (p 2 L) !;

!(

;

(

(p Rp p LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

$

30

0

0

0

0

0

0

w

w

(p S[p: p00 !^p0 ] pp LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ))):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the composition of theunary relations' complements , given as5.1.2


8 8 2 S R ) ! (Q 2 S R ) !(Q0

(

!(

0


$

S[Qx: Q00 x!^Q0 x] QQ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(Q)):

There will be a member Iw(cd te) of the set S0 Iw of a natural logic's particular world identifying unary relations constituting a candidateworld identifying unary relation , that is, there will be a member ofthe set of elements of the logic underlying a natural logic constitutinga derivative law of the existence of the candidate world identifying unaryrelations , given as

6.1.1

:


31

!(

9Iw

(cd:te) ;

(Iw(cd:te)(Iw(cd:te)

(Iw(cd:te)

2 S R) ^2 S I) ^2SI )^00

0

w

):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative propositional law of the worldof candidate elements , given as6.1.2


8Iw 8Ix8QT8pT(Iw2 S I) ! (Ix 2 S I) !(QT 2 S R ) !(pT 2 L) !(cd:te) ;

^

^

!(

;

;

;

0

(cd:te)

^

0

0

^


(

!

^00 00 0(p^T R[p:`8w;8x;(w2w(cd:te) )!(x2w)!Ìw 0 w!(Ìx 0 x!`Q0 x)0 ] Iw(cd:te) Ix QT LLe Lp LD LI S R S p S D S I S Iw )

$

^00 00 0(p^T R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw(cd:te) Ix QT LLe Lp LD LI S R S p S D S I S Iw )

)):

6.2.1

There will be a member of the set of elements of the logic underlying32

a natural logic constituting a derivative propositional law of the worldof candidate elements , given as(L R(nt:l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )


^

^

!(

;

;

;

0

(cd:te)

^

0

0

^


(

!

^00 00 0(p^T R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw(cd:te) Ix QT LLe Lp LD LI S R S p S D S I S Iw )00 00 0(p^T T^ LLe Lp LD LI S R S p S D S I S Iw )

)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative propositional law of the worldof candidate elements , given as

6.2.2



^

^

;

;

;

0

(cd:te)

^

!(

0

0

^


33

!

!

(^00 00 0(p^T R[p:`8w;8x;(w2w(cd:te) )!(x2w)!Ìw 0 w!(Ìx 0 x!`Q0 x)0 ] Iw(cd:te) Ix QT LLe Lp LD LI S R S p S D S I S Iw )00 00 0(p^T T^ LLe Lp LD LI S R S p S D S I S Iw )

)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the basic propositions'conjunctive normalisation , given as

6.3.1


8Iw8Iw 8Ix8QT 8Q8pn 8pp8p

!(

(cd:te) ;

;

^

;

;

;

;

;

;

2 S I) !(Iw 2 S I ) ! (Ix 2 S I ) !(QT 2 S R ) ! (Q 2 S R ) !(pn 2 L) ! (pp 2 L) !(p 2 L) !(Iw(cd:te)

0

^

0

0

0

0


!

(pn R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw(cd:te) Ix Q LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )00 00 0(pp R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw Ix Q^T LLe Lp LD LI S R S p S D S I S Iw )

((p R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw Ix Q LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

$

(p S[p: p00 ^p0 ] pn pp LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ))

34

!

!

!

):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the basic propositions'disjunctive normalisation , given as6.3.2


8Iw8Iw 8Ix8QT 8Q8pnatural 8pp8p(Iw2 S I) ! (Iw 2 S I) !(Ix 2 S I ) ! 2 S R) !(QT 2 S R ) ! (Q(pnatural 2 L) ! (pp 2 L) !(p 2 L) !

!(

(cd:te) ;

;

^

;

;

;

;

;

;

0

(cd:te)

0

0

^

0

0


!

LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(pnatural R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw(cd:te) Ix Q00 00 0(pp R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw Ix Q^T LLe Lp LD LI S R S p S D S I S Iw )

(

LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(p R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw Ix Q

$

natural LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(p S[p: p0 !p00 ] pp p)):

35

!

!

There will be a member p^T of the set L of a natural logic's propositions constituting a basic conjunctively necessary proposition ,that is, there will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the existence of the basicconjunctively necessary propositions , given as

7.1.1


9pT (pT 2 L) ^^

;

^

!(

^

(pT R[p: b:c c:j:v:ly nc:ss:ry prp:s:n] LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )):

p^T will also be called a conjunctively tautological proposition , or atautological proposition , or a conjunctive tautology , or a tautology .There will be a member p^F of the set L of a natural logic's propositions constituting a basic conjunctively forbidden proposition ,that is, there will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the existence of the basicconjunctively forbidden propositions , given as7.1.2


9pF (pF 2 L) ^^

;

^

!(

^

(pF R[p: b:c c:j:v:ly frb:dd:n prp:s:n] LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )):

p^F will also be called a conjunctively contradictory proposition , or a contradictory proposition , or a conjunctive contradiction , or a contradiction.36

There will be a member p_T of the set L of a natural logic 'spropositions constituting a basic disjunctively necessary proposition , thatis, there will be a member of the set of elements of the logic underlying anatural logic constituting a derivative law of the existence of the basicdisjunctively necessary propositions , given as7.1.3


9pT (pT 2 L) ^_

;

_

!(

_

(pT R[p: b:c d:sj:v:ly nc:ss:ry prp:s:n] LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )):

p_T will also be called a disjunctively tautological proposition , or adisjunctive tautology .There will be a member p_F of the set L of a natural logic's propositions constituting a basic disjunctively forbidden proposition ,that is, there will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the existence of the basicdisjunctively forbidden propositions , given as7.1.4


9pF (pF 2 L) ^_

;

_

!(

_

(pF R[p: b:c d:sj:v:ly frb:dd:n prp:s:n] LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )):

p_F will also be called a disjunctively contradictory proposition , or adisjunctive contradiction .37

There will be a member Q^T of the set S0 R of a natural logic 'sunary relations constituting a basic conjunctively necessary unary relation ,that is, there will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the existence of the basicconjunctively necessary unary relations , given as

7.2.1


9QT (QT 2 S R) ^^

;

^

0

!(

^

(QT R[Q: b:c c:j:v:ly nc:ss:ry u:ry r:l:n] LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )):

Q^T will also be called a conjunctively tautological unary relation , or atautological unary relation . ^F of the set S0 R of a natural logicThere will be a member Q's unary relations constituting a basic conjunctively forbidden unaryrelation , that is, there will be a member of the set of elements of the logicunderlying a natural logic constituting a derivative law of the existenceof the basic conjunctively forbidden unary relations , given as

7.2.2

(L R(nt:l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ) ^ ;( Q ^ S0 R )Q

9

F

^

F

2

^

!(

(QF R[Q: b:c c:j:v:ly frb:dd:n u:ry r:l:n] LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

):

^F will also be called a conjunctively contradictory unary relation , orQa ctrry unary relation .38

_T of the set S0 R of a natural logic 'sThere will be a member Qunary relations constituting a basic disjunctively necessary unary relation ,that is, there will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the existence of the basicdisjunctively necessary unary relations , given as

7.2.3

(L R(nt:l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ) _ ;( Q _ S0 R )Q

9

T

T

2

^

!(

_ R[Q: b:c d:sj:v:ly nc:ss:ry u:ry r:l:n] LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(QT

):

_T will also be called a disjunctively tautological unary relation .QThere will be a member Q_F of the set S0 R of a natural logic 'sunary relations constituting a basic disjunctively forbidden unary relation ,that is, there will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the existence of the basicdisjunctively forbidden unary relations , given as7.2.4


9QF (QF 2 S R) ^_

;

_

0

!(

_

(QF R[Q: b:c d:sj:v:ly frb:dd:n u:ry r:l:n] LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )):

Q_F will also be called a disjunctively contradictory unary relation .7.3.1

There will be a member of the set of elements of the logic underlying39

a natural logic constituting a derivative law of the resolution of thebasic conjunctively necessary propositions , given as(L R(nt:l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

8pT (pT 2 L) !^

;

^

!(

(

00 00 0(p^T R[p: b:c c:j:v:ly nc:ss:ry prp:s:n] LLe Lp LD LI S R S p S D S I S Iw )^

0

0

0

0

0

(pT T^ LLe Lp LD LI S R S p S D S I S Iw )

!

)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the resolution of the basicconjunctively forbidden propositions , given as

7.3.2


8pF (pF 2 L) !^

;

^

!(

(

00 00 0(p^F R[p: b:c c:j:v:ly frb:dd:n prp:s:n] LLe Lp LD LI S R S p S D S I S Iw )^

0

0

0

0

0

(pF T_ LLe Lp LD LI S R S p S D S I S Iw )

!

)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the resolution of thebasic disjunctively necessary propositions , given as7.3.3

40


8pT (pT 2 L) !_

;

_

!(

(

00 00 0(p_T R[p: b:c d:sj:v:ly nc:ss:ry prp:s:n] LLe Lp LD LI S R S p S D S I S Iw )_

0

0

0

0

0

(pT T_ LLe Lp LD LI S R S p S D S I S Iw )

!

)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the resolution of the basicdisjunctively forbidden propositions , given as7.3.4


8pF (pF 2 L) !_

;

_

!(

(

00 00 0(p_F R[p: b:c d:sj:v:ly frb:dd:n prp:s:n] LLe Lp LD LI S R S p S D S I S Iw )_

0

0

0

0

0

(pF T^ LLe Lp LD LI S R S p S D S I S Iw )

!

)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the resolution of the basicconjunctively necessary unary relations , given as7.4.1


!(

41

8Iw 8Ix8QT 8pT(Iw2 S I) ! (Ix 2 S I) !(QT 2 S R ) ! (pT 2 L) !(cd:te) ;

^

^

;

;

;

0

(cd:te)

^

0

0

^

(Iw(cd:te) R[Iw : I

0

0

0

0

!

0

] LLe Lp LD LI S R S p S D S I S Iw )^00 00 0(p^T R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw(cd:te) Ix QT LLe Lp LD LI S R S p S D S I S Iw )w(cd:te)

(00 00 0(Q^T R[Q: b:c c:j:v:ly nc:ss:ry u:ry r:l:n] LLe Lp LD LI S R S p S D S I S Iw )^

0

0

0

0

0

(pT R[p: b:c c:j:v:ly nc:ss:ry prp:s:n] LLe Lp LD LI S R S p S D S I S Iw )

!

!

)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the resolution of the basicconjunctively forbidden unary relations , given as7.4.2


8Iw 8Ix8Q F 8pF(Iw2 S I) ! (Ix 2 S I) !

(QF 2 S R ) ! (pF 2 L) !(cd:te) ;

^

;

^

^

;

;

0

(cd:te)

!(

0

(Iw(cd:te) R[Iw : I

0

^

0

0

0

0

!

0

] LLe Lp LD LI S R S p S D S I S Iw )00 00 0^(p^F R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw(cd:te) Ix QF LLe Lp LD LI S R S p S D S I S Iw )(

w(cd:te)

^ R[Q: b:c c:j:v:ly frb:dd:n u:ry r:l:n] LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(QF^

0

0

0

0

0

(pF R[p: b:c c:j:v:ly frb:dd:n prp:s:n] LLe Lp LD LI S R S p S D S I S Iw ))):

42

!

!

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the resolution of the basicdisjunctively necessary unary relations , given as

7.4.3


8Iw 8Ix8Q T 8pT(Iw2 S I) ! (Ix 2 S I) ! 2 S R ) ! ((QpT 2 L) !T(cd:te) ;

_

_

;

;

;

0

(cd:te)

_

!(

0

0

_

(Iw(cd:te) R[Iw : I

0

0

0

0

!

0

] LLe Lp LD LI S R S p S D S I S Iw )00 00 0_(p_T R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw(cd:te) Ix QT LLe Lp LD LI S R S p S D S I S Iw )(

w(cd:te)

_ R[Q: b:c d:sj:v:ly nc:ss:ry u:ry r:l:n] LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(QT_

0

0

0

0

0

(pT R[p: b:c d:sj:v:ly nc:ss:ry prp:s:n] LLe Lp LD LI S R S p S D S I S Iw )

!

)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the resolution of thebasic disjunctively forbidden unary relations , given as

7.4.4


8Iw 8Ix8QF 8pF(Iw2 S I) ! (Ix 2 S I) !(cd:te) ;

_

;

_

(cd:te)

!(

;

;

0

0

43

!

(Q_F

2 S R) ! (pF 2 L) !0

_

(Iw(cd:te) R[Iw : I

0

0

0

0

!

0

] LLe Lp LD LI S R S p S D S I S Iw )_00 00 0(p_F R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw(cd:te) Ix QF LLe Lp LD LI S R S p S D S I S Iw )w(cd:te)

(00 00 0(Q_F R[Q: b:c d:sj:v:ly frb:dd:n u:ry r:l:n] LLe Lp LD LI S R S p S D S I S Iw )_

0

0

0

0

0

(pF R[p: b:c d:sj:v:ly frb:dd:n prp:s:n] LLe Lp LD LI S R S p S D S I S Iw )

!

)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the composition of thebasic conjunctively necessary propositions , given as

7.5.1


8p 8p8pT(p 2 L) ! (p 2 L) !(pT 2 L) !;

^

!(

;

;

^

(p Rp p LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(

!

00 00 0(p^T R[p: b:c c:j:v:ly nc:ss:ry prp:s:n] LLe Lp LD LI S R S p S D S I S Iw )

$

(p^p LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )T S[p: p00 _p0 ] p)):

44

!

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the composition of thebasic conjunctively forbidden propositions , given as

7.5.2


8p 8p8pF(p 2 L) ! (p 2 L) !(pF 2 L) !;

!(

;

^

;

^


!

00 00 0(p^F R[p: b:c c:j:v:ly frb:dd:n prp:s:n] LLe Lp LD LI S R S p S D S I S Iw )

$

(p^p LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )F S[p: p00 ^p0 ] p)):

7.5.3 There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the composition of thebasic disjunctively necessary propositions , given as


8p 8p8pT(p 2 L) ! (p 2 L) !(pT 2 L) !;

_

!(

;

;

_


!

00 00 0(p_T R[p: b:c d:sj:v:ly nc:ss:ry prp:s:n] LLe Lp LD LI S R S p S D S I S Iw )

45

$

(p_p LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )T S[p: p00 ^p0 ] p)):

7.5.4 There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the composition of thebasic disjunctively forbidden propositions , given as


8p 8p8pF(p 2 L) ! (p 2 L) !(pF 2 L) !;

_

!(

;

;

_


!

00 00 0(p_F R[p: b:c d:sj:v:ly frb:dd:n prp:s:n] LLe Lp LD LI S R S p S D S I S Iw )

$

(p_p LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )F S[p: p00 _p0 ] p)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the composition of thebasic conjunctively necessary unary relations , given as7.6.1

46


8 88QT 2 S R ) ! (Q 2 S R ) !(Q(QT 2 S R ) !^

!(

;

0

^

0

0

R Q LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(QQ(

!


$

00 00 0

(Q^T S[Qx: Q00 x_Q0 x] QQ LLe Lp LD LI S R S p S D S I S Iw )

)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the composition of thebasic conjunctively forbidden unary relations , given as7.6.2


8 88Q F 2 S R ) ! (Q 2 S R ) !(Q 2 S R) !(QF^

;

0

^

!(

0

0


!

^ R[Q: b:c c:j:v:ly frb:dd:n u:ry r:l:n] LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(QF

$

^ S[Qx: Q00 x^Q0 x] QQ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(QF)

47

):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the composition of thebasic disjunctively necessary unary relations , given as

7.6.3


8 88Q T 2 S R ) ! (Q 2 S R ) !(Q 2 S R) !(QT_

!(

;

0

_

0

0


!

_ R[Q: b:c d:sj:v:ly nc:ss:ry u:ry r:l:n] LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(QT

$

_ S[Qx: Q00 x^Q0 x] QQ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(QT)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the composition of thebasic disjunctively forbidden unary relations , given as7.6.4


8 8

48

!(

8QF 2 S R ) ! (Q 2 S R ) !(Q(QF 2 S R ) !_

;

0

0

_

0


!

00 00 0(Q_F R[Q: b:c d:sj:v:ly frb:dd:n u:ry r:l:n] LLe Lp LD LI S R S p S D S I S Iw )

$

00 00 0

(Q_F S[Qx: Q00 x_Q0 x] QQ LLe Lp LD LI S R S p S D S I S Iw )

)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the disjunctive dominanceof the basic disjunctively forbidden propositions , given as

7.7.1


8p 8pF(p 2 L) ! (pF 2 L) !;

_

!(

;

_

00 00 0(p_F R[p: b:c d:sj:v:ly frb:dd:n prp:s:n] LLe Lp LD LI S R S p S D S I S Iw )_

_

0

0

0

0

0

(pF S[p: p00 _p0 ] pF p LLe Lp LD LI S R S p S D S I S Iw )

!

):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the disjunctive neutralityof the basic disjunctively necessary propositions , given as

7.7.2

49


8p 8pT(p 2 L) ! (pT 2 L) !;

_

!(

;

_

00 00 0(p_T R[p: b:c d:sj:v:ly nc:ss:ry prp:s:n] LLe Lp LD LI S R S p S D S I S Iw )_

0

0

0

0

0

(p S[p: p00 _p0 ] pT p LLe Lp LD LI S R S p S D S I S Iw )

!

):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the conjunctive dominanceof the basic conjunctively forbidden propositions , given as

7.7.3


8p 8pF(p 2 L) ! (pF 2 L) !;

^

!(

;

^

00 00 0(p^F R[p: b:c c:j:v:ly frb:dd:n prp:s:n] LLe Lp LD LI S R S p S D S I S Iw )^

^

0

0

0

0

0

(pF S[p: p00 ^p0 ] pF p LLe Lp LD LI S R S p S D S I S Iw )

!

):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the conjunctive neutralityof the basic conjunctively necessary propositions , given as7.7.4


8p 8pT;

^

;

50

!(

(p

2 L) ! (pT 2 L) !^

00 00 0(p^T R[p: b:c c:j:v:ly nc:ss:ry prp:s:n] LLe Lp LD LI S R S p S D S I S Iw )^

0

0

0

0

0

(p S[p: p00 ^p0 ] pT p LLe Lp LD LI S R S p S D S I S Iw )

!

):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the disjunctive dominanceof the basic disjunctively forbidden unary relations , given as

7.8.1

(L R(nt:l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ) ; Q_ ;Q

8 8F 2 S R ) ! (Q 2 S R ) !(QF0

_

!(

0

00 00 0(Q_F R[Q: b:c d:sj:v:ly frb:dd:n u:ry r:l:n] LLe Lp LD LI S R S p S D S I S Iw ) LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(Q_ S[Qx: Q00 x_Q0 x] Q_ Q

F

!

F

):

7.8.2 There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the disjunctive neutralityof the basic disjunctively necessary unary relations , given as

(L R(nt:l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ); Q _;Q

8 8T 2 S R ) ! (Q 2 S R) !(QT0

_

!(

0

_ R[Q: b:c d:sj:v:ly nc:ss:ry u:ry r:l:n] LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(QT S[Qx: Q00 x_Q0 x] Q _Q LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(QT

51

!

):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the conjunctive dominanceof the basic conjunctively forbidden unary relations , given as7.8.3

(L R(nt:l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ) ^;Q; Q

8 8F 2 S R) !(Q 2 S R ) ! (QF0

^

!(

0

^ R[Q: b:c c:j:v:ly frb:dd:n u:ry r:l:n] LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(QF

^ Q LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(Q^ S[Qx: Q00 x^Q0 x] QF

!

F

):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the conjunctive neutralityof the basic conjunctively necessary unary relations , given as7.8.4


8Q 8QT(Q 2 S R ) ! (QT 2 S R ) !;

^

0

!(

;

^

0

00 00 0(Q^T R[Q: b:c c:j:v:ly nc:ss:ry u:ry r:l:n] LLe Lp LD LI S R S p S D S I S Iw )^

0

0

0

0

0

(Q S[Qx: Q00 x^Q0 x] QT Q LLe Lp LD LI S R S p S D S I S Iw )):

52

!

For any ordered pair fp00 ; p0 g of members of set L of a naturallogic 's propositions there will be a member p of the set constituting adisjunctive proposition with respect to the pair , that is, there will bea member of the set of elements of the logic underlying a natural logicconstituting a derivative law of the existence of the disjunctive propositions, given as8.1.1


8p 8p9p(p 2 L) ! (p 2 L) !(p 2 L) ^00

;

0

!(

;

;

00

0

(p S[p: p00 _p0 ] p00 p0 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )):

For any ordered pair fp00 ; p0 g of members of set L of a naturallogic 's propositions there will be a member p of the set constitutinga conjunctive proposition with respect to the pair , that is, there willbe a member of the set of elements of the logic underlying a naturallogic constituting a derivative law of the existence of the conjunctivepropositions , given as

8.1.2


8p 8p9p(p 2 L) ! (p 2 L) !(p 2 L) ^00

;

0

!(

;

;

00

0

(p S[p: p00 ^p0 ] p00 p0 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

53

):

0 g of members of set S0 R of a natural 00 ; QFor any ordered pair fQ of the set constitutinglogic 's unary relations there will be a member Qa disjunctive unary relation with respect to the pair , that is, therewill be a member of the set of elements of the logic underlying a naturallogic constituting a derivative law of the existence of the disjunctive unaryrelations , given as

8.2.1

(L R(nt:l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ) 00 ; Q 0;Q

8 89Q 2 S R ) ! (Q 2 S R) !(Q 2 S R) ^(Q

!(

;

00

0

0

0

0

S[Qx: Q00 x_Q0 x] Q 00 Q 0 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(Q):

For any ordered pair fQ00 ; Q0 g of members of set S0 R of a naturallogic 's unary relations there will be a member Q of the set constitutinga conjunctive unary relation with respect to the pair , that is, therewill be a member of the set of elements of the logic underlying a naturallogic constituting a derivative law of the existence of the conjunctive unaryrelations , given as8.2.2


8Q 8Q00

;

0

;

54

!(

9Q

;

2 S R) ! (Q 2 S R) !(Q 2 S R ) ^(Q

00

0

0

0

0

(Q S[Qx: Q00 x^Q0 x] Q00 Q0 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )):

8.3.1 There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the composition of thedisjunctive propositions , given as


8p 8p8p 8p8p(p 2 L) ! (p 2 L) !(p 2 L) ! (p 2 L) !(p 2 L) !000

;

;

00

0

!(

;

;

;

00

00

0

0

!(p Rp p LLe Lp LD LI S R S p S D S I S I ) !

(p00 Rp p00 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )0

0

0

0

0

0

0

w

(

(p S[p: p00 _p0 ] p00 p0 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

$

(p S[p: p00 !^p0 ] p00 p0 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ))):

55

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the composition of theconjunctive propositions , given as

8.3.2


8p 8p8p 8p(p 2 L) ! (p 2 L) !(p 2 L) ! (p 2 L) !00

0

;

;

!(

;

;

00

0

(p S[p: p00 !^p0 ] p00 p0 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(

!

(p S[p: p00 ^p0 ] p00 p0 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

$

(p Rp p LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ))):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the composition of thedisjunctive unary relations , given as8.4.1

(L R(nt:l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ) 00 ; Q00 ;Q

8 88Q 8Q8Q 2 S R ) ! (Q 2 S R ) !(Q 2 S R ) ! (Q 2 S R ) !(Q 2 S R) !(Q0

0

;

!(

;

;

00

0

00

0

0

0

0

0

0

00 R Q00 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(QQ

56

!

0 R Q0 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(QQ(

!

S[Qx: Q00 x_Q0 x] Q 00 Q 0 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(Q

$

S[Qx: Q00 x!^Q0 x] Q00 Q0 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(Q)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the composition of theconjunctive unary relations , given as

8.4.2


8Q 8Q8Q 8Q(Q 2 S R ) ! (Q 2 S R ) ! 2 S R ) ! (Q 2 S R ) !(Q00;

0

;

!(

;

;

00

0

0

0

0

0

S[Qx: Q00 x!^Q0 x] Q00 Q0 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(Q(

!

(Q S[Qx: Q00 x^Q0 x] Q00 Q0 LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

$

)


):

9.1.1

For any subset L_ of set L of a natural logic 's propositions there57

will be a member p of L constituting a disjoint proposition with respectto L_ , that is, there will be a member of the set of elements of the logicunderlying a natural logic constituting a derivative law of the existenceof the disjoint propositions , given as(L R(nt:l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

8L(pwr) 8L9p(L(pwr) 2 S) ! (L 2 S) !(p 2 L) ^(L(pwr) R[S : pwr st S] L) !(L 2 L(pwr) ) !_

;

!(

;

;

_

0

;

_

(p S[p: d:sjnt prp:s:n; L_ ] L_ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )):

For any subset L^ of set L of a natural logic 's propositions therewill be a member p of L constituting a conjoint , or joint , propositionwith respect to L^ , that is, there will be a member of the set of elementsof the logic underlying a natural logic constituting a derivative law ofthe existence of the joint propositions , given as9.1.2


89p

8

L(pwr) ; L^ ;

!(

;

2 S) ! (L 2 S) !(p 2 L) ^(L(pwr) R[S : pwr st S] L) !(L 2 L(pwr) ) !(L(pwr)

^

0

;

^

(p S[p: jnt prp:s:n; L^ ] L^ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

58

):

For any subset S0 R _ of set S0 R of a natural logic 's unary relations of S0 R constituting a disjoint unary relationthere will be a member Q0 _with respect to S R , that is, there will be a member of the set of elementsof the logic underlying a natural logic constituting a derivative law ofthe existence of the disjoint unary relations , given as9.2.1


8S R9Q

0 (pwr)

;

8S R

0 _

!(

;

;

2 ! (S R 2 S) !2 ^(pwr)(S RR[S : pwr st S] S R ) !(pwr)(S R 2 S R)!0 (pwr)

(S RS)0 S R)(Q0

0 _

0

0 _

;

0

0

_

000 00 0S(Q[Q: d:sjnt u:ry r:l:n; S0 R _ ] S R LLe Lp LD LI S R S p S D S I S Iw )

):

For any subset S0 R ^ of set S0 R of a natural logic 's unary relationsthere will be a member Q of S0 R constituting a joint , or conjoint , unaryrelation with respect to S0 R ^ , that is, there will be a member of the set ofelements of the logic underlying a natural logic constituting a derivativelaw of the existence of the joint unary relations , given as9.2.2


59

!(

8S R(pwr) 8S R9Q(pwr)(S R2 S) ! (S R 2 S) !(Q 2 S R ) ^(pwr)(S RR[S : pwr st S] S R ) !(pwr)(S R 2 S R)!0

0 ^

;

;

;

0 ^

0

0

0

0

0 ^

;

0

0

^

(Q S[Q: jnt u:ry r:l:n; S0 R ^ ] S0 R LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative rst law of the resolution ofthe disjoint propositions , given as9.3.1


8L(pwr) 8L8p(L(pwr) 2 S) ! (L 2 S) !(p 2 L) !(L(pwr) R[S : pwr st S] L) !(L 2 L(pwr) ) !_

;

!(

;

;

_

0

;

_

(p S[p: d:sjnt prp:s:n; L_ ] L_ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(

8p(p

_

_

2 L) !2L )!

;

(p_

_

(p_ T_ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

)

$

(pT_ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )):

60

!

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative second law of the resolution ofthe disjoint propositions , given as

9.3.2


8L(pwr) 8L8p(L(pwr) 2 S) ! (L 2 S) !(p 2 L) !(L(pwr) R[S : pwr st S] L) !(L 2 L(pwr) ) !_

;

!(

;

;

_

0

;

_

(p S[p: d:sjnt prp:s:n; L_ ] L_ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(

9p

_

_

(p

!

2 L) ^2L )^

;

(p_

_

(p_ T^ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

)

!

(pT^ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative rst law of the resolution ofthe joint propositions , given as9.3.3


61

!(

8L(pwr) 8L8p(L(pwr) 2 S) ! (L 2 S) !(p 2 L) !(L(pwr) R[S : pwr st S] L) !(L 2 L(pwr) ) !^

;

;

;

^

0

;

^

(p S[p: jnt prp:s:n; L^ ] L^ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(

8p

^

^

(p

!

2 L) !2L )!

;

(p^

^

(p^ T^ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

)

$

(pT^ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative second law of the resolution ofthe joint propositions , given as

9.3.4


8L(pwr) 8L8p(L(pwr) 2 S) ! (L 2 S) !(p 2 L) !(L(pwr) R[S : pwr st S] L) !(L 2 L(pwr) ) !;

^

!(

;

;

^

0

;

^

(p S[p: jnt prp:s:n; L^ ] L^ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(

9p

^

;

(p^

2 L) ^62

!

(p^

2L )^^

(p^ T_ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

)

!

(pT_ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the resolution of thedisjoint unary relations , given as

9.4.1


888 88Iw 8Ix8Q 8p(L(pwr) 2 S) ! (L 2 S) !(pwr)(S R2 S) ! (S R 2 S) !(Iw2 S I) ! (Ix 2 S I) ! 2 S R ) ! ((Qp 2 L) !(pwr)(S RR[S : pwr st S] S R ) !(pwr)(LR[S : pwr st S] L) !(pwr)(S R 2 S R)!(pwr)(L 2 L)!(pwr)0 _S0 R; S R ;L(pwr) ; L_ ;(cd:te) ;

;

!(

;

;

_

0 _

0

0

(cd:te)

0

0

0

0

0

0 _

0

;

;

0

_

(Iw(cd:te) R[Iw : I(

8p

w(cd:te)

0

2 L) !(p 2L )$(9Q (Q 2 S R) ^_

;

(p_

_

_

_

;

_

0

0

0

0

] LLe Lp LD LI S R S p S D S I S Iw )

0

63

!

_(Q

2SR )^0 _

_ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(p_ R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw(cd:te) Ix Q)

)

!

LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(p R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw(cd:te) Ix Q(

_

000 00 0S(Q[Q: d:sjnt u:ry r:l:n; S0 R _ ] S R LLe Lp LD LI S R S p S D S I S Iw )

(p S[p: d:sjnt prp:s:n; L_ ] L_ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

!

)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative law of the resolution of thejoint unary relations , given as

9.4.2


888 88Iw 8Ix8Q 8p(L(pwr) 2 S) ! (L 2 S) !(pwr)(S R2 S) ! (S R 2 S) !(Iw2 S I) ! (Ix 2 S I) !(Q 2 S R ) ! (p 2 L) !(pwr)(S RR[S : pwr st S] S R ) !(pwr)(LR[S : pwr st S] L) !(pwr)(S R 2 S R)!(pwr)(L 2 L)!(pwr)0 ^S0 R; S R ;L(pwr) ; L^ ;(cd:te) ;

;

!(

;

;

^

0 ^

0

0

(cd:te)

0

0

0

0

0

0 ^

;

0

;

0

^

64

!

(Iw(cd:te) R[Iw : I(

8p

0

w(cd:te)

0

0

0

0

] LLe Lp LD LI S R S p S D S I S Iw )

!

2 L) !(p 2 L ) $ (9Q (Q 2 S R) ^(Q 2 S R ) ^^

;

(p^

^

^

^

^

;

^

0

0 _

(p^ R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw(cd:te) Ix Q^ LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

)

)

!

(p R[p:`8w;9x;(w2w(cd:te) )!(x2w)^Ìw 0 w!(Ìx 0 x^`Q0 x)0 ] Iw(cd:te) Ix Q LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(

^

(Q S[Q: jnt u:ry r:l:n; S0 R ^ ] S0 R LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )^

0

0

0

0

0

!

(p S[p: jnt prp:s:n; L^ ] L LLe Lp LD LI S R S p S D S I S Iw ))):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative third law of the compositionof the disjoint propositions , given as

9.5.1


88Lb 8La8L8pb 8pa8p(L(pwr) 2 S) !(Lb 2 S) ! (La 2 S) !(L 2 S) !L(pwr) ;_

_

;

_

;

;

;

;

;

_

_

_

65

!(

!

2 L) ! (pa 2 L) !(p 2 L) !(L(pwr) R[S : pwr st S] L) !(Lb 2 L(pwr) ) !(La 2 L(pwr) ) !(L 2 L(pwr) ) !

(pb

0

;

___

!(pa S[p: d sjnt prp s n L ] La LLe Lp LD LI S R S p S D S I S I ) !(p S[p: d sjnt prp s n L ] L LLe Lp LD LI S R S p S D S I S I ) !

00 00 0(pb S[p: d:sjnt prp:s:n; L_ ] L_b LLe Lp LD LI S R S p S D S I S Iw ):

: : ;

:

: : ;

_

_

(

_(L_ S[S: S00 [S0 ] L_b La )

_

0

_

0

0

0

0

0

0

0

0

0

w

w

!

(p S[p: p00 _p0 ] pb pa LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative third law of the composition ofthe joint propositions , given as

9.5.2


88Lb 8La8L8pb 8pa8p(L(pwr) 2 S) !(Lb 2 S) ! (La 2 S)! (L 2 S) !(pb 2 L) ! (pa 2 L) !L(pwr) ;^

^

^

;

;

;

;

;

;

^

^

^

66

!(

(p

2 L) !

(L(pwr) R[S0 : pwr st;S] L)(pwr) )(L^b L(L^L(pwr) )a(L^ L(pwr) )

222

!!!

!

!(pa S[p: jnt prp s n L ] La LLe Lp LD LI S R S p S D S I S I ) !(p S[p: jnt prp s n L ] L LLe Lp LD LI S R S p S D S I S I ) !

00 00 0(pb S[p: jnt prp:s:n; L^ ] L^b LLe Lp LD LI S R S p S D S I S Iw ): : ;

: : ;

^

^

0

^

^

(

^(L^ S[S: S00 [S0 ] L^b La )

0

0

0

0

0

0

0

0

0

w

w

!

(p S[p: p00 ^p0 ] pb pa LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ))):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative second law of the compositionof the disjoint propositions , given as9.6.1



_

!(

;

;

_

0

;

_

(

(L_ R[S: s:ngle el:t s:t; x] p)

_

!

(p S[p: d:sjnt prp:s:n; L_ ] L LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ))

67

):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative second law of the compositionof the joint propositions , given as9.6.2



^

!(

;

;

^

0

;

^

(

(L^ R[S: s:ngle el:t s:t; x] p)^

!

(p S[p: jnt prp:s:n; L^ ] L LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ))):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative rst law of the composition ofthe disjoint propositions , given as9.6.3


8

8

L(pwr) ; L_ ;

68

!(

8pT_

;

2 S) ! (L 2 S) !(pT 2 L) !(L(pwr) R[S : pwr st S] L) !(L 2 L(pwr) ) !(L(pwr)

_

_

0

;

_

_00 00 0(p_T S[p: d:sjnt prp:s:n; L_ ] L LLe Lp LD LI S R S p S D S I S Iw )

((L_ R[S: :mpt: s:t] )_

!

!

(pT R[p: b:c d:sj:v:ly nc:ss:ry prp:s:n] LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ))):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative third law of the composition ofthe joint propositions , given as

9.6.4


8 88pT(L(pwr) 2 S) ! (L 2 S) !(pT 2 L) !(L(pwr) R[S : pwr st S] L) !(L 2 L(pwr) ) !L(pwr) ; L^ ;^

!(

;

^

^

0

;

^

^00 00 0(p^T S[p: jnt prp:s:n; L^ ] L LLe Lp LD LI S R S p S D S I S Iw )

((L^ R[S: :mpt: s:t] )^

!

!

(pT R[p: b:c c:j:v:ly nc:ss:ry prp:s:n] LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )

)

69

):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative third law of the compositionof the disjoint unary relations , given as9.7.1


88S R b 8S R a8S R8Q b 8Q a8Q(pwr)(S R2 S) !(S R b 2 S) ! (S R a 2 S) !(S R 2 S) ! b 2 S R ) ! (Q a 2 S R) !(Q 2 S R) !(Q(pwr)(S RR[S : pwr st S] S R ) !(pwr)(S R b 2 S R)!(pwr)(S R a 2 S R)!(pwr)(S R 2 S R)!(pwr)S0 R;_

0

;

0 _

;

_

0

;

!(

;

;

;

;

;

0

0

_

_

0

;

;

0 _

0

0

0

0

0

0

0

_

0

_

0

;

;

0 _

;

0

0

!] S R a LLe Lp LD LI S R S p S D S I S I ) !] S R LLe Lp LD LI S R S p S D S I S I ) !_

000 00 0b S(Q[Q: d:sjnt u:ry r:l:n; S0 R _ ] S R;b LLe Lp LD LI S R S p S D S I S Iw )

a S(Q[Q: d:sjnt u:ry r:l:n; S0 R _S(Q[Q: d:sjnt u:ry r:l:n; S0 R _(

_

_

0

_

0

;

0 _

_

0

!

0

0

0

0

0

0

(S0 R S[S: S00 [S0 ] S0 R;b S0 R;a ) S[Qx: Q00 x_Q0 x] Q bQ a LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(Q)

70

0

0

w

w

):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative third law of the compositionof the disjoint unary relations , given as9.8.1


88S R b 8S R a8S R8Q b 8Q a8Q(pwr)(S R2 S) !(S R b 2 S) ! (S R a 2 S) !(S R 2 S) ! b 2 S R ) ! (Q a 2 S R) !(Q 2 S R) !(Q(pwr)(S RR[S : pwr st S] S R ) !(pwr)(S R b 2 S R)!(pwr)(S R a 2 S R)!(pwr)(S R 2 S R)!(pwr)S0 R;_

0

;

0 _

;

_

0

;

!(

;

;

;

;

;

0

0

_

_

0

;

;

0 _

0

0

0

0

0

0

0

_

0

_

0

;

;

0 _

;

0

0

!] S R a LLe Lp LD LI S R S p S D S I S I ) !] S R LLe Lp LD LI S R S p S D S I S I ) !_

000 00 0b S(Q[Q: d:sjnt u:ry r:l:n; S0 R _ ] S R;b LLe Lp LD LI S R S p S D S I S Iw )

a S(Q[Q: d:sjnt u:ry r:l:n; S0 R _S(Q[Q: d:sjnt u:ry r:l:n; S0 R _(

_

_

0

_

0

;

0 _

_

0

!

0

0

0

0

0

0

(S0 R S[S: S00 [S0 ] S0 R;b S0 R;a ) S[Qx: Q00 x_Q0 x] Q bQ a LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(Q)

71

0

0

w

w

):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative third law of the composition ofthe joint unary relations , given as9.8.2


88S R b 8S R a8S R8Qb 8Qa8Q(pwr)(S R2 S) !(S R b 2 S) ! (S R a 2 S) !(S R 2 S) !(Qb 2 S R ) ! (Qa 2 S R ) !(Q 2 S R ) !(pwr)(S RR[S : pwr st S] S R ) !(pwr)(S R b 2 S R)!(pwr)(S R a 2 S R)!(pwr)(S R 2 S R)!(pwr)S0 R;^

0

;

0 ^

;

^

0

;

!(

;

;

;

;

;

0

0

^

^

0

;

;

0 ^

0

0

0

0

0

0

0

^

0

^

0

;

;

0 ^

0

;

0

!] S R a LLe Lp LD LI S R S p S D S I S I ) !] S R LLe Lp LD LI S R S p S D S I S I ) !^

(Qb S[Q: jnt u:ry r:l:n; S0 R ^ ] S0 R;b LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw )(Qa S[Q: jnt u:ry r:l:n; S0 R ^(Q S[Q: jnt u:ry r:l:n; S0 R ^(

^

^

0

^

0

;

0 ^

0

^

!

0

0

0

0

0

0

0

(S0 R S[S: S00 [S0 ] S0 R;b S0 R;a )(Q S[Qx: Q00 x^Q0 x] Qb Qa LLe Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ))

72

0

w

w

):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative second law of the compositionof the disjoint unary relations , given as

9.9.1


8S R(pwr) 8S R8Q(pwr)(S R2 S) ! (S R 2 S) !

(Q 2 S R ) !(pwr)(S RR[S : pwr st S] S R ) !(pwr)(S R 2 S R)!0

0 _

;

!(

;

;

0 _

0

0

0

0 _

(

0

;

0

0

!

_

(S0 R R[S: s:ngle el:t s:t; x] Q)0 _00 00 0S(Q[Q: d:sjnt u:ry r:l:n; S0 R _ ] S R LLe Lp LD LI S R S p S D S I S Iw )

)):

There will be a member of the set of elements of the logic underlyinga natural logic constituting a derivative second law of the compositionof the joint unary relations , given as

9.9.2


8

8

(pwr)0 ^S0 R; S R ;

73

!(

8Q

;

2 S) ! (S R 2 S) !(Q 2 S R ) !(pwr)(S RR[S : pwr st S] S R ) !(pwr)(S R 2 S R)!0 (pwr)

(S R

0 ^

0

0

0

0 ^

0

;

0

(

^

(S0 R R[S: s:ngle el:t s:t; x] Q)

!^

(Q S[Q: jnt u:ry r:l:n; S0 R ^ ] S0 R LLe Lp LD L

logic an axiomatisation

Documents

language s elements

language s atoms

language willbe

language willbe

language s narrative

language s propositions

language s relations1

set of elements dened