A Logic for Disorderly Domains

Gwynne Taraska

A dissertation

submitted in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

University of Washington

2009

Program Authorized to Offer Degree:

Department of Philosophy

1

1. Philosophical Motivations

Classical first-order logic is not made to reason about disorderly objects.

1 In none of

its interpretations can an object be incomplete (neither F nor ~F) or inconsistent

(both F and ~F). Neither can an object be nonexistent (beyond the reach of !). In this

dissertation, I present a family of first-order logics called !NL." NL#s base logic,

NLB, permits objects that are incomplete, inconsistent, or nonexistent, whereas its

extensions are made to reason about more orderly domains.2 Its final extension is

classical.3 I describe NL in more detail in the next chapter; here, I present its

philosophical motivations.

Successful but Disorderly Theories

Science, social science, and mathematics abound with theories to which classical

logic is ill suited. Here are some examples from science.

1 I follow the misleading tradition of referring to the first-$%&'%()%'&*+,-'(+,.+/./0(,0(!+.,00*+,.(.$1*+2"

2 NL therefore permits but does not compel the inclusion of disorderly objects.

3 The final extension differs from classical logic only in allowing its domain to be empty.

2

1. In Bohr!s theory of the atom, bound electrons both radiate and do not radiate

energy.4 Applying classical logic, which is explosive, to Bohr!s theory would

result in triviality.5

2. In quantum mechanics, taking the Dirac !-functions as members of the Hilbert

space results in contradiction.6

3. In the famous two-slit experiment of quantum mechanics, reasoning

classically results in the mistaken prediction that the beam of light passing

through both slits will produce a superposition of the individual patterns. A

logic that admits inconsistency without explosion does not predict this.7

4. Frisch recently has argued that classical electrodynamics is inconsistent.8

Even if the world itself is classical and these theories are ultimately false, we will

want to draw inferences from them if their predictions are accurate. We therefore

have a pragmatic need for a logic that can accommodate the disorderly.

Names that Fail to Refer to Existing Objects

Classical logic notoriously struggles in the face of names that fail to refer to existing

objects.9 (It is common to refer to these as "empty# names, although whether this

4 $%%&'()%*+&,--,.&,/-0&123(!*&+3%2(4&)*&52&6257%(&)5&8*%.&98+&:%&;286<&3=>%&8sed a nonclassical logic

in 1913. 5 A logic L is explosive iff (p"~p !L q).

6 See Priest and Routley 1989, 376-377.

7 For the classical proof that results in the mistaken prediction, see Priest and Routley 1989, 377-378.

Orthodox quantum logic provides an alternative solution to the problem of the two-slit experiment; it

does not admit inconsistency but rejects distribution. 8 Frisch 2004.

3

short form is apt is a matter of discussion.) Consider the name Apollo. We cannot

provide a name letter a as a symbolization and say that a fails to refer, as the

interpretation function in classical logic is total for singular terms. Neither can we

provide a as a symbolization and say that a refers to Apollo, as the domain includes

only existing objects.10

One possibility is to follow Russell. We would understand the name Apollo as an

abbreviation of a definite description such as the sun god, which, following Russell!s

analysis of definite descriptions, we would understand as there exists at least one

and at most one sun god. For the sake of simplicity, let the predicate letter F

symbolize is a sun god. Apollo, then, ultimately would appear in classical logic as

!x[Fx " #y(Fy$y=x)] (Russell 1905; see also Quine 1972).

A second possibility is to follow Frege and let all empty names designate a single

artificial referent. We then could symbolize Apollo as a, whose interpretation would

be a member of the domain, say the number 0 (Frege 1892; see also Carnap 1956).

Neither attempt to make classical logic applicable to inferences containing empty

names has been an unqualified success. Russell has been criticized for his violence to

natural language, as he replaces names with existential generalizations, and Frege

9 Classical logic struggles equally in the face of definite descriptions that fail to refer to existing

objects. I limit this discussion to names, as NL does not include a description calculus. I leave this

expansion of NL for a future project. 10

We have the option of doing this and understanding ! "#$ "$ %&'()*+$ ,-").(/('0$ 0".1'0$ .1")$ ")$

%'2(#.')3'+$,-").(/('04$"#$5$6(#3-#s in the next chapter. This option, however, will alienate those who

(reasonably) want to say that Apollo not only fails to exist but fails to have any being at all.

4

has been criticized for his counterintuitive truth-values, as he holds the claims Apollo

is Zeus and Apollo is less than 1 to be true.11

Unlike classical logic, NL does not require reconciliation with empty names. It

has no trouble assessing inferences that contain Apollo, even if Apollo is incomplete

or inconsistent, and it does so without breaking with natural language or producing

incongruous truth-values. I explain how NL does this toward the end of the next

chapter.

The Accommodation of Meinongian Metaphysics

Because of Meinong and Russell!s famous disagreement at the turn of the last

century, those who hold that there are nonexistent objects are called Meinongians

and those who hold that there are only existent objects are called Russellians.12

Meinongians hold that there are plenty of perfectly good objects such as Sherlock

Holmes, Pegasus, the golden mountain, and the round square that fail to exist but

nevertheless are;13

Russellians hold that Holmes and Pegasus neither exist nor are"

#they$ are not objects at all.

Historically, Meinongians have been dismissed as unreasonable eccentrics. This is

in large part because of the following three objections.

11

See, for example, Burge 1991, 190. 12

%&''())*'+,-*.(,/0/1*)+23(-*+*#405&'6*'(/'(*07*4(+),68$9*.(,/0/1*'+,-*%&''())*:+-*+*;4(<&-,2(*#,/*

7+=04*07*6:(*+26&+)$9*+/-*'0*,6*>(/6?*@((A*(?1?A*B+4'0/'*CDEFA*CGA*CE?* 13

The historical Meinong divided objects into three sets: existing objects, merely subsisting objects,

and objects with no being at all. See Findlay 1933, 113. I here consider modern Meinongianism,

which favors only two sets and jettisons the idea of objects with no being at all"05<(26'*#07*>:,2:*,6*

,'*64&(*6:+6*6:(4(*+4(*/0*'&2:*05<(26'$*H.(,/0/1*CDIFA*EJK?*@((*B+4'0/'*CDEF*+/-*L+)6+*CDEJ?

5

An Objection from Quine

Although you seem to be alone (let us suppose), Meinongians maintain that there are

many nonexistent fat men beside you, as well as many nonexistent bald men. How

many fat men are beside you? How many bald ones? Are some of the bald ones

identical to some of the fat ones? These men and their nonexistent compatriots result

in a disorganized ontology, as they lack clear being and identity conditions (Quine

1948, 23).

An Objection from Russell

Meinongians maintain that !the golden mountain" denotes an object that is golden.

By parity of reasoning, they must maintain that !the existent present King of France"

denotes an object that exists (Russell 1905, 483). Given that France is no longer a

monarchy, Meinongianism contradicts empirical facts.14

Another Objection from Russell

Meinongians maintain that the round square is a nonexistent object, but this is #apt to

infringe the law of contradiction$ (Russell 1905, 483). Here is a reconstruction of

Russell"s argument:

14

%&'() (**+() ,-) .*) /&0,) 12((*33) &0() '4) +'45) /&*4) &*) 0662(*()7*'4-48) -9) +0'4,0'4'48) #,&0,) ,&*)

*:'(,*4,);<*(*4,)='48)-9)><046*)*:'(,(?)045)03(-)5-*()4-,)*:'(,$) @ABCD?)EFGHI)%&0,) '(?)7*'4-48)+2(,)

say that the existent present King of France exists, as he is committed to the truth of sentences of the

form the F G is F , but he also must say that the existent present King of France does not exist, as he

knows that France is not a monarchy.

6

1. Suppose for the sake of reductio: The round square is round.

2. Suppose for the sake of reductio: The round square is square.

3. For all x, if x is square, then it is not the case that x is round.

4. From 2 and 3: It is not the case that the round square is round.

5. From 1 and 4: The round square is round and it is not the case that the

round square is round.15

There are, however, plausible responses to all of these objections. Meinongian

theories are typically two-sorted: Parsons, for example, distinguishes between two

sorts of properties, and Zalta distinguishes between two sorts of predication.

According to Parsons, ordinary properties such as being blue and being a mountain

are nuclear, whereas intentional properties (e.g., being thought about by Meinong),

ontological properties (e.g., being existent16

), and modal properties (e.g., being

possible) are extranuclear (Parsons 1980, 23). Nonexistent objects are those objects

that fail to have the extranuclear property of existence. According to Zalta, an object

might either exemplify or encode a property.17

Nonexistent objects fail to exemplify

existence (Zalta 1983, 12); they encode the properties that determine them and they

exemplify the properties that do not. The golden mountain, for example, encodes

15

This argument is implicit in Russell 1905. It is similarly reconstructed in Zalta 1988, 116 and

Parsons 1980, 38. 16

Parsons also introduces a nuclear existence; see Parsons 1980, 43. 17

Zalta uses includes for encodes in McMichael and Zalta 1980. The exemplifying / encoding

(including) distinction seems to be the same distinction that Van Inwagen has in mind when he says

!"#!$ %&'()*!+$ +&,)!-,)+$ have properties, and sometimes properties are ascribed !&$ !"),.$ /0#1$Inwagen 1977, 305).

7

being golden and exemplifies being thought about by Meinong. Existent objects, on

the other hand, do not encode!all of their properties are exemplified.

It is this two-sortedness that allows Parsons and Zalta to provide being and

identity conditions. According to Parsons, there is an object that corresponds to

every set of nuclear properties (Parsons 1980, 26), and objects with exactly the same

nuclear properties are identical (Parsons 1980, 28). According to Zalta, there is a

nonexistent object that corresponds to every set of properties, which encodes the

properties in the set. Objects are identical just in case 1) they are both existent and

exemplify the same properties, or 2) they are both nonexistent and encode the same

properties (Zalta 1983, 12-13 and Zalta 1988, 19).

Meinongians, therefore, have something intelligible to say to Quine. According to

both Zalta and Parsons, there are infinitely many nonexistent fat men in the room: for

Zalta, there are as many as there are sets of properties that include is fat, is a man,

and is in the room, and for Parsons, there are as many as there are sets of nuclear

properties that include is fat, is a man, and is in the room (minus one, if you are a fat

man). Infinitely many of the nonexistent fat men are also bald.

The two-sortedness also allows Parsons and Zalta to keep the existent present

King of France from (really) existing. According to Parsons, the existent present

King of F rance fails to determine an object, as only nuclear properties do this. "He#

therefore is not an object at all and therefore is not an existent one.18

According to

18

Parsons also introduces nuclear existence. If this is the sort of existence we have in mind in the existent present King of F rance, then we have determined an object that indeed exists, but not in the

8

Zalta, the existent present King of France encodes existence, but fails to exemplify it

(Zalta 1983, 50; McMichael and Zalta, 311).

Russell!s second objection is equally nonlethal. Parsons maintains that the third

premise is true when the domain of discourse is existent objects but false when the

domain is unrestricted. "Since there has not been a serious attempt to establish (3) in

its complete generality,# Parsons says, "I suggest that the evidence for it is meager,

and there is no need to accept it. Not all square objects fail to be round; the round

square provides an example of this# (Parsons, 40). Meinongians, therefore, may

avoid Russell!s conclusion by rejecting the third premise, although I encourage them

to accept the conclusion and claim that "there has not been a serious attempt to

establish the law of noncontradiction in i$%&'()*+,$,&-,.,/0+1$23#

Not only does Meinongianism easily escape these objections, but we might argue

with Parsons that it is the more natural worldview:

If we forget or inhibit our philosophical training for a moment, we are all

prepared to cite examples of nonexistent objects: Pegasus, Sherlock

Holmes, unicorns, centaurs.... Those are all possible objects, but we can

find examples of impossible ones too; Quine!s example of the round square

cupola on Berkeley College will do. It is an impossible object, and it

certainly doesn!t exist, so it seems to be an example of an impossible

nonexistent object. With so many examples at hand, what is more natural

full-blooded extranuclear way. No matter which sort of existence appears in the definite description,

$4,.5&67%%,++!%&(89,'$1(.&1%&8+(':,;3&<,,&=0/%(.%5&>?-44.

9

than to conclude that there are nonexistent objects!lots of them! (Parsons,

2)

I am fairly certain that Holmes isn!t, and should I become convinced that he is, I

would argue that we should boldly usher him into our one ontological category and

say that he exists. I will not argue for a Russellian object theory here, however. The

point of this section is that there are reasonable people on both sides of the debate,

and classical logic cannot easily accommodate a Meinongian worldview. Existential

Generalization is valid in classical logic (Fa !!xFx), and the claim "there are

nonexistent objects# is logically false (!~!x~!y[x=y]).19

NL, on the other hand, is a

simple system that can be used regardless of one$s metaphysical theory.

The Possibility that the World Behaves Nonclassically

Last, we may come to find that parts of the world itself are disorderly!that some of

the theories described above are not only useful or reasonable but also true. Whether

there are in fact objects that are incomplete, inconsistent, or nonexistent is a question

not for logic but for empirical science and metaphysics. Perhaps the micro-objects of

quantum mechanics really exist; perhaps the round square, the golden mountain, and

the infinitely many nonexistent fat men of Meinongian mythology really have being;

perhaps there are even existent inconsistent macro-objects such as the box described

19

Again, there is the option of understanding ! %&' %' "()*+,#' -.%+/*0*)12' (./' /3*&' 4*55' 6.1/%*5' %'

7)*+8+,*%+$&')9:1)&&*;)':84)1<'&))'/3)'next chapter.

10

in Priest 1997 that is both empty and full. We can hope. And should we meet a

disorderly object, we will want to have a logic to reason about it.

11

2. Description of NL

Description of N L B: Propositional L evel

The logics of NL have two exclusive and exhaustive values: every statement is either

true or false and no statement is both. The base logic, NLB, permits the incomplete

and inconsistent by severing the traditional link between negation and falsity: it lacks

the classical clause v(p)=1 iff v(~p)=0.1 There are consequently interpretations of

NLB in which v(Fa)=0 and v(~Fa)=0, and there are interpretations of NLB in which

v(Fa)=1 and v(~Fa)=1. As NLB, unlike classical logic, is not explosive, the

admission of inconsistent objects does not result in triviality.

NLB therefore lacks the version of the law of non-contradiction that states never

shall p and ~p be true, and it lacks the version of the law of excluded middle that

states always shall p or ~p be true. NLB retains, however, the versions of these laws

that state never shall p be true and false and always shall p be either true or false.2

NLB consequently can be a friendly logic for dialetheists!those who hold that there

1 The loss of the classical clause in NLB does not deprive negation of its significance. In NLB, I

"#$%&'()#$*+~Fa, #-(*)'*+.Fa/ is false, 0"(*)'*+F is absent in a1, As will be seen in the third chapter,

$-"02%*#%3)(4-#*)#$*5%*6-&3)#/s laws hold in all of the logics of NL. 2 For a discussion of the many formulations of the law of non-contradiction, see Grim 2004.

12

are true contradictions!if they claim not that some statements are true and false, but

that some statements and their negations are true.3

Abandoning the former versions of these laws and retaining the latter versions

enables NLB to be faithful to the natures of inconsistent and incomplete objects.

Other nonclassical logics4 that admit the inconsistent or incomplete typically retain

classical negation but abandon the requirements that 0 and 1 be exclusive or

exhaustive.5 I believe, however, that NLB has the better approach. Consider an

object a that is inconsistent with respect to F . Does a have F? Does a not have F?

Nonclassical logics that retain classical negation say yes and no in each case (that is,

they assign 1 and 0 to both Fa and ~Fa), but I think no, in each case, is a mistake.

One does not say no to an inconsistent object.6 Consider now an object b that is

incomplete with respect to G. Does b have G? Does b not have G? Nonclassical

logics that retain classical negation shrug in each case (that is, they assign neither 1

nor 0 to either Fa or ~Fa), but I think no, in each case, is better. One does not say yes

to an incomplete object. In orderly domains, where negation and falsity travel arm in

arm, there is no harm in running them together. Elsewhere, however, the notions

should be disentangled.

3 I think this is the formulation dialetheists should adopt, as I explain in the next paragraph.

"#ialetheism$%&'(%'(%)'*+%,-.)/0'12-*(%'(%1&3%0'4%-,%*-*-contradiction; again, see Grim 2004. 4 5%/(3%"*-*60'((26'0%0-726$8%"'013.*'1293%0-726$8%'*:%393*%1&3%)3*'62*7%":392'*1%0-726$%2*13.6&'*73';0+8%

as is customary. 5 See Chapter 4 for a comparison of NL and its deviant relatives.

6 I sometimes suspect that those who say yes and no are picturing the object as flickering!perhaps

quickly!!with respect to F . But this is the wrong image; a firmly has and does not have the property

simultaneously.

13

Description of N L B: Predicate L evel

The domains of discourse for the logics of NL are possibly empty. This is an

advantage, as the claims Something is and Something exists need not be true as a

matter of logic.7

The base logic, NLB, allows us to reason about nonexistent objects by giving us

somewhere to put them!somewhere beyond the reach of the classical quantifiers !

and ". It introduces a second pair of quantifiers, ! and ", to range over objects

existent and nonexistent alike.8 NLB therefore has a primary domain, #, and a

subdomain, #":

F igure 1. Domains of NL. ""#$#$%&$'()*$"+,('($(-%&+&$).$# such that $#/$""#$#$%&$'()*$"+,('($%&$).$# such that

$#/ "!#$# %&$ '()*$ "(0('1$ (-%&+(.+$ +,%.2$ %&$ &34,$ +,)+$ $#/$ ).*$ "!#$#$ %&$ '()*$

"(0('1+,%.2$ %&$ &34,$ +,)+$$#5$6,('($)'($ +,('(78'($ %.+('9'(+)+%8.&$87$:;B in which an

object a does not exist but nevertheless is: "x(x=a)=0 and$"x(x=a)=1.

7 So !NL "x(x=x).

8 NLB is thus friendly to the Meinongian tradition, as # < #" can house Jane Eyre and the golden

mountain. NLB %&$)=&8$7'%(.*=1$+8$+,($:1>1)$?)%@(&%A)$'()=%&+$+')*%+%8.B$C,8&($8.+8=821$%.4=3*(&$D8+,$

98&%+%0($(.+%+%(&$ED,>0)F$).*$.(2)+%0($(.+%+%(&$8'$G)D&(.4(&H$E)D,>0)F5$I(($J)++)$KLLK5

#" = {x: "y(y=x)}

#$= {x: "y(y=x)}

14

Classical logic would be able to accommodate these putative objects if we were to

understand ! as a being quantifier, rather than an existence quantifier, but NLB would

have the advantage when it comes to expressive power. It can say, for example, as

we might want to say in the case of Jane Eyre, that something married Rochester

(!xFxa); that something with the property of existence married Rochester (!x[Fxa "

E!x]); and that no really existing thing married Rochester (~!xFxa).9

Apollo Revisited We first symbolize Apollo with a name letter such as a. How to proceed depends on

our metaphysics!NL does not enforce a particular ontology. Those with

Meinongian worldviews!who hold that empty names are not really empty but refer

to nonexistent objects!will let the interpretation of a be a member of "!"!. Those

with Russellian worldviews!who hold that names that fail to refer to existing

objects fail to refer simpliciter!will leave "!"! empty and will let a be undefined.

Both approaches produce the appropriate results. With the Meinongian approach, it

will be false that a exists, true that a nevertheless is, and, depending on the

interpretation, true that a is an archer. With the Russellian approach, it will be false

9 "#$%&$%'( )**( +,-#( ./*,)( )#0( +0.%$%'( *1( 203$4)0%-05( -*%6,-04( )*( 4),701.-)$*%8( 4*( 9( #*70( )#040(

examples suffice. I think they make clear that I understand E!a as akin to a has the property of being located, and I understand the quantified !x(x=a) as akin to a is real. I have nothing even approaching

a definition for the Meinongian notion of being short of existence!I suspect, in fact, that being and

existence are synonymous, which is why I am partial to the extension NLCD, which requires that

everything!incomplete, inconsistent, or otherwise!exists. (For this extension, see the next section.)

:.;4*%4(0%-*,;.'04(,4()*(047*,40()#0(6$4)$%-)$*%(/<(*110;$%'(03.+7=04>(#0(,404()#0(?*;6(203$4)45(4*(

)#.)(@$)(includes all the ordinary physical objects that we normally take to exist, and it does not include

,%$-*;%48('*=6(+*,%).$%48(?$%'06(#*;4048( ;*,%6(4A,.;04(B(:0'.4,48(*;(C#0;=*-&(D*=+04E(F:.;4*%4(

1980, 11). These things merely are.

15

that a exists and false that a is. It also will be false that a is an archer!in fact, any

simple statement containing a will be false.

Description of Extensions

Again, NL permits us to reason about disorderly objects, but it does not insist on it.

NLB"#$%&'%(#)*($NLC1 requires that objects be complete; its extension NLC2 requires

that objects be consistent; and its extension NLCD requires that objects exist.10

As

these requirements are not exclusive, NLB has four extensions besides. The middle

logic in the diagram below is classical.

F igure 2. Extensions of NLB.

10

+,-.$)#$/*0$+1*2234#%$*/ 5*63)(#7.

16

3. Language and Semantics

Primitive Symbols

a, b Individual constants, with or without numerical subscripts

a*, b* Individual constants, with or without numerical subscripts

F1, G

1 One-place predicates, with or without numerical subscripts

F2, G

2 Two-place predicates, with or without numerical subscripts

. .

. .

. .

Fn, G

n N-place predicates, with or without numerical subscripts

E! One-!"#$%&'%()*+%,$%-&!.%/)$#+%

x, y Individual variables, with or without numerical subscripts

!, ! Quantifiers

(,) Punctuation

v, " Binary connectives

~ Unary connective

G rammar

1. An n-place predicate followed by n terms is a formula. Individual constants and

variables are terms.

2. If # is a formula and $ is a variable, then ~#, !$#, and !$# are formulas.

3. If # and % are formulas, then (# v %) and (#"%) are formulas.

17

A formula is a sentence just in case every variable ! is in the scope of an !

quantifier.

Definitions

(" # $) =df ~(~" v ~$)

!!" =df ~"!~"

%!" =df ~&!~"

Interpretations

An interpretation M of a set ' of sentences is a triple <#, #%, I>, where # !"#$%&

'()*+,&(-&'+./(01.%23&+.&*&4(..+567&%)4#7&.%#&(-&(58%/#.&*,'&#% !"#$%&.05'()*+,23&+.&

a subset of #. The interpretation function, I, assigns a pair <(, )> to each n-place

predicate that occurs in the sentences of Z. Both ( and ) are sets of n-tuples of

elements of #. ( is called the extension of the predicate, and ) is called the

antiextension. The interpretation function is partial for singular terms. If I is defined

for a constant a, I(a) * #, and if I is defined for a constant a*, I(a*) * #%.

T ruth in an Interpretation1

Every sentence is assigned either 0 or 1 and not both.

1 p and q range over sentences; ! ranges over variables; " and $ range over formulas; c ranges over

individual constants; and F ranges over predicates.

18

Literals

v(Fc)=1 iff I(c) is an element of the extension of F . So v(E!c)=1 iff I(c) is an

element of the extension of E!.

v(~Fc)=1 iff I(c) is an element of the antiextension of F .

v(Fc1c2)=1 iff < I(c1), I(c2)> is an element of the extension of F .

v(~Fc1c2)=1 iff < I(c1), I(c2)> is an element of the antiextension of F .

And so on for 3+-place predicates.

Disjunctions and Negated Disjunctions

v(p v q)=1 iff v(p)=1 or v(q)=1

v(~(p v q))=1 iff v(~p)=1 and v(~q)=1

Conjunctions and Negated Conjunctions

Given the definition of !,

v(p ! q)=1 iff v(p)=1 and v(q)=1

v(~(p ! q))=1 iff v(~p)=1 or v(~q)=1

Negated Negations

v(~~p)=1 iff v(p)=1

Conditionals and Negated Conditionals

v(p"q)=1 iff v(p)=0 or v(q)=1

19

v(~(p!q))=1 iff v(p)=1 and v(~q)=1.

Generalizations and Negated Generalizations

We expand the vocabulary of Z to include a new constant a. For all o " !, let Mao be

an interpretation that assigns o to a and is otherwise as M.

v("#$)=1 in an interpretation M of Z iff for all o " !, v($[#/a])=1 in Mao.

v(~"#$)=1 in an interpretation M of Z iff for some o " !, v(~$[#/a])=1 in

Mao.

v(##$)=1 in an interpretation M of Z iff for some o " !, v($[#/a])=1 in Mao.

v(~##$)= 1 in an interpretation M of Z iff for all o " !, v(~$[#/a])=1 in

Mao.

We expand the vocabulary of Z to include a new constant a*. For all o " !%, let

Ma*

o be an interpretation that assigns o to a* and is otherwise as M.

v(&#$)=1 in an interpretation M of Z iff for all o " !%, v($[#/a*])=1 in Ma*

o.

v(~&#$)=1 in an interpretation M of Z iff for some o " !%, v(~$[#/a*])=1 in

Ma*

o.

v(%#$)=1 in an interpretation M of Z iff for some o " !%, v($[#/a*])=1 in

Ma*

o.

v(~%#$)= 1 in an interpretation M of Z iff for all o " !%, v(~$[#/a*])=1 in

Ma*

o.

20

The classical ! and " therefore quantify over the subdomain, whereas ! and "

quantify over the entire domain. Here again is the diagram from the last chapter.

#

#

#

#

#

#

F igure 1. Domains of NL.

Semantic Consequence and Tautologousness

A sentence p is a consequence in NL of a set of sentences # iff there is no NL

interpretation in which the value of every member of # is 1 and the value of p is 0.

A sentence p is a tautology of NL iff there is no interpretation in which v(p)=0.

Informal Understandings

Elements of $! are understood to be existent objects, whereas elements of $ ! $!

"#$%&'($#)*++(%*+%,$%'+'$-.)*$'*%+,/$0*)1%2!$%34%*5$#$6+#$4%.)%#$"(%2*5$#$%$xists an $

such that %37% 2"$%3% .)% #$"(% 2*5$#$% .)% "'% $ such that %37 2"$%3% .)% #$"(% 2$8$#9%

existent thing is such that %37%"'(%2!$%3%.)%#$"(%2$8$#9*5.':%.)%)&05%*5"*%%31%

$! = {x: !y(y=x)}

$#= {x: "y(y=x)}

16

3. Language and Semantics

Primitive Symbols

a, b Individual constants, with or without numerical subscripts

a*, b* Individual constants, with or without numerical subscripts

F1, G

1 One-place predicates, with or without numerical subscripts

F2, G

2 Two-place predicates, with or without numerical subscripts

. .

. .

. .

Fn, G

n N-place predicates, with or without numerical subscripts

E! One-!"#$%&'%()*+%,$%-&!.%/)$#+%

x, y Individual variables, with or without numerical subscripts

!, ! Quantifiers

(,) Punctuation

v, " Binary connectives

~ Unary connective

G rammar

1. An n-place predicate followed by n terms is a formula. Individual constants and

variables are terms.

2. If # is a formula and $ is a variable, then ~#, !$#, and !$# are formulas.

3. If # and % are formulas, then (# v %) and (#"%) are formulas.

17

A formula is a sentence just in case every variable ! is in the scope of an !

quantifier.

Definitions

(" # $) =df ~(~" v ~$)

!!" =df ~"!~"

%!" =df ~&!~"

Interpretations

An interpretation M of a set ' of sentences is a triple <#, #%, I>, where # !"#$%&

'()*+,&(-&'+./(01.%23&+.&*&4(..+567&%)4#7&.%#&(-&(58%/#.&*,'&#% !"#$%&.05'()*+,23&+.&

a subset of #. The interpretation function, I, assigns a pair <(, )> to each n-place

predicate that occurs in the sentences of Z. Both ( and ) are sets of n-tuples of

elements of #. ( is called the extension of the predicate, and ) is called the

antiextension. The interpretation function is partial for singular terms. If I is defined

for a constant a, I(a) * #, and if I is defined for a constant a*, I(a*) * #%.

T ruth in an Interpretation1

Every sentence is assigned either 0 or 1 and not both.

1 p and q range over sentences; ! ranges over variables; " and $ range over formulas; c ranges over

individual constants; and F ranges over predicates.

18

Literals

v(Fc)=1 iff I(c) is an element of the extension of F . So v(E!c)=1 iff I(c) is an

element of the extension of E!.

v(~Fc)=1 iff I(c) is an element of the antiextension of F .

v(Fc1c2)=1 iff < I(c1), I(c2)> is an element of the extension of F .

v(~Fc1c2)=1 iff < I(c1), I(c2)> is an element of the antiextension of F .

And so on for 3+-place predicates.

Disjunctions and Negated Disjunctions

v(p v q)=1 iff v(p)=1 or v(q)=1

v(~(p v q))=1 iff v(~p)=1 and v(~q)=1

Conjunctions and Negated Conjunctions

Given the definition of !,

v(p ! q)=1 iff v(p)=1 and v(q)=1

v(~(p ! q))=1 iff v(~p)=1 or v(~q)=1

Negated Negations

v(~~p)=1 iff v(p)=1

Conditionals and Negated Conditionals

v(p"q)=1 iff v(p)=0 or v(q)=1

19

v(~(p!q))=1 iff v(p)=1 and v(~q)=1.

Generalizations and Negated Generalizations

We expand the vocabulary of Z to include a new constant a. For all o " !, let Mao be

an interpretation that assigns o to a and is otherwise as M.

v("#$)=1 in an interpretation M of Z iff for all o " !, v($[#/a])=1 in Mao.

v(~"#$)=1 in an interpretation M of Z iff for some o " !, v(~$[#/a])=1 in

Mao.

v(##$)=1 in an interpretation M of Z iff for some o " !, v($[#/a])=1 in Mao.

v(~##$)= 1 in an interpretation M of Z iff for all o " !, v(~$[#/a])=1 in

Mao.

We expand the vocabulary of Z to include a new constant a*. For all o " !%, let

Ma*

o be an interpretation that assigns o to a* and is otherwise as M.

v(&#$)=1 in an interpretation M of Z iff for all o " !%, v($[#/a*])=1 in Ma*

o.

v(~&#$)=1 in an interpretation M of Z iff for some o " !%, v(~$[#/a*])=1 in

Ma*

o.

v(%#$)=1 in an interpretation M of Z iff for some o " !%, v($[#/a*])=1 in

Ma*

o.

v(~%#$)= 1 in an interpretation M of Z iff for all o " !%, v(~$[#/a*])=1 in

Ma*

o.

20

The classical ! and " therefore quantify over the subdomain, whereas ! and "

quantify over the entire domain. Here again is the diagram from the last chapter.

#

#

#

#

#

#

F igure 1. Domains of NL.

Semantic Consequence and Tautologousness

A sentence p is a consequence in NL of a set of sentences # iff there is no NL

interpretation in which the value of every member of # is 1 and the value of p is 0.

A sentence p is a tautology of NL iff there is no interpretation in which v(p)=0.

Informal Understandings

Elements of $! are understood to be existent objects, whereas elements of $ ! $!

"#$%&'($#)*++(%*+%,$%'+'$-.)*$'*%+,/$0*)1%2!$%34%*5$#$6+#$4%.)%#$"(%2*5$#$%$xists an $

such that %37% 2"$%3% .)% #$"(% 2*5$#$% .)% "'% $ such that %37 2"$%3% .)% #$"(% 2$8$#9%

existent thing is such that %37%"'(%2!$%3%.)%#$"(%2$8$#9*5.':%.)%)&05%*5"*%%31%

$! = {x: !y(y=x)}

$#= {x: "y(y=x)}

21

Comparison of N L B and C lassical Logic

NL is similar to classical logic insofar as it has two exclusive and exhaustive values.

In contrast to classical logic, the domain of discourse is possibly empty and the

interpretation function is partial for singular terms. NLB, in addition, severs the

traditional link between falsity and negation: it lacks the classical clause v(~p)=1 iff

v(p)=0 and allows objects to fall in both the extension and the antiextension of a

predicate!or in neither the extension nor the antiextension of a predicate. That is,

NLB permits objects that are inconsistent or incomplete.

It follows that Fa v ~Fa is not logically true in NLB. Neither is ~(Fa ! ~Fa), which

is equivalent. In a model in which ! contains an object a that is incomplete with

respect to F, v(Fa)=0 and v(~Fa)=0, so v(Fa v ~Fa)=0. (The negation of this sentence

is also false.)

In addition, Fa ! ~Fa is not logically false. Neither is ~(Fa v ~Fa), which is

equivalent. In a model in which ! contains an object a that is inconsistent with

respect to F, v(Fa)=1 and v(~Fa)=1, so v(Fa ! ~Fa)=1. (The negation of this sentence

is also true.)

The inference (p v q), ~p " q (disjunctive syllogism) fails in NLB. In a model, for

example, in which a is inconsistent with respect to F and not in the extension of G,

v(Fa v Ga)=1, v(~Fa)=1, but v(Ga)=0.

22

The inference p ! ~p " q (explosion) fails also: in the model above, v(Fa !

~Fa)=1, but v(Ga)=0.2

Modus tollens likewise fails. In a model, for example, in which a is inconsistent

with respect to G and not in the antiextension of F, v(Fa#Ga)=1 and v(~Ga)=1, but

v(~Fa)=0.

Modus ponens holds, however, as do double negation, De !"#$%&'()*%+(,)%&-)./0)

distributive laws.

Modus Ponens: p#q, p$= q

Double Negation: p % ~~q

De !"#$%&'(1)~(p ! q) % ~p v ~q

De !"#$%&'(1)~(p v q) % ~p ! ~q

De !"#$%&'(1)(p ! q) % ~(~p v ~q)

De Morgan'(1)(p v q) % ~(~p ! ~q)

Distribution: p v (q ! r) % (p v q) ! (p v r)

Distribution: p ! (q v r) % (p ! q) v (p ! r)

Here is an informal proof that ~(~p v ~q)$= NLB (p ! q).

Suppose v(~[~p v ~q])=1. It follows from the truth conditions for negations of

disjunctions that v(~~p)=1 and v(~~q)=1. By the conditions for negations of

negations, it follows that v(p)=1 and v(q)=1. Finally, it follows by the

conditions for conjunctions that v(p ! q)=1.

2 As explosion fails, NLB is a paraconsistent logic. As p "q v ~q fails, NLB is paracomplete.

23

The classical inferences of adjunction, simplification, and addition hold in NLB as

well.

Adjunction: p, q!= NLB p " q

Simplification: p " q!= NLB p; p " q!= NLB q

Addition: p!= NLB p v q; p!= NLB q v p

Evaluating Sentences in Sample Models

1. In a model in which a is in the extension of F as well as its antiextension, and b is

in only the extension of F, !xFx is true, ~!xFx is false, "xFx is true, and ~"xFx

is true.

Extension of F Antiextension of F a + +

b + !

2. In the following model, !xFx is true, ~!xFx is false, "xFx is false, and ~"xFx is

false.

Extension of F Antiextension of F a ! !

b + !

3. In the following model, !xFx is true, ~!xFx is true, "xFx is false, and ~"xFx is

true.

Extension of F Antiextension of F a + +

b ! +

24

4. In the following model, Fa ! Ga is true, ~(Fa ! Ga) is true, Fa v Ga is true, and

~(Fa v Ga) is false.

Extension of F Antiextension of F Extension of G Antiextension of G a + + + !

5. In the following model, Fa ! Ga is false, ~(Fa ! Ga) is false, Fa v Ga is true, and

~(Fa v Ga) is false.

Extension of F Antiextension of F Extension of G Antiextension of G a ! ! + !

6. In the following model, Fa ! Ga is false, ~(Fa ! Ga) is true, Fa v Ga is false, and

~(Fa v Ga) is false.

Extension of F Antiextension of F Extension of G Antiextension of G a ! + ! !

Beyond N L B

Adding Completeness: NLC1

When working in a complete domain, where every n-place predicate is such that

every n-tuple of objects is in either its extension or antiextension, we may add the

clause If v(p)=0, then v(~p)=1. Call this logic NLC1.

p v ~p is logically true in NLC1. Here is an informal proof.

Either v(p)=1 or v(p)=0. If v(p)=1, then v(p v ~p)=1. If v(p)=0, then, by the

completeness clause, v(~p)=1, and so v(p v ~p)=1.

~(p ! ~p) is also logically true.

25

Either v(p)=1 or v(p)=0. If v(p)=1, then v(~~p)=1, and so, by the conditions

for negations of conjunctions, v(~[p ! ~p])=1. If v(p)=0, then, by the

completeness clause, v(~p)=1, and so, by the conditions for negations of

conjunctions, v(~[p ! ~p])=1.

Adding Consistency: NLC2

When working in a consistent domain, where every n-place predicate is such that no

n-tuple of objects falls in both its extension and antiextension, we may add the clause

If v(p)=1 then v(~p)=0. Call this logic NLC2.

p ! ~p is logically false in NLC2, and so we gain explosion. Here is an informal

proof.

If v(p)=1, then, by the consistency clause, v(~p)=0, and so, by the truth

conditions for conjunctions, v(p ! ~p)=0. If v(p)=0, then, by the conditions for

conjunctions, v(p ! ~p)=0.

NLC2 also gains disjunctive syllogism.

Suppose v(p v q)=1 and v(~p)=1. By the consistency clause, v(p)=0, and so

v(q)=1.

26

Collapse of Domains: NLCD

When working in a domain that contains only existent objects, where every object

falls in the subdomain, we may add the clause v(!!")=1 iff v(#!")=1. Call this

logic NLCD.

*********

As the three clauses are not exclusive, eight logics are generated. Here again is the

diagram of NLB!"# $%&$'"()'"*# Each circle represents the addition of one of the

clauses. The result of adding all three to NLB is classical.3

F igure 2. Extensions of NLB.

3 +,$#)'-.#/(00$1$'2$#3$&4$$'#56#7'/#2-7""(27-#-)8(2#("#&,7&#56!"#/)97('#("#:)""(3-.#$9:&.*