1 Introduction
I present a
logic
that expresses gradual shifts in domain of discourse.
The motivation is to capture certain peculiar phenomena
about concepts/objects and other concepts/objects^{1}^{1}1
These two terms will not be strongly
distinguished in this work. An object may exist by itself, but
to reason about relation between objects or
just to speak about them,
it is, as we presume,
concepts referring to the objects that
we reason/speak about.
as their attributes. The first such phenomenon is that
extension of a concept alters when
it becomes an attribute to other concepts.
Also a concept that is specified another
concept as its attribute becomes an intension [Carnap47, Montague74, Church51] of
the concept which itself is an extensional concept.
Consider
for instance ‘brooch’ed ‘hat’ and ‘brooch’.
The ‘brooch’ in the former is an attribute to ‘hat’.
By definition, if
anything is an attribute
to something at all, it must be found among all that can
be an attribute to it.
Whereas the extension of
‘brooch’ in the latter is only
delimited by our understanding about ‘brooch’,
that in the former as an attribute of
‘hat’ is delimited also by our understanding
about the concept ‘hat’ (needless to say, only if
the understanding
of ours should permit ‘brooch’ as an attribute at all).
But this is not all. The attribute in turn
specialises the ‘hat’ to which it is an attribute:
the ‘brooch’ed ‘hat’ forms an intension of the ‘hat’,
and itself becomes an extensional concept
“brooch’ed ‘hat”.
The shift in extension is not typically
observed in formal logics, be they
temporal, epistemic, modal etc.
Some exceptions that challenge the norm are
a kind of spatiotemporal
logics [Gabelaia05, Muller98] and
some kinds of context logics (Cf.
[McCarthy93, Guha03]) in the line of [Buvac93, Nayak94]. In [Gabelaia05] for instance,
Gabelaia et al. consider the definition
of EU at a point of time and at another point of
time. If some countries are merged into the current
EU, then the term EU will remain EU at the
future time reference as it is now, but the
spaces that the two EU occupy are not the same.
Similar phenomena are occurring
in the relation between concepts/objects and their
attributes. However, unlike the case of
the spatiotemporal logics,
there is no external and global space for them:
there are only those spaces generated by the
(extension of) concepts
themselves. The stated (re)action of
intension/extension within an attributed concept/object
is another intriguing feature that has not been
formalised before.
Another point about the concept is that
a concept in itself, which is to say,
an atomic concept which
does not itself possess
any other concepts as its attributes,
is almost certainly imperceptible,^{2}^{2}2Let
us arbitrarily suppose the concept hat, and let
us conduct an experiment as to if we could
perceive the hat in itself as something
differing from nothingness for which any scenario where it comes with
an attribute is inconceivable.
To begin with, if the
word hat evoked in our mind any specific hat with
specific colour and shape, we first remove the colour
out of it. If the process should make it transparent,
we then remove the transparentness away from it.
And if there should be still some things that
are by some means perceivable as having originated
from it, then because they are an attribute of the hat,
we again remove any one of them. If the humanly
no longer detectable
something is not nothingness is not itself
contradictory, then there must be still some quality
originating in the hat that makes the something
differ from nothingness. Now the question is whether
the something can be perceived at all to be different
from nothingness. Intuition speaks otherwise. and hence almost certainly cannot be
reasoned about. Typically, however, formal/symbolic logics assume smallest entities
upon which other expressions are formed.
In this work I challenge the assumption, and
materialise the observation that we cannot tell apart
 that is,
we cannot know  whether
a soregarded atomic entity is atomic or is just
atomic enough not to be considered nonatomic.
I present a logic in which every entity is nonatomic,
reflecting our intuition about the concept.^{3}^{3}3The utility
of logical
nonatomicity
is noted in a recent work
[Jung15] in programming community.
The idea of logical nonatomicity
in formal/symbolic logic itself, however,
appears previously
in
the immediately preceding
work
to the current paper, namely,
in [Arisaka14tech1]; as well as, and more bluntly, in its variation as
a technical report.
Strikingly we can represent both extensional
shifts and nonatomicity using the familiar classical
logic only (but
the results extend to other
Boolean logics); with
many domains of discourse.^{4}^{4}4The use of
multiple domains of discourse is also notable
in contextual logics.
Connections to those will be mentioned
at the end of this work.
The idea is as follows. We shall
define a binary connective over classical logic
instances. As an example, reads as; Hat is,
and under the presupposition
that Hat is, Brooch is as its attribute.
In this simple expression there are
two domains of discourse: one in which
Hat in is being discussed;
and one in which Brooch in
is being
discussed.
The second domain of discourse
as a whole is delimited by the (extension of)
Hat
that gives rise to it. is an intension of
Hat, and itself forms
an extensional concept. The nonatomicity of concepts
is captured without breaking the properties
of classical logic. The ideas are that
every concept has attributes,
but that the attributes are not discussed
in the same domain of discourse as the concept is
discussed in. From within the domain
of discourse discussing Hat
in ,
it cannot be perceived whether it has or has not
attributes, i.e. whether it is atomic or is
just atomic enough not to be considered nonatomic.
We can also explain some reasonably common
everyday linguistic
phenomenon with this connective. Let us turn to
an example.
Episode.
There is
a tiny hat shop in our town, having the following in stock:

[leftmargin=0.5cm]

3 types of hats: orange hats, green hats ornamented with some brooch, and blue hats decorated with some white hatshaped accessory. Only the green and the blue hats are displayed in the shop.

2 types of shirts: yellow and blue shirts. Only the blue shirts are displayed in the shop.
A young man has come to the hat shop.
After a while he asks the shop owner, a lady of many a year of experience
in hatmaking;
“Have you got a yellow hat?”
Knowing that it is not in her shop, she answers;
“No, I do not have it in stock,” negating
the possibility that there is one in stock at her shop at the present
point of time. Period.
“What is she actually denying about?”
is our question, however.
It is plausible that, in delivering the answer,
the question posed may have allowed her to infer that the young man was
looking for
a hat, a yellow hat in particular. Then the answer
may be followed by she saying; “…but I do have hats with
different colours including ones not currently displayed.”
That is, while she denies the presence of a yellow hat,
she still presumes the availability of hats of which she
reckons he would like to learn.
It does not appear so unrealistic to suppose such a thought
of hers that he may be ready to
compromise his preference for a yellow hat with
some nonyellow one,
possibly an orange one in stock, given its comparative closeness
in hue to yellow.
Now, what if the young man turned out to be a
townfamous collector of yellow articles? Then it may be
that from his question she had divined instead that
he was looking for something yellow, a yellow hat in
particular, in which case her answer could have
been a contracted form of “No, I do not have it
in stock, but I do have a yellow shirt nonetheless (as
you are looking after, I suppose?)”
Either way, these somewhatappearingtobe partial
negations contrast with
the classical negation with which her
answer can be interpreted only as that she does not have a yellow
hat, nothing less, nothing more, with no restriction in the range
of possibilities outside it.
The explanation that I wish to provide is that in the
first case she actually means , presuming
the main concept Hat
but negating Yellow as its attribute in a different
domain of discourse in which its attributes
can be discussed; and in the second case
she actually means
with the main
concept Yellow presumed but Hat denied as its attribute.
Like this manner, gradual classical logic
that I propose here
can capture partial negation, which is known as
contrariety in the preFregean term logic
from the Aristotle’s era [Horn89],
as well as nowadays more
orthodox contradictory negation.
Here we illustrated attribute
negation. Complementary, we may also consider
object negation of the kind
,
as well as more orthodox negation of the sort
which I call attributedobject negation.
My purpose
is to assume attributed concepts/objects^{5}^{5}5In the rest
I simply write attributed objects, assuming
that it is clear that we do not strongly distinguish
the two terms in the context of this work. as primitive
entities and to analyse the logical behaviour of
in interaction with
the other familiar connectives in classical logic. In
this logic,
the sense of ‘truth’, a very fundamental property of classical
logic, gradually shifts^{6}^{6}6This should not
be confused with the idea of many
truths in a single domain of discourse [Hajek10, Gottwald09].
by domains of discourse moving deeper into
attributes of (attributed) objects.
As for
inconsistency, if there is an inconsistent argument within
a discourse on attributed objects, wherever it may be that
it is occurring,
the reasoning part of which is inconsistent cannot be
said to be consistent. For this reason it remains in gradual classical logic
just as strong as is in standard classical logic.
1.1 Structure of this work
Shown below is the organisation of this work. The basic conceptual core is formed in Section 1, Section 2, which is put into formal semantics in Section 3. Decidability of the logic is proved in Section 4. After the foundation is laid down, more advanced observations will be made about the objectattribute relation. They will be found in Section 5. Section 6 concludes with prospects.

[leftmargin=0.3cm]

Development of gradual classical logic (Sections 1 and Section 2).

A formal semantics of gradual classical logic and a proof that it is not paraconsistent/inconsistent (Section 3).

Decidability of gradual classical logic (Section 4).

Advanced materials: the notion of recognition cutoffs, and an alternative presentation of gradual classical logic (Section 5).

Conclusion (Section 6).
2 Gradual Classical Logic: Logical Particulars
In this section we shall look into logical particulars of gradual classical logic. Some familiarity with propositional classical logic, in particular with how the logical connectives behave, is presumed. Mathematical transcriptions of gradual classical logic are found in the next section.
2.1 Logical connective for object/attribute and interactions with negation
We shall dedicate the symbol to represent the objectattribute relation. The usage of the new connective is fixed to take the form . It denotes an attributed object. is more generic an object than ( acting as an attribute to makes more specific). The schematic reading is as follows: “It is true that is, and it is true that it has as its attribute.” Now, this really is a shortform of the following expression: “It is true by some sense of truth X reigning over the domain of discourse discussing that is judged existing in the domain of discourse,^{7}^{7}7As must be the case, a domain of discourse defines what can be talked about, which itself does not dictate that all the elements that are found in the domain are judged existing. and it is true by some sense of truth Y reigning over the domain of discourse discussing as an attribute to that is judged having as its attribute.” Also, this reading is what is meant when we say that “It is true that is,” where the sense of the truth Z judging this statement has relation to X and Y, in order for compatibility. I take these sideremarks for granted in the rest without explicit stipulation. Given an attributed object , expresses its attributed object negation, its object negation and its attribute negation. Again the schematic readings for them are, respectively;

[leftmargin=0.3cm]

It is false that is.

It is false that is, but it is true that some non is which has an attribute of .

It is true that is, but it is false that it has an attribute of .
The presence of negation flips “It is true that …” into “It is false that …” and vice versa. But it should be also noted how negation acts in attribute negations and object/attribute negations. Several specific examples constructed parodically from the items in the hat shop episode are;

[leftmargin=0.5cm]

: It is true that hat is, and it is true that it has the attribute of being yellow (that is, it is yellow).

: It is true that yellow is, and it is true that it has hat as its attribute.

: It is true that hat is, but it is false that it is yellow.

: It is false that hat is, but it is true that yellow object (which is not hat) is.

: Either it is false that hat is, or if it is true that hat is, then it is false that it is yellow.
2.2 Object/attribute relation and conjunction
We examine specific examples first involving and (conjunction), and then observe what the readings imply.

[leftmargin=0.5cm]

: It is true that hat is, and it is true that it is green and brooched.

: for one, it is true that hat is, and it is true that it is green; for one, it is true that hat is, and it is true that it is brooched.

: It is true that hat and shirt are, and it is true that they are yellow.

: for one, it is true that hat is, and it is true that it is yellow; for one, it is true that shirt is, and it is true that it is yellow.
By now it has hopefully become clear that by existential facts as truths I do not mean how many of a given (attributed) object exist: in gradual classical logic, cardinality of objects (Cf. Linear Logic [DBLP:journals/tcs/Girard87]) is not what it must be responsible for, but only the facts themselves of whether any of them exist in a given domain of discourse, which is in line with classical logic.^{8}^{8}8 That proposition A is true and that proposition A is true mean that proposition A is true; the subject of this sentence is equivalent to the object of its. Hence they univocally assume a singular rather than a plural form, as in the examples inscribed so far. The first and the second, and the third and the fourth, then equate.^{9}^{9}9I will also touch upon an alternative interpretation in Section 5, as an advanced material: just as there are many modal logics with a varying degree of strength of modalities, so does it seem that more than one interpretations about in interaction with the other logical connectives can be studied. Nevertheless, it is still important that we analyse them with a sufficient precision. In the third and the fourth where the same attribute is shared among several objects, the attribute of being yellow ascribes to all of them. Therefore those expressions are a true statement only if (1) there is an existential fact that both hat and shirt are and (2) being yellow is true for the existential fact (formed by existence of hat and that of shirt). Another example is in Figure 1.
2.3 Object/attribute relation and disjunction
We look at examples first.

[leftmargin=0.5cm]

: It is true that hat is, and it is true that it is either hatted or brooched.

: At least either that it is true that hat is and it is true that it is hatted, or that it is true that hat is and it is true that it is brooched.

: It is true that at least either hat or shirt is, and it is true that whichever is existing (or both) is (or are) yellow.

: At least either it is true that hat is and it is true that it is yellow, or it is true that shirt is and it is true that it is yellow.
Just as in the previous subsection, here again 1) and 2), and 3) and 4) are equivalent. However, in the cases of 3) and 4), we have that the existential fact of the attribute yellow depends on that of hat or shirt, whichever is existing, or that of both if they both exist.^{10}^{10}10In classical logic, that proposition A or proposition B is true means that at least one of the proposition A or the proposition B is true though both can be true. Same goes here.
2.4 Nestings of object/attribute relations
An expression of the kind is ambiguous. But we begin by listing examples and then move onto analysis of the readings of the nesting of the relations.

[leftmargin=0.5cm]

: It is true that hat is, and it is true that it is brooched. It is true that the object thus described is green.

: It is true that hat is, and it is true that it has the attribute of which it is true that hat is and that it is white. (More simply, it is true that hat is, and it is true that it is whitehatted.)

: Either it is false that hat is, or else it is true that hat is but it is false that it is yellow.^{11}^{11}11 This is the reading of . If it is false that hat is, then it is true that brooched object (which obviously cannot be hat) is. If it is true that hat is but it is false that it is yellow, then it is true that the object thus described is brooched.
Note that to say that Hat Brooch (brooched hat) is being green, we must mean to say that the object to the attribute of being green, i.e. hat, is green. It is on the other hand unclear if green brooched hat should or should not mean that the brooch, an accessory to hat, is also green. But common sense about adjectives dictates that such be simply indeterminate. It is reasonable for (Hat Brooch) Green, while if we have (Hat Large) Green, ordinarily speaking it cannot be the case that the attribute of being large is green. Therefore we enforce that amounts to in which disjunction as usual captures the indeterminacy. No ambiguity is posed in 2), and 3) is understood in the same way as 1).
2.5 Two nullary logical connectives
Now we examine the nullary logical connectives and which denote, in classical logic, the concept of the truth and that of the inconsistency. In gradual classical logic denotes the concept of the presence and denotes that of the absence. Several examples for the readings are;

[leftmargin=0.5cm]

: It is true that yellow object is.

: It is true that hat is, and it is true that it has the following attribute of which it is true that it is yellow object.

: It is true that nothingness is, and it is true that it is yellow.

: It is true that hat is.

: It is true that hat is, and it is true that it has no attributes.

: It is true that nothingness is, and it is true that it has no attributes.
It is illustrated in 1) and 2) how the sense of the ‘truth’ is delimited by the object to which it acts as an attribute. For the rest, however, there is a point which is not so vacuous as not to merit a consideration, and to which I in fact append the following postulate.
Postulate 1
That which does not have any attribute cannot be distinguished from nothingness for which any scenario where it comes with an attribute is inconceivable. Conversely, anything that remains once all the attributes have been removed from a given object is nothingness.
With it, the item 3) which asserts the existence of nothingness is contradictory. The item 4) then behaves as expected in that Hat which is asserted with the presence of attribute(s) is just as generic a term as Hat itself is. The item 5) which asserts the existence of an object with no attributes again contradicts Postulate 1. The item 6) illustrates that any attributed object in some part of which has turned out to be contradictory remains contradictory no matter how it is to be extended: a cannot negate another . Cf. the footnote 2 for the plausibility of the postulate.
3 Mathematical mappings: syntax and semantics
In this section a semantics of gradual classical logic is formalised. We assume in the rest of this document;

[leftmargin=0.3cm]

denotes the set of natural numbers including 0.

and are two binary operators on Boolean arithmetic. The following laws hold; , , , and .

, , , and are metalogical connectives: conjunction, disjunction,^{12}^{12}12 These two symbols are overloaded. Save whether truth values or the ternary values are supplied as arguments, however, the distinction is clear from the context in which they are used. material implication, negation, existential quantification and universal quantification, whose semantics follow those of standard classical logic. We abbreviate by .

Binding strength of logical or metalogical connectives is in the order of decreasing precedence. Those that belong to the same group are assumed having the same precedence.

For any binary connectives , for any and for that are some recognisable entities, is an abbreviation of.

For the unary connective , for some recognisable entity is an abbreviation of . Further, for some and some recognisable entity is an abbreviation of .

For the binary connective , for some three recognisable entities is an abbreviation of .
On this preamble we shall begin.
3.1 Development of semantics
The set of literals in gradual classical logic is denoted by
whose elements are referred to by with or without
a subscript. This set has a countably many
number of literals.
Given a literal ,
its complement is denoted by which is
in . As usual, we have .
The set where
and are the two nullary logical
connectives is denoted by . Its elements
are referred to by with or without
a subscript. Given , its
complement is denoted by which
is in . Here we have
and
.
The set of formulas is denoted by
whose elements, with or without
a sub/superscript,
are finitely constructed from the following grammar;
We now develop semantics. This is done in two parts: we do not outright
jump to the definition
of valuation (which we could, but which we simply do not
choose for succinctness of the
proofs of the main results).
Instead, just as we only need
consider negation normal form in classical logic because every
classical logic formula definable
has a reduction into a normal form, so
shall we first define rules for formula reductions
(for any ):

[leftmargin=0.3cm]

( reduction 1).

( reduction 2).

( reduction 3).

( reduction 4).

( reduction 1).

( reduction 2).

( reduction 3).

( reduction 4).

( reduction 5).
Definition 1 (Domain function/valuation frame)
Let denote the setunion of (A) the set of finite sequences of elements of and (B) a singleton set denoting an empty sequence. We define a domain function . We define a valuation frame as a 2tuple: , where is what we call local interpretation and is what we call gloal interpretation. The following are defined to satisfy for all and for all .
 Regarding domains of discourse


[leftmargin=0.3cm]

For all , is closed under complementation, and has at least and .

 Regarding local interpretations


[leftmargin=0.3cm]

^{13}^{13}13Simply for a presentation purpose, we use a dot such as for to show that is an element of in which is the preceding constituent and the following constituent of . When , we assume that . Same applies in the rest. ( valuation of ).

(That of ).

(That of a literal). 
(That of a complement). 
(Synchronization condition on interpretation; this reflects the dependency of the existential fact of an attribute to the existential fact of objects to which it is an attribute).

 Regarding global interpretations


[leftmargin=0.3cm]

(Noncontradictory valuation). 
(Contradictory valuation).

Note that the global interpretation is completely characterised by the local interpretation. What we will need in the end are global interpretations; local interpretations are for intermediate value calculations for the ease of presentation of the semantics and of proofs of the main results. In the rest, we assume that any literal that appears in a formula is in a domain of discourse.
Definition 2 (Valuation)
Suppose a valuation frame . The following are defined to hold for all and for all :

[leftmargin=0.3cm]

.

.

.
The notions of validity and satisfiability are as usual.
Definition 3 (Validity/Satisfiability)
A formula is said to be satisfiable in a valuation frame iff ; it is said to be valid iff it is satisfiable for all the valuation frames; it is said to be invalid iff for some valuation frame ; it is said to be unsatisfiable iff it is invalid for all the valuation frames.
3.2 Study on the semantics
We have not yet formally verified some important points.
Are there, firstly, any formulas that do not reduce into
some valueassignable formula? Secondly, what if
both
and ,
or both
and
for some under
some ? Thirdly, should it happen
that
for any formula , given a valuation frame?
If the first should hold, the semantics 
the reductions and valuations as were presented in the previous
subsection  would not assign a value (values) to every member
of even with the reduction rules
made available. If the second should hold,
we could gain , which would relegate this gradual logic
to a family of paraconsistent logics  quite out of keeping
with my intention. And the third should never hold, clearly.
Hence it must be shown that these unfavoured situations
do not arise. An outline to the completion of the proofs is;

to establish that every formula has a reduction through and reductions into some formula for which it holds that , to settle down the first inquiry.

to prove that any formula to which a value 0/1 is assignable without the use of the reduction rules satisfies for every valuation frame (a) that and ; and (b) either that or that , to settle down the other inquiries partially.

to prove that the reduction through reductions and reductions on any formula is normal in that, in whatever order those reduction rules are applied to , any in the set of possible formulas it reduces into satisfies for every valuation frame either that , or that , for all such , to conclude.
3.2.1 Every formula is 0/1assignable
We state several definitions for the first objective of ours.
Definition 4 (Chains/Unit chains/Unit chain expansion)
A chain is defined to be any formula such that for . A unit chain is defined to be a chain for which for all . We denote the set of unit chains by . By the head of a chain , we mean ; and by the tail . and by the tail some formula satisfying (1) that is not in the form for some and (2) that for some . By the tail of a chain , we then mean some formula such that for some as the head of . Given any , we say that is expanded in unit chains only if any chain that occurs in is a unit chain.
Definition 5 (Formua length)
Let us define a function as follows.

.

.

.
Then we define the length of to be .
Definition 6 (Maximal number of nesting)
Let us define a function.

.

.

.
Then we define the maximal number of nesting for to be .
We now work on the main results.
Lemma 1 (Linking principle)
Let and be two formulas in unit chain expansion. Then it holds that has a reduction into a formula in unit chain expansion.
Apply reductions 2 and 3 on into a formula in which the only occurrences of the chains are , , …, for some and some . Then apply reductions 4 and 5 to each of those chains into a formula in which the only occurrences of the chains are: , , …, , …, , …, for some and some . To each such chain, apply reduction 1 as long as it is applicable. This process cannot continue infinitely since any formula is finitely constructed and finitely branching by any reduction rule, and since, on the assumptions, we can apply induction on the number of elements of occurring in , . The straightforward inductive proof is left to readers. The result is a formula in unit chain expansion.
Lemma 2 (Reduction without negation)
Any formula in which no occurs reduces into some formula in unit chain expansion.
By induction on the formula length. For inductive cases, consider what actually is:

[leftmargin=0.5cm]

or : Apply induction hypothesis on and .

: Apply induction hypothesis on and to get where and are formulas in unit chain expansion. Then apply Lemma 1.
Lemma 3 (Reduction)
Any formula reduces into some formula in unit chain expansion.
By induction on the maximal number of nesting, and asubinduction on the formula length. We quote Lemma 2 for the base cases. For the inductive cases, assume that the current lemma holds true for all the formulas with of up to . Then we conclude by showing that it still holds true for all the formulas with of . Now, because any formula is finitely constructed, there exist subformulas in which occur no . By Lemma 2, those subformulas have a reduction into a formula in unit chain expansion. Hence it suffices to show that those formulas with already in unit chain expansion reduce into a formula in unit chain expansion, upon which inductive hypothesis applies for a conclusion. Consider what is:

: then apply reduction 1 on to remove the occurrence.

: apply reduction 2. Then apply induction hypothesis on and .

: apply reduction 3. Then apply induction hypothesis on and .

: apply reduction 4. Then apply induction hypothesis on .
Lemma 4
For any in unit chain expansion, there exists such that for any valuation frame.
Since a value 0/1 is assignable to any element of by Definition 2, it is (or they are if more than one in {0, 1}) assignable to . Hence we obtain the desired result for the first objective.
Proposition 1
To any corresponds at least one formula in unit chain expansion into which reduces. It holds for any such that for any valuation frame.
For the next subsection, the following observation about the negation on a unit chain comes in handy. Let us state a procedure.
Definition 7 (Procedure recursiveReduce)
The procedure given below takes as an input a formula in unit chain expansion.^{14}^{14}14Instead of stating in lambda calculus, we aim to be more descriptive in this work for notsotrivial a function or a procedure, using a pseudo program. Description of

[leftmargin=0.5cm]

Replace in with , and with . These two operations are simultaneous.

Replace all the nonchains in simultaneously with .

For every chain in with its head for some and its tail , replace with .

Reduce via reductions in unit chain expansion.
Then we have the following result.
Proposition 2 (Reduction of negated unit chain expansion)
Let be a formula in unit chain expansion. Then reduces via the and reductions into . Moreover is the unique reduction of .
For the uniqueness, observe that only reductions and reduction 5 are used in the reduction of , and that at any point during the reduction, if there occurs a subformula in the form , the subformula cannot be reduced by any reduction rules. Then the proof of the uniqueness is straightforward.
3.2.2 Unit chain expansions form a Boolean algebra
We make use of disjunctive normal form in this subsection for a simplification of proofs.
Definition 8 (Disjunctive/Conjunctive normal form)
A formula is defined to be in disjunctive normal form only if . Dually, a formula is defined to be in conjunctive normal form only if .
Now, for the second objective of ours, we prove that , recursiveReduce, and form a Boolean algebra,^{15}^{15}15http://en.wikipedia.org/wiki/Boolean_algebra for the laws of Boolean algebra. from which follows the required outcome.
Proposition 3 (Annihilation/Identity)
For any formula in unit chain expansion and for any valuation frame, it holds (1) that ; (2) that ; (3) that ; and (4) that .
Lemma 5 (Elementary complementation)
For any for some , if for a given valuation frame it holds that , then it also holds that ; or if it holds that , then it holds that . These two events are mutually exclusive.
For the first one,
implies that
.
So we have;
by the
definition of .
Meanwhile,
.
Therefore for the given valuation frame.
For the second obligation,
implies that
. Again
by the definition of ,
we have the required result. That these two events
are mutually exclusive is trivial.
Proposition 4 (Associativity/Commutativity/Distributivity)
Given any formulas in unit chain expansion and any valuation frame , the following hold:

(associativity 1).

(associativity 2).

(commutativity 1).

(commutativity 2).

(distributivity 1).

(distributivity 2).
Make use of Lemma 5 to note that each for is assigned one and only one value . Straightforward with the observation.
Proposition 5 (Idempotence and Absorption)
Given any formula in unit chain expansion, for any valuation frame it holds that (idempotence); and that (absorption).
Both are assigned one and only one value . Trivial to verify. We now prove the laws involving recursiveReduce.
Lemma 6 (Elementary double negation)
Let denote for some . Then for any valuation frame it holds that .
. Here, assume that the right hand side of the equation which is in conjunctive normal form is ordered, the number of terms, from left to right, strictly increasing from 1 to . Then as the result of a transformation of the conjunctive normal form into disjunctive normal form we will have 1 (the choice from the first conjunctive clause which contains only one term ) 2 (a choice from the second conjunctive clause with 2 terms and ) … (k 1) clauses. But almost all the clauses in will be assigned 0 (trivial; the proof left to readers) so that we gain .
Proposition 6 (Complementation/Double negation)
For any in unit chain expansion and for any valuation frame, we have and that (complementation). Also, for any in unit chain expansion and for any valuation frame we have (double negation).
By Proposition 4,
has a disjunctive normal form:
for some ,
some
and some.
Then we have that;
,
which, if transformed into a disjunctive normal form,
will have [a choice from
]
[a choice from
]
clauses. Now if
, then we already have the required
result. Therefore suppose that .
Then it holds that .
By Lemma 5, this is equivalent to
saying that . But then a clause in disjunctive normal form
of exists,
which is assigned 1.
Dually for .
For ,
by Proposition 4,
has a disjunctive normal form:
for some ,
some and
some . Then;
.
But by Lemma 6
for each appropriate and
. Straightforward.
Theorem 1
Denote by the set of the expressions comprising all for . Then for every valuation frame, it holds that defines a Boolean algebra.
Follows from earlier propositions and lemmas.
3.2.3 Gradual classical logic is neither paraconsistent nor inconsistent
To achieve the last objective we assume several notations.
Definition 9 (Subformula notation)
Given a formula , we denote by the fact that occurs as a subformula in . Here the definition of a subformula of a formula follows that which is found in standard textbooks on mathematical logic [Kleene52]. itself is a subformula of .
Definition 10 (Small step reductions)
By for some formulas and we denote that reduces in one reduction step into . By we denote that the reduction holds explicitly by a reduction rule (which is either of the 7 rules). By
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