Articular models for materiality and necessity ∗
November 30, 2009
Abstract
We present a class of models (A-models) in which the sentences of a propositional language are represented as simple hypergraphs on the power set of a
universe, and in which entailments are modeled by relations between hypergraphs. We attempt to build an intuitive understanding of hypergraph from
various familiar semantic idioms as applied to modal logic, in such a way that
it naturally emerges as a unifying descriptive language for those idioms. Finally we extend the A-model theoretic semantic analysis to modal language
and higher-degree system with respect to implicational connective. A brief
discussion was given on the hypergraph representation of logical connectives
as they can be understood as either relations or operators. In this paper we
take necessity in modal language as a unary relation on simple hypergraph
and introduce an implicational connective of material aspects as a binary
function on a pair of simple hypergraphs.
1
Some standard pictures redrawn
Asked what a proposition is, a philosophically-minded logician might answer, ‘a
subset of the universe of a model’, because the meaning of a sentence is given by
distinguishing the set of states in which it is true from the set of states in which it
is false, and in philosophical parlance a proposition is the meaning of a sentence.
It is trite semantics that, when propositions are understood in this way, there might
be more propositions than there are sentences to express them. Given a countable
language, a model on a denumerable universe must have uncountably many propositions. That standard semantic conception of a proposition is one of the pictures
that we will wish to redraw. However, at least to begin with, we use the word
proposition (set of propositions) interchangeably with subset (set of subsets), and
distinguish those propositions or sets of propositions that are expressed in a model
∗
The research presented in this paper is supported by SSHRC research grant # 410-2008-2330.
1
from those that are not. Whereas entailments are taken to be relationships between
(sets of) sentences and sentences with respect to a model, it will be convenient to
speak of propositions and sets of propositions, whether expressed or unexpressed,
as semantically entailing some propositions and not others.
Given this semantic idiom1 , a binary relational frame F = hU, Ri can be described
as a structure in which each object, or ‘point’, in U is assigned a primordially necessary proposition, R(x), which is identical to the set {y | Rxy}. In a model on such
a frame R(x) might not be expressed by any sentence of the language; nevertheless,
the expressed necessities at x in the model are the propositions expressed by sentences of the language and semantically entailed by R(x). The set of necessities at x
in a particular model is the theory expressed by the filter of expressed propositions
semantically entailed by R(x). For reasons that will become clear in the sequel, we
are interested in structures in which all points in the model share a single primordial necessity.
The simplest instance in the literature is provided by the so-called ‘leibnizian’ account of necessity. In this account, every point is, in effect, assigned the same
primordially necessary proposition, namely U. On this scheme, any sentence expressed by U in a model is necessary at every point in that model. Alternatively,
we can understand it as the special case of the relational account in which R is
universal. To this case we now turn.
The first-order condition of universality
(x)(y)Rxy.
(Univ)
is not modally definable. That is, there is no modal formula whose class of frames
is the class of f.o. models for (Univ). To demonstrate this, simply note that modal
validity is preserved by the operation of forming disjoint unions of frames, but
(Univ) is not. (Consider the disjoint union of two singleton reflexive relational
frames.) So although S5 is complete with respect to this class of frames, it does
not correspond to it, that is, Cuniv is not the largest class of frames with respect to
which S5 is sound.
Now if our aim were only to devise semantic idiomata for modal logics in which
all points share some single primordial necessity, either the leibnizian or the neighbourhood or the binary relational idiom would afford us the means. In the first
case, we need only some designated subset of U. In the second case, the class of
1
By semantic idiom we mean a class of structures together with a truth theory.
2
neighbourhood frames would be that in which the neighbourhood function N is
constant and N (x) is a singleton set. The relational idiom affords more freedom of
movement. We might for example consider the class of frames for which R satisfies
pre-universality:
(x)(y)(z)(Rxy → Rzy).
(pUniv)
Again, (pUniv) is not preserved by the formation of disjoint unions of frames, (As
for universality, consider the disjoint union of two singleton reflexive relational
frames.) Such a frame is not a frame for S5, though it validates
♦p → ♦p
(5)
(p → p)
(T [ )
(p → ♦p)
(B[ )
p ∧ q → (p ∧ q)
(K)
and the rules of monotonicity and of normality
∀α, ∀β, ` α → β ⇒ ` α → β
(RM)
` α ⇒ ` α
(RN)
preserve validity.
We call the system S5[ , that adjoins to a complete axiomatisation of PL the principles (K), (T [ ), (B[ ), (5), (RM), and (RN).
If R is non-empty, then it is serial, and if R is symmetric, then the Brouwersche
formula
p → ♦p
(B)
is valid. Thus, if R is non-empty and symmetric, then R is universal, and S5 is
restored.
But the bare requirement for a common primordial necessity (CPN) is only that R
satisfies (pUniv). That (pUniv) is not a modally definable restriction on R suggests
that such frames constitute a distinct semantic idiom. A CPN frame is a pair, F =
hU, Pi where U , ∅ and P ⊆ U. A CPN model is a pair, M = hF , Vi, where V is
3
a valuation.
The satisfaction condition for α in a CPN model is
M |= α ⇔ P ⊆ ~αM ; else M |=
/ α
(CPN)
S5[ is determined by the class of all CPN frames.
Proof. Soundness is straightforward. For completeness, let US5[ be the set of all
maximal S5[ -consistent sets, and define PS5[ as the set of maximal S5[ -consistent
sets, Σ such that for all α ∈ Φ, α → α ∈ Σ.
1.1
Enter hypergraphs
Mention of three further semantic idioms and their interconnections will round out
our introductory illustration and provide a bridge to our main exposition. These are
idioms which provide primordial sets of propositions. The most primitive of these
is the neighbourhood idiom. The function, N : U → ℘℘(U) of a neighbourhood
frame (N -frame), F = hU, N i on this sort of understanding, merely assigns to
a point x a set of primordially necessary propositions called neighbourhoods. In a
model, those sentences of the language are necessary at x that express one of x’s
primordial necessities. We can understand the universality of R as the special case
of the neighbourhood account in which N is constant, and assigns to each point
the singleton collection, {U}. In the more general case, a constant N assigning
a non-empty set of propositions merely guarantees that propositions expressed by
modal sentences are either universal or empty. If the set of neighbourhoods is a
filter, then the set of sentences expressing neighbourhoods is a classical theory,
that is, a deductively closed set. The system K, (the principles of which are [K],
[RM], and [RN]) is determined by the class of all powerset filters. What places this
idiom in the second of our groups rather than the first is that N (x) can be naturally
understood as a hypergraph on ℘(U).
Definition 1. Let S be a set. H is a hypergraph on S if and only if H ⊆ ℘(S ). The
elements of H are called hyperedges; the elements of which are called vertices.
Definition 2. A simple hypergraph H = {E1 , E2 , . . . , En } is a hypergraph such
that if ∀Ei , E j ∈ H, Ei ⊂
/ E j.
The class H(U) of hypergraphs on ℘(U) do not in general satisfy the simplicity
requirement. However, although there may well be logically interesting systems
that correspond to well-known mathematical properties of hypergraphs, they do
4
not readily present themselves. We leave the reader to reflect upon the system of
modal logic that corresponds to the class of N -frames in which N (x) is a simple
hypergraph, that is, a hypergraph H satisfying
∀e, e0 ∈ H, e ⊂
/ e0 .
The second idiom of this group is provided by n-ary relational semantics. In an
n-ary relational frame, as in [4], the n-ary relation, R induces a function fR that
assigns to every point, x, a hypergraph H(x) on U. The truth-condition
∀x,
M
x
α ⇔ ∀y1 , . . . , y(n−1) , Rxy1 . . . y(n−1) ⇒ ∃n−1
i=1 i :
M
yi
α.
is equivalently stated as
∀x,
M
x
α ⇔ τ(H(x)) [(n−1) α.
(Σ [(n−1) α ⇔ ∀π ∈ Π(n−1) , ∃c ∈ π : c |= α.) The set τ(H(x)) of propositions is
(n − 1)-coherent where
n = min ∈ {1 ≤ i ≤ ω | ∃π ∈ Π(i−1) (τ(H(x))) : ∀c ∈ π, ∩c , ∅}
i
In consequence, the set (x)M of propositions expressed by necessities at the point
x in a model M is (n − 1)-coherent. Since it is also closed under [RM] and [RN],
(x)M is referred to as an (n − 1)-filter (expressed by an (n − 1)-theory).
In 1975, Schotch proposed a generalization of the neighbourhood idiom in which
the truth-condition for a modal formula, α in a model, M was weakened to require (in the language of this paper) only that one of the propositions assigned to a
point x in the underlying frame semantically entail the proposition expressed by α
in the model. In effect, this would require that for some edge, e of H(x), e ⊆ ~α.
The logic corresponding to this new “environ” idiom is the closure of PL under the
rule [RM]. Jennings proposed a further idiomatic restriction by which, in effect,
H(x) is always simple. The simplicity restriction not being modally definable, the
resulting distinct idiom was called “locale” semantics. A locale frame (L -frame),
F is a pair hU, L i in which L is a function assigning to each point, x, a family
of sets L (x), the set of x-locales. It is easy to see that if for every x, L (x) , ∅,
then [RN] preserves validity, and that if for every x, L (x) is a singleton, then [K]
is valid. A moment’s reflection reveals that a L -model M = hU, L , Vi satisfying
the condition that for every x, |L (x)| = n is equivalent to a n-ary relational model
M = hU, R1 , . . . , Rn , Vi where Ri xy iff y ∈ ai where ai ∈ L (x). The interpretation
of α is the truth-condition
M
x
α ⇔ ∃ni=1 i : ∀y, Ri xy ⇒
5
M
y
α.
The class of all n-ary relational frames determines the system K n which has [RM],
[RN] and all instances of the schema
α1 ∧ . . . ∧ αn →
W
((α j ∧ αk )(1 ≤ j , k ≤ n)).
(K n )
It should be equally evident that if, for every x, L (x) is n−coherent, that τ(L(x)),
possibly with some diagonalisation, will yield a set of n-tuples. Formally, let
a1 , . . . , ai . . . be an enumeration of L (x). Then hx, y1 , ...yn i is in R iff ∀ai ∈ L (x),
∃ni=1 j : y j ∈ ai . Thus a locale frame satisfying the n−coherence restriction will
generate an equivalent n + 1-ary relational frame. The class of all n−ary relational
frames determines the system Kn which has [RM], [RN] and all instances of the
schema
W
α1 ∧ ... ∧ αn → ( (α j ∧ αk )(1 ≤ j , k ≤ n)).
2
(Kn )
Propositions as Hypergraphs
In the previous examples, the language of hypergraphs has contributed little more
than a convenient formal descriptive language for a few semantic idioms, as the
language of propositions has provided a convenient, and possibly suggestive reading for others. In [1] and [2], hypergraphs played a more central role: in those
papers powerset hypergraphs are the propositions. Semantics for a number of firstdegree propositional systems are discussed in [2]. In what follows we extend those
discussions to first-degree modal systems. The basic definitions are as follows.
2.1
Syntax
The language L of the paraconsistent modal logics is
α ::= p|¬α|(α ∨ α)|α
2.2
Hypergraph Semantics
A hypergraph model (A-model) is a triple M = hS, H, H i where
1. S , ∅ is a set;
2. H ⊆ ℘℘℘(S );
3. H : At → {H | H ∈ H & H is a simple hypergraph}.
6
That is, to each pi , H assigns a simple hypergraph H (pi ) on ℘(S), we write as H i .
The account of H , which extends H to Φ, requires some preliminary definitions.
Definition 3. If A ⊆ ℘(S ), then b is an intersector of A iff ∀a ∈ A, b ∩ a , φ.
Definition 4. If A ⊆ ℘(S ), then τ(A) = {b | b is a minimal intersector of A}.
Definition 5. If A ⊆ ℘(S ), then [A] = {a | a ∈ A}.
Simplification Not all of the set-theoretic operations introduced below preserve
simplicity. However, for present purposes, every non-simple (‘uncluttered’) hypergraph H has an equivalent simple hypergraph, ?H, which is obtained by casting
out superedges. Our attention is focused upon simple hypergraphs.
Definition 6. ∀H ∈ H, ?H is H − {E ∈ H | ∃E 0 ∈ H : E 0 ⊂ E}.
Definition 7. H t H 0 = ?{{a ∪ b} | a ∈ H, b ∈ H 0 }
H u H 0 = ?{a | a ∈ H or a ∈ H 0 }
H = H{[Bi ] | Bi ∈ τ(H)}.
H extends H to Φ as follows:
H i = H (pi )
H¬α = H
Hα∨β = Hα t Hβ
Hα∧β = Hα u Hβ .
The distinct systems studied in [2] are presented as binary systems, that is, as codifications of distinct kinds of provability. There are good methodological reasons
for representing such systems this way rather than as the first-degree fragments of
system possessed of a distinctive conditional connective: principally, they codify
relations between propositions construed as hypergraphs rather than as functions
mapping pairs of hypergraphs to hypergraphs. To be sure, first-degree classical
propositional logic is representable as a codification of the subset relation between
propositions construed as sets. But it can also be represented as a function mapping
pairs of sets to sets, namely the function f⊆
f⊆ (a, b) = {x | {x} ∩ a ⊆ b}.
Accordingly we can define as a satisfaction condition for α → β at x, the condition {x} ⊆ f⊆ (~α, ~β). In classical semantics, what can be expressed in the
language of points and membership can be expressed in the language of singleton
propositions and inclusion. In a semantic idiom such as the one we are introducing,
7
one that takes propositions to be hypergraphs, we could forsake singleton propositions {x} for unit hypergraphs {{x}}, and the inclusion relation ‘⊆’ would then be
replaced by a subsumption relation ‘v’. However, corresponding methodological
reasons urge circumspection.
2.3
First-degree propositional systems
In [2] distinct entailment relations were associated with distinct hypergraph-theoretic
relations between propositions represented as hypergraphs. The systems based
upon the resulting hypergraph-structures along with their defining hypergraph relations are presented in table 1 and table 2. Table 2 permits comparisons between
codifications of these and binary versions of other well-known systems.
Natural Deductive Rules
PL
FDE
FDAE
FDPE
LP
AI
E
p a` ¬¬p
p ∧ (a ∨ r) a` (p ∧ r) ∨ (q ∧ r)
¬(p ∨ q) a` ¬p ∧ ¬q
p` p∨q
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
¬¬p ` p
X
X
X
p∧q` p
X
X
p∧q` p∨q
X
X
X
p ∨ (q ∧ ¬q) ` p
[MP]
p ` q ⇒ ¬q ` ¬p
[Mon]
[Re f ]
[T rans]
X
X
X
X
X
X
×
X
X
X
X
X
×
X
X
X
X
X
X
X
X
X
p∧q` p∨q
p ` p ∨ ¬p
×
X
X
X
X
X
×
×
×
X
X
×
X
X
X
X
X
X
×
X
X
X
X
X
Table 1: Syntax table
Systems
FDE
FDAE
FDPE
Entailment relation
∀e0 ∈ H 0 , ∃e ∈ H such that e ⊆ e0
FDE + (V(H 0 ) ≤ V(H))
FDE + (∀e0 ∈ V(H 0 ), i f e0 ∈ |x|, then ∃e ∈ V(H) such that e ∈ |x|)
(|x| is the pair {x, x})
Table 2: Semantics table
8
2.4
A-frames
The systems studied in [1] include binary versions of familiar systems such as
FDE (first-degree entailment) and some hitherto unstudied variant systems corresponding to altered accounts of subsumption. We illustrate the general character
of hypergraph frames by reference to the FDE definition of subsumption. But the
illustration is just that. Other modal systems will be obtained by varying the subsumption relation. In keeping with the usual way in which modal logics are studied
classically, in all of the systems we consider here, the definition of semantic entailment and the satisfaction-conditions for necessity invoke the same subsumption
relation. Obviously hybrid systems result from adopting distinct subsumption relations for propositional systems and for their modal extensions.
Definition 8. ∀H, H 0 ∈ H, H v H 0 , (H is subsumed by H 0 ) iff ∀b ∈ H 0 , ∃a ∈ H
such that a ⊆ b.
Definition 9. ∀α, β, α |= β (α semantically entails β ) iff ∀M = hU, H i, Hα v Hβ .
Alternatively, we say that α ` β is valid. So, mutatis mutandis, for Γ |= α.
It will be useful to repeat from [2] a remark about this semantic idiom: H in the
class of articular models introduced above is a set of hypergraphs on the subset
algebra b(S ) of a set S. However, the semantic analysis of FDE requires no such
particularity. The locus of generalisation is in the definition of subsumption, which
we here rehearse.
∀H, H 0 ∈ H, H v H 0 , (H is subsumed by H 0 ) iff ∀b ∈ H 0 , ∃a ∈ H such that a ⊆ b.
Since the hypergraphs of H are powerset hypergraphs, the edges of these hypergraphs are themselves hypergraphs on the underlying set. Thus the subset relation
holding between an edge E of the subsumed and E 0 the subsuming hypergraphs is
actually a subgraph relation requiring the following:
∀v ∈ E, ∃v0 ∈ E 0 : v = v0 .
The generalisation replaces the identity with an inequality:
∀v ∈ E, ∃v0 ∈ E 0 : v ≤ v0
thereby generalizing the relation between edges:
Definition 10. E E 0 i f f ∀v ∈ E, ∃v0 ∈ E 0 : v ≤ v0 ,
which yields a new, generalised relation of subsumption*:
9
Definition 11. ∀H, H 0 ∈ H, H v∗H 0 iff ∀B ∈ H 0 , ∃A ∈ H such that ∀a ∈ A, ∃b ∈ B
and a b.
Determination. FDE is determined with respect to the class of articular models
where the entailment relation is subsumption*.
Definition 12. An articular frame (A − f rame) is a pair, F = hU, Hi where H is a
designated hypergraph on ℘(U). A model on F adds a function H , defined as for
hypergraph models and extended to modally augmented Φ by the clause:
M |= α ⇐ H v H (α); else M |=
/ α.
(FDE)
As we have said, the v here is that of FDE. It is notable that if the v of PL is
adopted, then not only does the resulting underlying system revert to PL, but the
resulting class of frames is just the class of locale frames satisfying ∀x, L(x) , ∅,
modulo the altered representation of propositional wffs. The system FDE is the
system that adjoins to the codification of FDE, the rule of aggregation
∀α, ∀β, α ∧ β ` (α ∧ β)
(Ke)
and rules of monotonicity and of normality
3
3.1
∀α, ∀β, α ` β ⇒ α ` β
(RMe)
∀α, ` α ⇒ ` α
(RNe)
Materiality and Necessity
Materiality
FDE bears a family resemblance with the first-degree fragment of NR, one of the
modal logic systems built upon the relevance logic system R [6]. While the morphological similarity gives us motivation to compare the two systems, the comparison of these higher-degree systems with classical logic mentioned in section 2.2
reveals some common aspects of higher-degree systems worth noting. But first for
the comparison.
It is straightforward to see, that presented in the axiomatic manner, the first-degree
modal fragment of NR is the same as FDE where entailment is a relation between
two semantic representations. However, in R, → is an operator, as is ∧ and ∨. So
10
if we were to add a connective → to the language of FDE so as to facilitate further
comparison, we must interpret the → so that α → β can be given a hypergraph
representation.
Suppose we interpret → as the following binary function taking hypergraphs as
inputs and generating a hypergraph as output,
Hα→β = H¬α t Hβ
then we have a higher-degree system, FDE # , built upon FDE, where the following
binary transforms of R-axioms in the axiomatization of the implicational fragment
by Alonzo Church [3] are valid.
WI 1 p ` p;
WI 2 p → (p → q) ` p → q;
WI 3 p → (q → r) ` q → (p → r);
WI 4 (p → q) ` (r → p) → (r → q).
It is not difficult to detect the material nature of the → given, along with which
comes the following CL-theorems2 :
p`q→p
(I)
¬p ` p → q
(II)
Both R and I, II are theorems of FDE # . Now it may be tempting to think
that we are taking a detour via FDE # to classical logic, but a moment’s reflection
reveals that the following two formulae are disqualified to be theorems of FDE,
and thereby that of FDE # :
p ∧ ¬p ` q
(III)
p ` q ∨ ¬q
(IV)
Note that in FDE [1], we did not define the verum (>) or the falsum (⊥), for it is a
natural consequence of hypergraph semantics that there are only contingent formulae, in the sense that no formula entails or is entailed by every formula. Therefore,
in the axiomatic presentation of FDE # , the axioms must be binary as in [1].
2
CL refers to classical logic.
11
3.2
Necessity
Natural language presents us with a choice among at least three alternative semantic parsings of claims of necessity
(1) (Necessarily α) is true;
(2) α is (necessarily true);
(3) Necessarily (α is true).
(1) is a metalinguistic claim about the sentence ‘necessarily α’ in the object language; (2) is a metalinguistic claim about the object language sentence ‘α’; (3) is
a metalinguistic claim about a metalinguistic claim. The usual treatment of necesM
sity adopts the first parsing as in x α; in a first degree system that adopts ‘|=’,
it seems natural to adopt the third parsing, and to this end we introduce the notation ‘|A’ to mark the necessity of the truth of α in a model. Adopting this parsing
creates special considerations, in particular, it creates difficulty for the treatment of
nested modalities in higher degree modal systems. However, in this paper we are
concerned with first degree modal systems.
In the last section, we distinguished entailment from implication. In our axiomatic
presentation of FDE # , we use ‘ `0 and ‘ →0 to mark the difference. Given the
completeness proof, in the context of hypergraph semantics, the former is a relation between two hypergraphs (the hypergraph that represents a set of formulae is
the meet of the hypergraphs that represent its members), and the latter is a binary
function that takes two hypergraphs as inputs and generates a hypergraph as an
output. Entailment as a relation between a set of formulae Σ and a single formula α
is a property of the pair hΣ, αi. Unlike logical operators under which a certain semantic unit is closed, be it sets or hypergraphs, reiteration of relations can only be
considered as higher-order relations. Therefore, we speak of higher-degree system
only in the sense of reiterating operators, such as ¬, ∧ and →, whereas entailment
is construed as a property for pairs that consist of sets of formulae and single formulas. Following this line of thought, necessity, taken as a property of a formula,
can be interpreted thus in a class of articular models where
Definition 13. |A α ⇔ ∀e ∈ Hα , ∃v ∈ e such that ∃v0 ∈ e : v0 = v.
Such a property dictates in an articular model what the hypergraph of α should be
like in order to justify α to be necessary. It is not an interpretation for ‘necessarily
α’ which would answer for the symbol α. The difference between property and
function can be illustrated by a remark made by Kneale & Kneale in [5] on material
and strict implication:
12
When Whitehead and Russell spoke of P ⊃ Q as a statement of material implication, they seemed to have confused together two different
questions, namely (i) ‘What justifies inference from the proposition
that-P to the proposition that-Q?’ and (ii) What is the weakest additional premiss which in conjunction with the premiss that-P suffices
for inference to the conclusion that-Q?’. . . P ⊃ Q, this formula is constructed by use of the propositional signs P and Q, but it is not about
the propositions which they express.. . . On the other hand, Lewis’s formula P J Q is supposed to be about the propositions expressed by P
and by Q precisely because it is conceived as a proper answer to the
first of the questions noticed above.
Just as Lewis’ ‘ J0 , our notion of necessity here is also conceived to answer the
first, instead of the second, question.
The system FDEM consists of the axioms and rules of FDE in addition to
|A α, |A β
|A (α ∧ β)
(K ∧ )
|A α, α ` β
|A β
(RM)
α`β
|A (α → β)
(RN)
and
Again we prove the completeness of FDEM by the Henkin method.
The burden of proof for the fundamental theorem lies in the generalization of H ∗
to Φ; that for the non-modalized formulae has already been done in [1], what is left
to prove is the following:
Lemma. ∀α ∈ Φ, M kA α ⇒ |A α.
Proof. Since |A α, according to the definition of H ∗ ,
Hα∗ = {{|δ1i |, |δ2i |, . . . , |δni |} | ∀i, |δi | ∈ A (α) & ∃i, j : δi = ¬δ j }
To put it in more concrete terms, Hα∗ is of the following form
Hα∗ = {{. . . , |p1 |, |¬p1 |, . . .}, . . . , {. . . , |pn |, |¬pn |, . . .}}
By definition 13, ∃α0 ∈ Φ such that
13
Hα∗0 = {{|p1 |, |¬p1 |}, . . . , {|pn |, |¬pn |}}
By definition 13 and metatheorem 2, we have
α0 a` (¬p1 ∨ p1 ) ∧ . . . ∧ (¬pn ∨ pn )
which is equivalent with
α0 a` (p1 → p1 ) ∧ . . . ∧ (pn → pn )
Since
p`p
By RN we have
|A (p → p)
By K ∧ and rules of FDE, we have
|A ((p1 → p1 ) ∧ . . . ∧ (pn → pn ))
which is simply
|A α0
But
α0 ` α
By RM, we have
|A α
The rest is standard work. Therefore,
Completeness. α β ⇒ α ` β
FDEM is complete with respect to the class of modal articulate models.
14
References
[1] R. E. Jennings & Y. Chen. Articular models for first degree entailment. forthcoming, 2009.
[2] R. E. Jennings & Y. Chen. Articular models for first degree paraconsistent
systems. forthcoming, 2009.
[3] A. Church. The weak theory of implication. Kontrolliertes Denken, Fest. gabe
zum 60 Geburtstag von Prof. W. Britzelmayr, 1951.
[4] P. K. Schotch & R. E. Jennings. Inference and necessity. Journal of Philosophical Logic, 9, 1980.
[5] W. Kneale & M. Kneale. The Development of Logic. Oxford: Clarendon Press,
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[6] E. Mares.
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