Download FORMALIZATION OF A PLAUSIBLE INFERENCE

Bulletin of the Section of Logic
Volume 33/1 (2004), pp. 41–52
Szymon Frankowski
FORMALIZATION OF A PLAUSIBLE INFERENCE
The key intuition behind plausibility in the sense of Ajdukiewicz [1] is
that it admits reasonings wherein the degree of strength of the conclusion
(i.e. the conviction it is true) is smaller then that of the premises. In
consequence in the formal description of plausible inference, the standard
notions of consequence and rule of inference cannot be countenanced as
entirely suitable.
Accordingly, for a formal description of plausible inference, the notions
of rule of inference and consequence adequate for the deductive case are
not fully appropriate.
The main difference between the deductive and the plausible inferences
lies in the fact that the condition:
(*) if X ` α1 , . . . , X ` αn and {α1 , . . . , αn } ` α then X ` α.
holds for the former but is dispensable for the latter. In terms of the operations on the set of formulas, this means that the condition C(C(X)) ⊆
C(X) may not hold for a proper plausible consequence relation. Nothwithstanding, nothing prevents us from accepting other Tarski conditions, the
notion of plausible consequence being a generalization of Tarskian consequence operation.
On the other hand, the notion of p-consequence operation turns out
to be closely related to the so-called quasi-consequence (q-consequence)
operation considered by Malinowski [4]. Single matrices for the operations
of both types contain two sets of designated values: one of possible values
(degrees of truth) for the premisses, the other for the conclusion. In a
matrix determining a p-consequence operation, the set of possible values
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Szymon Frankowski
for a conclusion is included in the set of the values for premisses (contrary
to the q-consequence case); the values from the smaller set represent the
higher degrees of truth, yet the premisses might be justified better than a
conclusion in a weak-deductive reasoning.
1. p-conseqence operation, p-derivability relation
Definition 1. By a p-consequence operation for a sentential language L
we mean any function Z : 2L → 2L that is subject to the conditions (for
all X ⊆ L, α ∈ L):
(i) X ⊆ Z(X);
(ii) Z(X) ⊆ Z(Y ) whenever X ⊆ Y .
Moreover, if a p-consequence operation Z fulfils the condition
S
(iii) Z(X) = {Z(Xf ) : Xf ∈ Fin, Xf ⊆ X}, (Fin is the family of
all finite sets of formulas), then Z will be called finitary.
We can additionaly define the property of structurality for
p-consequence Z: Z is structural iff eZ(X) ⊆ Z(eX) for every endomorphism e ∈ End(L).
Obviously, any consequence operation is a p-consequence.
Definition 2. By a p-inference for the language L we shall understand
any finite sequence (a1 , . . . , an ), n ≥ 1, of ordered pairs from the Cartesian
product L × {∗, 1}. By a p-rule of inference for the language L we mean
an arbitrary nonempty set of p-inferences for L.
For example, the sequence (< p → q, ∗ >, < p, 1 >, < q, ∗ >) is a
p-inference while the set {(< (α → β, x1 >, < α, x2 >, < β, ∗ >) : α, β ∈
L, x1 , x2 ∈ {∗, 1}, x1 = 1 or x2 = 1} is a p-rule of inference.
Definition 3. Given the language L = (L, f1 , . . . , fn ) by a p-proof of a
formula α from a set X of formulas based on a set R of p-rules of inference
is a p-inference (a1 , . . . , ak ), k ≥ 1, for L such that
Formalization of a Plausible Inference
43
(i) pr1 (ak ) = α;
(ii) for all i = 1, 2, . . . , k, either pr1 (ai ) ∈ X and pr2 (ai ) = 1 or there
exists a p-rule r ∈ R and a p-inference (b1 , . . . , bj ) ∈ r such that ai = bj
and {b1 , . . . , bj−1 } ⊆ {a1 , . . . , ai−1 }. (pr1 , pr2 are the first and the second
projections on L × {∗, 1}, respectively).
We say that a formula α is p-derivable from a set of formulae X by
the p-rules from R (X k–R α in symbols) iff there is a p-proof of α from X
on the basis of R. The relation k–R will be called a p-derivability relation
determined by the set of p-rules R. We shall write (a1 , . . . , ak , < βi , xi>m
i=1 ,
c1 , . . . , cn ) instead (a1 , . . . , ak , < β1 , x1 >, . . . , < βm , xm >, c1 , . . . , cn ).
Let (a1 , . . . , an ) be p-inference. Let for each i ∈ {1, . . . , n} : A∗ (i) =
{pr1 (al ) : 1 ≤ l ≤ i & pr2 (al ) = ∗} and A1 (i) = {pr1 (al ) : 1 ≤ l ≤
i & pr2 (al ) = 1}. Now for any p-consequence Z on the language L let us
distinguish the following set R(Z) of p-rules of inference:
r ∈ R(Z) iff for any Y ⊆ L and p-inference (a1 , . . . , an ) ∈ r, the
conditions: A∗ (n − 1) ⊆ Z(Y ), Z(Y ∪ A1 (n − 1)) = Z(Y ) imply that
(pr2 (an ) = ∗ ⇒ pr1 (an ) ∈ Z(Y ))&(pr2 (an ) = 1 ⇒ Z(Y, pr1 (an )) = Z(Y )).
If Z is finitary, it coincides with the p-derivability relation determined
by the set R(Z):
Theorem 1. For any finitary p-consequence Z on the language L, any
X ⊆ L and α ∈ L : α ∈ Z(X) iff X k–R(Z) α.
Proof: (⇒): Assume first that α ∈ Z(X). Thus α ∈ Z(α1 , . . . , αn−1 ),
where {α1 , . . . , αn−1 } ⊆ X for Z being finitary. Consider 1-element pn−1
rule of inference: r0 = {(< αi , 1 >i=1
, < α, ∗ >)}. Then the p-inference
n−1
(< αi , 1 >i=1 , < α, ∗ >) is a p-proof of α from X on the basis of R(Z),
where the last element of the proof is obtained by r0 . Moreover – obviously
r0 ∈ R(Z).
(⇒): Assume that X k–R(Z) α. It follows that there is a p-proof
(a1 , . . . , ak ) of α from X on the basis of R(Z) such that pr1 (ak ) = α. We
have to show that pr1 (ak ) ∈ Z(X). It is sufficient to prove by induction
that for all i ∈ {1, . . . , k}:
(*) A∗ (i) ⊆ Z(X) & Z(X ∪ A1 (i)) = Z(X).
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Szymon Frankowski
First we show that (*) holds for i = 1. Indeed, we have: either (a)
pr1 (a1 ) ∈ X or (b) there exists r ∈ R(Z) such that (a1 ) ∈ r. If (a)
holds then A∗ (1) = ∅ and moreover, Z(X ∪ A1 (1)) = Z(X) for A1 (1) =
{pr1 (a1 )}. If (b) holds then from the definition of the set R(Z) applied for
the set X and p-inference (a1 ), it follows that (pr2 (a1 ) = ∗ ⇒ pr1 (a1 ) ∈
Z(X))&(pr2 (a1 ) = 1 ⇒ Z(X, pr1 (a1 )) = Z(X)). If pr2 (a1 ) = ∗, then
A∗ (1) = {pr1 (a1 )} ⊆ Z(X) and A1 (1) = ∅, and if pr2 (a1 ) = 1, then
A∗ (1) = ∅ and Z(X ∪ A1 (1)) = Z(X) for A1 (1) = {pr1 (a1 )}.
Now assume that (*) holds for i ∈ {1, . . . , k}. To show that it also
holds for i + 1, consider the disjunction: (c) pr1 (ai+1 ) ∈ X or (d) for some
r ∈ R(Z) there is a p-inference (b1 , . . . , bm , ai+1 ) ∈ r such that
(1) {b1 , . . . , bm } ⊆ {a1 , . . . , ai }.
If (c) holds, then A∗ (i + 1) = A∗ (i), therefore A∗ (i + 1) ⊆ Z(X) by
inductive assumption; moreover, A1 (i + 1) = A1 (i) ∪ {pr1 (ai+1 )}, so from
the inductive assumption it follows that Z(X ∪ A1 (i + 1)) = Z(X).
If (d) holds, first let us put for any j ∈ {1, . . . , m} : B∗ (j) = {pr1 (bl ) :
1 ≤ l ≤ j & pr2 (bl ) = ∗} and B1 (j) = {pr1 (bl ) : 1 ≤ l ≤ j & pr2 (bl ) = 1}.
By the definition of the set R(Z) follows that
(2) (B∗ (m) ⊆ Z(X ∪ A1 (i)) & Z(X ∪ A1 (i) ∪ B1 (m)) = Z(X ∪
A1 (i))) ⇒ [(pr2 (ai+1 ) = ∗ ⇒ pr1 (ai+1 ) ∈ Z(X ∪ A1 (i))) & (pr2 (ai+1 ) =
1 ⇒ Z(X ∪ A1 (i), pr1 (ai+1 )) = Z(X ∪ A1 (i)))].
Since, as a result of (1) it follows that B∗ (m) ⊆ A∗ (i) and B1 (m) ⊆
A1 (i), then by the inductive assumption: B∗ (m) ⊆ Z(X) ⊆ Z(X ∪ A1 (i))&
Z(X ∪ A1 (i) ∪ B1 (m)) = Z(X ∪ A1 (i)). Therefore from (2):
(3) pr2 (ai+1 ) = ∗ ⇒ pr1 (ai+1 ) ∈ Z(X ∪ A1 (i)) and
(4) pr2 (ai+1 ) = 1 ⇒ Z(X ∪ A1 (i), pr1 (ai+1 )) = Z(X ∪ A1 (i)).
In case pr2 (ai+1 ) = ∗ : A∗ (i + 1) = A∗ (i) ∪ {pr1 (ai+1 )}. However by
inductive assumption, A∗ (i) ⊆ Z(X), and from (3): pr1 (ai+1 ) ∈ Z(X ∪
A1 (i)) = Z(X), thus A∗ (i + 1) ⊆ Z(X). Moreover, A1 (i + 1) = A1 (i) so
Z(X ∪A1 (i+1)) = Z(X) by the inductive assumption. In case pr2 (ai+1 ) =
1 : A∗ (i + 1) = A∗ (i) ⊆ Z(X), and A1 (i + 1) = A1 (i) ∪ {pr1 (ai+1 )}. So
Z(X ∪ A1 (i + 1)) = Z(X ∪ A1 (i), pr1 (ai+1 )) = Z(X ∪ A1 (i)) = Z(X) due
to (4) and the inductive assumption. 2
Formalization of a Plausible Inference
45
2. p-matrices
Let L = (L, f1 , . . . , fn ) be a sentential language and let M = (M, F1 , . . . ,
Fn , D1 , D∗ ) be a relational stucture such that M = (M, F1 , . . . , Fn ) is an
algebra similar to L and D1 , D∗ are sets such that ∅ 6= D1 ⊆ D∗ ⊆ M .
Call such a structure M a p-matrix for the language L. Let us define for
any p-matrix M the following operation ZM : 2L → 2L :
Definition 4. For every X ⊆ L, α ∈ L :
α ∈ ZM (X) iff ∀h ∈ Hom(L, M)[h(X) ⊆ D1 ⇒ hα ∈ D∗ ].
We can show that ZM is a structural p-consequence operation on the
language L. However, it is easy to show that when D1 6= D∗ , the Tarski
condition put on consequence operation: ZM (ZM (X)) ⊆ ZM (X), does not
hold (cf. for example [1]). For example, consider the language L = (L, →)
with a binary conective “→” and a p-matrix M = ({0, 1/3, 2/3, 1}, →,
{2/3, 1}, {1/3, 2/3, 1}), where for all x, y ∈ {0, 1/3, 2/3, 1} : x → y =
min(1, 1−x+y). It is easy to, check, that the modus ponens rule of inference
is a valid rule of ZM , in the sense that for any α, β ∈ L : β ∈ ZM (α, α → β).
Then s ∈ ZM (ZM (q → r, q, q → (r → s)) and s 6∈ ZM (q → r, q, q → (r →
s)) (where s, q, r ∈ V ar, s 6= q); put for example hs = 0, hq = hr = 2/3.
More generally, for any class M = {Mt : t ∈ T } of p-matrices for the
language L = (L, f1 , . . . , fn ) let us define a structural p-consequence ZM :
2L → 2L determined
T by M as follows: α ∈ ZM (X) iff ∀t ∈ T (α ∈ ZMt (X)),
that is ZM (X) = {ZMt (X) : t ∈ T }.
Definition 5. For a given language L = (L, f1 , . . . , fn ), a p-rule of
inference r is said to be valid for a p-consequence ZM (or simply, is a p-rule
of ZM ), where M = {Mt : t ∈ T } is a class of p-matrices of the form Mt =
(Mt , Ft,1 , . . . , Ft,n , D1t , D∗t ) iff for each t ∈ T and for each homomorphism ht
from L into the algebra of Mt and for any n-tuple (a1 , . . . , an ) ∈ r(n ≥ 1) :
t
ht (pr1 (an )) ∈ Dpr
whenever for each i ∈ {1, 2, . . . , n−1}, ht (pr1 (ai )) ∈
2 (an )
t
t
Dpr2 (ai ) (in case n = 1, ht (pr1 (a1 )) ∈ Dpr
). The set of all valid p-rules
2 (a1 )
for a p-consequence ZM will be denoted by R(M).
In that way one can obtain the following
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Szymon Frankowski
Corollary. For any p-rule r of a p-consequence ZM , and for any sequence (a1 , . . . , an ) ∈ r : pr1 (an ) ∈ ZM (pr1 (a1 ), . . . , pr1 (an−1 ))(n = 1 :
pr1 (a1 ) ∈ ZM (∅)).
The following theorem corresponds to the well-known fact concerning
that ordinary consequence operations namely that a derivability relation
determined by the inferential base composed of all the rules of inference of
a given consequence operation is weaker than that consequence operation.
Whenever, given consequence operation is finitary, it coincides with the
derivability relation.
Theorem 2. For any class of p-matrices M = {(Mt , Ft,1 , . . . , Ft,n , D1t , D∗t ) :
t ∈ T } and any X ⊆ L, α ∈ L : (i) X k–R(M) α ⇒ α ∈ ZM (X). (ii) If ZM
is finitary: X k–R(M) α iff α ∈ ZM (X).
Proof: (i) Assume that X k–R(M) α. Then there exists a p-proof
(a1 , . . . , ak ) of α from the set X on the basis of the set R(M) of all prules of ZM such that pr1 (ak ) = α. To show that pr1 (ak ) ∈ ZM (X) choose
any t ∈ T and suppose that ht (X) ⊆ D1t . It is sufficient to show (by induct
tion) that for each j = 1, . . . , k : (∗)ht (pr1 (aj )) ∈ Dpr
. If j = 1, either
2 (aj )
(a) pr1 (a1 ) ∈ X and pr2 (a1 ) = 1, or (b) for some p-rule r ∈ R(M), (a1 ) ∈ r.
t
If (a) holds from the assumption it follows that ht (pr1 (a1 )) ∈ Dpr
. If
2 (a1 )
(b) holds then (*) is valid since r is a p-rule of ZM .
Now it is enough to show the following condition:
t
for any j ∈ {2, . . . , k} : ht (pr1 (aj )) ∈ Dpr
whenever ht (pr1 (a1 )) ∈
2 (aj )
t
t
Dpr2 (a1 ) , . . . , ht (pr1 (aj−1 )) ∈ Dpr2 (aj−1 ) .
t
t
Suppose that ht (pr1 (a1 )) ∈ Dpr
, . . . , ht (pr1 (aj−1 )) ∈ Dpr
.
2 (a1 )
2 (aj−1 )
Then either pr1 (aj ) ∈ X and pr2 (aj ) = 1, or for some p-rule r ∈ R :
(aj1 , . . . , ajs ) ∈ r and js = j, {j1 , . . . , j(s−1) } ⊆ {1, . . . , j − 1}. Obviously,
t
in both cases: ht (pr1 (aj )) ∈ Dpr
.
2 (aj )
(ii) To prove that ZM is finitary and α ∈ ZM (X). Then for some
finite Xf ⊆ X : α ∈ ZM (Xf ). Let us put Xf = {χ1 , . . . , χn }. Hence
we have for all t ∈ T : ht α ∈ D∗t whenever ht χ1 , . . . , ht χn ∈ D1t for any
homomorphism ht from L into Mt . Thus there exists r ∈ R(M) of the form
{(< χi , 1 >ni=1 , < α, ∗ >)}, and the p-inference (< χi , 1 >ni=1 , < α, ∗ >) is
a p-proof of α from X on the basis of R(M). 2
Formalization of a Plausible Inference
47
In general case we have the following:
Theorem 3. For every structural p-consequence Z there exists a class Π
of p-matrices such that Z = ZΠ .
Proof:
Consider a structural p-consequence Z.
Put Π
{(L, Y, Z(Y )) : Y ⊆ L}. Everyone can show that Z = ZΠ . 2
:=
The representation of any structural p-consequence by a class of matrices as well as Theorems 1, 2 offers opportunity to ask about relation
between two sets of p-rules: R(ZM ) and R(M). A simple example shows
that R(ZM ) 6⊆ R(M). Consider any 1-element class of p-matrices of the
form: M = {({0, 1, 2}, F1 , . . . , Fn , {2}, {0, 1, 2})}. Then, for any X ⊆ L :
ZM (X) = L. So for each formula α, the 1-element p-rule {(< α, 1 >)} is an
element of R(ZM ). However, {(< p, 1 >)} 6∈ R(M), where p is a sentential
variable, since there exists a homomorphism h from the language into the
algebra ({0, 1, 2}, F1 , . . . , Fn ) such that hp 6= 2.
The following theorem says that the converse inclusion holds.
Theorem 4. For any class of p-matrices M = {(Mt , Ft,1 , . . . , Ft,n , D1t , D∗t ) :
t ∈ T } : R(M) ⊆ R(ZM ).
Proof: Assume that r ∈ R(M) and (a1 , . . . , an ) ∈ r. Suppose that for a
set of formulas X:
(1) A∗ (n − 1) ⊆ ZM (X) and
(2) ZM (X ∪ A1 (n − 1)) = ZM (X).
Consequently it follows that
t
(3) ∀t ∈ T ∀ht ∈ Hom(L, Mt )[ht (pr1 (a1 )) ∈ Dpr
&...&
2 (a1 )
t
t
ht (pr1 (an−1 )) ∈ Dpr2 (an−1 ) ⇒ ht (pr1 (an )) ∈ Dpr2 (an ) ].
(i)
Consider:
pr2 (an )
=
1.
To show that
ZM (X, pr1 (an )) = ZM (X). So let β ∈ Z(X, pr1 (an )), that is
(4) ∀t ∈ T ∀ht ∈ Hom(L, Mt )(ht (X, pr1 (an )) ⊆ D1t ⇒ ht β ∈ D∗t ).
To show that β ∈ ZM (X) take advantage of (2) and suppose that for
a t ∈ T and an ht ∈ Hom(L, Mt ) : ht (X ∪ A1 (n − 1)) ⊆ D1t . Then for each
α ∈ A1 (n − 1)) : ht (α) ∈ D1t . Furthermore,
(5) ht (X) ⊆ D1t .
It implies that for each α ∈ A∗ (n − 1) : ht (α) ∈ D∗t , due to (1). In that
t
t
way ht (pr1 (a1 )) ∈ Dpr
& . . . &ht (pr1 (an−1 )) ∈ Dpr
which yields
2 (a1 )
2 (an−1 )
t
t
ht (pr1 (an )) ∈ Dpr2 (an ) , by (3) that is ht (pr1 (an )) ∈ D1 . Hence it follows
that ht β ∈ D∗t .
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Szymon Frankowski
(ii) Now let pr2 (an ) = ∗. Then one can prove that pr1 (an ) ∈ ZM (X)
using (2) and under the assumption that ht (X ∪A1 (n−1)) ⊆ D1t , obtaining
t
– along the lines (1), (5) and (3) that ht (pr1 (an )) ∈ Dpr
. 2
2 (an )
The main difference between the classes R(Z) and R(M ) consists in
the fact that we do not have any procedure to check whether r ∈ R(Z) or
r 6∈ R(Z), and R(M ) exists only for structural p-consequences.
3.
A p-consequence related to n-valued
L
à ukasiewicz logic
Now, let us provide an example of p-derivability relation and p-consequence.
Consider sentencial language L = (L, →, ∨, ∧, ¬). Let R be the set of
following p-rules:
A1 = {(< α, 1 >) : α ∈ Axn },
A2 = {(< [(α →n−2 ¬α) ∧ (¬α → α)] → α, ∗ >) : α ∈ L},
where n > 2 is an odd natural number, Axn is the set of axioms adequate
for n-valued L
à ukasiewicz logic L
à n (cf. [6], n-valued L
à ukasiewicz logic is
axiomatizable for all natural n),
p(M P )1 = {(< α → β, 1 >, < α, 1 >, < β, 1 >) : α, β ∈ L},
p(M P )2 = {(< α → β, 1 >, < α, ∗ >, < β, ∗ >) : α, β ∈ L},
p(M P )3 = {(< α → β, ∗ >, < α, 1 >, < β, ∗ >) : α, β ∈ L}.
We show that R is adequate with respect to following p-matrix:
M∗n = ({0, 1/n − 1, 2/n − 1, . . . , n − 2/n − 1, 1}, →, ∨, ∧, ¬, {1},
{n − 1/2(n − 1), . . . , n − 2/n − 1, 1}),
where Mn = (Mn , {1}), Mn = ({0, 1/n − 1, 2/n − 1, . . . , n − 2/n − 1, 1}, →,
∨, ∧, ¬), is n-valued L
à ukasiewicz matrix: x → y = min(1, 1 − x + y), x ∨ y =
max(x, y), x ∧ y = min(x, y), ¬x = 1 − x, x →k y = min(1, k(1 − x) + y).
Soundness Theorem: For all X ⊆ L, α ∈ L : X k–R α ⇒ α ∈ ZM∗n (X).
Proof: Since for any two sets of p-rules R1 , R2 for a given language:
k–R1 ⊆k–R2 whenever R1 ⊆ R2 , so accordingly Theorem 2, in order to
prove soundness theorem it is sufficient to show that every p-rule from R
is a rule of the p-consequence ZM∗n .
Formalization of a Plausible Inference
49
It is obvious that for any a ∈ A1 : ∀h ∈ Hom(L, Mn ), h(pr1 (a)) = 1,
which implies that A1 is a p-rule of ZM∗n . Now suppose that for some
homomorphism h and a formula α: [(hα →n−2 ¬hα) ∧ (¬hα → hα)] →
hα < 1/2. Then 1/2+hα < min{min[1, (n−2)(1−hα)+1−hα], min[1, 1−
(1 − hα) + hα]}. This yields 1/2 + hα < 1, that is hα < 1/2. However
1/2+hα < min{min[1, (n−1)(1−hα)], min[1, 2hα]}. Since when hα < 1/2
it follows that (n−1)(1−hα) > 1 and 2hα < 1, so 1/2+hα < min[1, 2hα] =
2hα but this implies that 1/2 < hα; a contradiction. In that way, for any
a ∈ A2 : ∀h ∈ Hom(L, Mn ), h(pr1 (a)) ∈ {n − 1/2(n − 1), . . . , n − 2/n − 1, 1}
therefore A2 is a p-rule of ZM∗n . In order to show that for i = 1, 2, 3, p(M P )i
is a p-rule of ZM∗n , one can verify for any h ∈ Hom(L, Mn ) the following 3
cases:
(i) hα = hα → hβ = 1, then obviously hβ = 1
(ii) hα → hβ = 1 and hα ≥ 1/2, then 1 − hα + hβ ≥ 1, so hβ ≥ hα ≥ 1/2
(iii) hα → hβ ≥ 1/2 and hα = 1, then 1 − hα + hβ ≥ 1/2 and consequently
hβ ≥ 1/2. 2
Completeness Theorem:
X k–R α.
For all X ⊆ L, α ∈ L : α ∈ ZMn∗ (X) ⇒
Proof: Let α ∈ ZM∗n (X). Then ∀h(Hom(L, Mn )[h(X) ⊆ {1} ⇒ hα ∈
{n−1/2(n−1), . . . , n−2/n−1, 1}]. Let us put α0 := [(α →n−2 ¬α)∧(¬α →
α)] ∨ α. Then one can obtain that ∀h ∈ Hom(L, Mn )[h(X) ⊆ {1} ⇒ hα0 =
1]. Namely, when h(X) ⊆ {1}, that is hα ≥ 1/2, the following holds true:
(1) h((α →n−2 ¬α) ∧ (¬α → α)) = min(min[1, (n − 1)(1 − hα)],
min[1, 2hα]) = 1 if hα 6= 1, and h((α →n−2 ¬α) ∧ (¬α → α)) = 0 if
hα = 1, which yields h([(α →n−2 ¬α) ∧ (¬α → α)] ∨ α) = 1. Thus
α0 ∈ L
à n (X). So from the completeness theorem for n-valued L
à ukasiewicz
logic it follows that there exists a proof (α1 , . . . , αr , α0 ) of α0 from the
set X on the basis of the rules Axn and modus ponens. So the sequence
(< αi , 1 >ri=1 , < α0 , 1 >) is a p-proof of α0 from the set X on the basis of
the set of p-rules {Axn , p(M P )1 }, i.e., on the basis of the set R.
Now let us consider a formula Ξ := β0 → (α0 → α), where β0 :=
[(α →n−2 ¬α) ∧ (¬α → α)] → α. Then Ξ ∈ L
à n (X). Indeed, suppose that
for some h : L 7→ Mn : h(X) ⊆ {1} and h(Ξ) 6= 1. Then from the earlier
results it follows that hα ≥ 1/2, hα0 = 1. Moreover, from the definition of
Ξ : hβ0 → (hα0 → hα) < 1. Hence, hα0 → hα < hβ0 and consequently,
1 → hα < hβ0 , which means that hα < hβ0 . Thus hα < 1. However, then
50
Szymon Frankowski
according to (1) and the definition of β0 : hβ0 = h((α →n−2 ¬α) ∧ (¬α →
α)) → hα = 1 → hα. In that way, hα < 1 → hα, and since hα < 1 so
1 − 1 + hα < 1 which means that 1 → hα = hα and consequently, hα < hα.
Finally, there exists a p-proof (< βi , 1 >si=1 , , < Ξ, 1 >) of the formula
Ξ from X on the basis of R, where (β1 , . . . , βs , Ξ) is an ordinary proof of
Ξ from X on the basis of Axn and modus ponens. Then the sequence:
(< β1 , 1 >, . . . , < βs , 1 >, < Ξ, 1 >, < β0 , ∗ > (A2 ), < α0 → α, ∗ >
(p(M P )2 ), < α1 , 1 >, . . . , < αr , 1 >, < α0 , 1 >, < α, ∗ > (p(M P )3 )) is a
p-proof of α from X on the basis of the set of p-rules R, which yields
X k–R α. 2
In particular, for n = 3 the set R of the following p-rules (cf. [3,
p. 179]) is adequate for the p-consequence determined by the p-matrix
M∗3 = ({0, 1/2, 1}, →, ∨, ∧, ¬, {1}, {1/2, 1}):
A1 is now the union of the sets:
{(< α → (β → α), 1 >) : α, β ∈ L},
{(< (α → β) → ((β → γ) → (α → γ)), 1 >) : α, β, γ ∈ L},
{(< (¬α → ¬β) → (β → α), 1 >) : α, β ∈ L},
{(< ((α → ¬α) → α) → α, 1 >) : α, β ∈ L},
A2 = {(< (α ↔ ¬α) → α, ∗ >) : α, β ∈ L},
p(MP)1 ∪ p(MP)2 ∪ p(MP)3 .
4. A p-consequence related to m-valued Post
logic
In the last example let us consider the p-inferences of the forms:
(< αi , 1 >ni=1 , < αn , 1 >) or (< α, ∗ >) for some formulae α1 , α2 , . . . , αn , α.
In other words, all p-inferences (a1 , . . . , an ) such that the condition: for
some k ∈ {1, . . . , n}, pr2 (ak ) = ∗ implies that n = 1.
Definition. The following matrix (cf. [5]) is said to be an m-element
Post’s matrix for the sentential language L = (L, ¬, ∨) : Mm
i = (({0, 1, . . . ,
m − 1}, n, a), {m − i, m − i + 1, . . . , m − 1}), where 0 < i < m, and n, a are
the unary and binary operations, respectively, defined as follows:
51
Formalization of a Plausible Inference

 m − 1, if x = 0
n(x) =

a(x, y) = max(x, y).
x − 1,
if x 6= 0,
By an m-element Post’s p-matrix for the sentential language L =
(L, ¬, ∨), we shall understand any structure of the form: Mm
i,r = (({0, 1, . . . ,
m − 1}, n, a), {m − i, m − i + 1, . . . , m − 1}, {m − r, m − r + 1, . . . , m − i,
m − i + 1, . . . , m − 1}), where 1 ≤ i < r < m and Mm
i = (({0, 1, . . . ,
m − 1}, n, a), {m − i, m − i + 1, . . . , m − 1}) is the m-element Post’s matrix.
Theorem 5. For any set of the ordinary rules of inference R adequate
for the consequence operation determined by the matrix Mm
i and for any
r ∈ {i + 1, . . . , m − 1}, there exists a set of p-rules Φr (R) such that for all
X ⊆ L, α ∈ L : X k–Φr (R) α iff α ∈ Z(X), where Z is a p-consequence
determined by the p-matrix Mm
i,r .
Proof: Assume that for the consequence C determined by the matrix
Mm
i , a set of the ordinary rules of inference R is such that
(1) X `R α iff α ∈ C(X), for all X ⊆ L, α ∈ L,
where `R is the ordinary provability relation determined by R. Let us put
Φr (R) := {(< αj , 1 >nj=1 ) : < {α1 , . . . , αn−1 }, αn >∈ ρ for some ρ ∈ R} ∪
Wm−r+1
{(< α ∨ j=m−1 ¬j α, 1 >, < α, ∗ >) : α ∈ L},
(⇒) (soundness): According to Theorem 2 it is sufficient to show that
every p-rule from Φr (R) is a p-rule of the p-consequence Z. The only case
worth checking is that
(2) hα ∈ {m − r, m − r + 1, . . . , m − i, m − i + 1, . . . , m − 1} when
Wm−r+i
h(α ∨ j=m−1 ¬j α) ∈ {m − i, m − i + 1, . . . , m − 1},
for all homomorphisms h from (L, ¬, ∨) into ({0, 1, . . . , m − 1}, n, a). To
this aim first consider the following tableau providing the values for some
superpositions of the operation n:
\x =
n(m−1) (x)
n(m−2) (x)
·
n(m−r+i) (x)
0
1
2
·
r−i
1
2
3
·
r−i+1
·
·
·
·
m−r−2
m−r−1
m−r
·
m−i−2
m−r−1
m−r
m−r+1
·
m−i−1
52
Szymon Frankowski
On the basis of the tableau one can establish that for any homomorphism h from (L, ¬, ∨) into ({0, 1, . . . , m − 1}, n, a) and any formula α the
following condition holds:
Wm−r+i
(3) hα ∈ {m−r, . . . , m−1} iff h(α∨ j=m−1 ¬j α) ∈ {m−i, . . . , m−1}.
Clearly, (2) follows directly from (3).
(⇐) (completeness): Assume that α ∈ Z(X). Consider any homomorphism h such that h(X) ⊆ {m − i, . . . , m − 1}. Then hα ∈ {m − r, . . . , m −
Wm−r+i
1}. So from (3): h(α ∨ j=m−1 ¬j α) ∈ {m − i, . . . , m − 1}. Thus α ∨
Wm−r+i
j=m−1 ¬j α ∈ C(X). In that way on the basis of (1) and the definition
of the set Φr (R) it follows that there exists a p-proof (a1 , . . . , ak ) of α ∨
Wm−r+i
→
j=m−1 ¬j α from X on the basis of Φr (R) such that pr2 ({a1 , . . . , ak }) = 1.
So the sequence (a1 , . . . , ak , ak+1 ), where pr1 (ak+1 ) = α, pr2 (ak+1 ) = ∗ is
a p-proof of the formula α from the set X on the basis of the set of p-rules
Φr (R). 2
References
[1] K. Ajdukiewicz, Pragmatic Logic, Reidel, Dordrecht-Boston
1974.
[2] J. Czelakowski, G. Malinowski, Key notions of Tarski’s methodology of deductive systems, Studia Logica 44 (1985), pp. 321–351.
[3] G. Malinowski, Historical Note, [in:] Selected Papers on
L
à ukasiewicz sentential calculi, WrocÃlaw 1977.
[4] G. Malinowski, Q-consequence operation, Reports on Mathematical Logic 24 (1990), pp. 49-59.
[5] G. Malinowski, Many-valued logics, Oxford Logic Guides 25,
Clarendon Press, Oxford 1993
[6] M. Tokarz, A method of axiomatization of L
à ukasiewicz logics, [in:]
Selected Papers on L
à ukasiewicz sentential calculi, WrocÃlaw 1977.
Department of Logic
L
à ódź University
[email protected]