Download 1. Positive implicational Hilbert`s logic H→ (see [9], ch.3)

Alexander S. Karpenko
LATTICES OF IMPLICATIONAL LOGICS
Recently there has been published several articles that establish an interrelationship among some propositional logics (see [7], [23] and especially
[16]). But it is worth emphasizing that already in 1972 the problem of classification of logical systems was considered by V.A. Smirnov in his book
[19].
The main goal of this paper is to analyze interrelationships between the
most interesting implicational logics. It is a very natural task because many
logical systems are distinguished only by their implicational fragments.
1. Positive implicational Hilbert’s logic H→
(see [9], ch.3)
We must give an accurate definition of implicational logic which is taken
from the paper by Y. Komori [10].
By an implicational formula (C formula) we mean a propositional formula which contains no connective other than →.
Definition. A set of C formulas is an implicational logic L if it satisfies
the following two conditions:
1) L is closed with respect to modus ponens, that is, p, p ⊃ q ∈ L implies
q ∈ L,
2) L is closed with respect to substitution for propositional variables by
C formulas.
An implicational logic L is positive implicational logic H→ (implicational
fragment of intuitionistic propositional logic [20]) if L contains the following
five formulas:
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I.
p→p
B. (q → r) → ((p → q) → (p → r))
C.
(p → (q → r)) → (q → (p → r))
W. (p → (p → q)) → (p → q)
K1 . (p → q) → (r → (p → q)),
where K1 is a substitutional instance of axiom K: p → (q → p).
Proposition 1.1. {I, B, C, W, K1 } is an independent axiomatization of
H→ .
Let us denote implicational logics by names of axioms which they comprise.. For instance, H→ is IBCW K1 .
Proposition 1.2. IBCW K1 ≡ BCW K ≡ B 0 W K ≡ SK, where
B 0 is (p → q) → ((q → r) → (p → r)),
S is (p → (q → r)) → ((p → q) → (p → r)).
Note. Axioms denoted as I, B, C, W, K, S correspond to primitive combinators I, B, C, W, K, S. These combinators are simple operators which
reorder brackets, cancel and/or duplicate terms they are applied to. For
instance, Ix = x, Bxyz = x(yz), for arbitrary terms x, y, z. Further combinators are formed from primitive ones, for example, B 0 xyz = x(zy) is
CB. Combinatorial completeness is proved for combinators (B, C, W, K),
(B 0 , W, K), (S, K). Due to the fact that every axiom of H→ corresponds to
the type–scheme of a combinator ([6], ch. 9.E), we can classify combinators
through logics and vice–versa [8].
2. Boolean lattice of sublogics of H→
Let us consider a set of all subsets of {I, B, C, W, K1 }. Since IBCW K1 is
an independent axiomatization of H→ (Proposition 1.1.), we have 32(= 25 )
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implicational logics which are sublogics of H→ . It is well–known that the
set of all subsets of some set forms Booelan lattice, i.e. distributive lattice
with complementation.
For simplicity let us take logic IB as the null of the lattice (of course,
the unit is logic H→ ). Thus, we have
H→
◦
¡@
¡
@
@
¡
@
¡
◦¡
R→ @
@
◦
¡@
@
¡
BCK
¡
BCI ◦@
@◦¡
@¡
¡@
@
@◦
¡ IBW K1
¡
@¡
¡@
@◦
IBW ¡
IBK1
¡
@
¡
@
¡
@◦¡
IB
fig. 1
In [5] H. Curry formulated the deduction theorem for logic IB. Logics
BCI and BCK (= IBCK1 ) were introduced in 1956 by C.A. Meredith
(see [13]). Recently a logic BCI attracts special attention since it is an
implicational fragment of Linear logic of Girard (see [3]). Logic R→ , i.e.
IBCW , is a weak positive implicational logic which was introduced in 1951
by A. Church. R→ is an implicational fragment of relevant logic R (see [1]).
3. The problem of extending H→ to T V→
Following Anderson and Belnap ([1], p.4), T V→ denotes implicational fragment of classical propositional logic. Pioneer investigations of the system with the classical implication as the only connective belong to A.
Tarski. Later on the so called Tarski–Bernays theorem (see [11], p.145)
became well–known which claims that an axiomatization of T V→ are formulas B 0 , K, P where P is Pearce’s law: (((p → q) → p) → p).
However, given the way I state the problem of extending H→ to T V→ ,
namely: is there formula X such that {I, B, C, W, K1 , X} is an independent
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axiomatization of T V→ ? It is evident that the formula P is absolutely
unsuitable for our purposes. A. Prior has shown ([18], p.318) that IB 0 CP ≡
B 0 KP . Since B 0 C ≡ BC, we have IBCP ≡ B 0 KP . Let’s note that
this result is already implicitly contained in Wajsberg’s work [21] (see also
[22]) where system B 0 KP is thouroughly investigated and a whole series of
formulas is proposed which equivalently can replace the axiom K. In [21]
Wajsberg gives the scheme of completeness proof for B 0 KP .
We can consider weaker formula than P , namely, Commutability of
Implication which we denote by D : ((p → q) → q) → ((q → p) → p).
However, even this formula is not a suitable candidate for X, since we have
Proposition 3.1. W, D ` P .
Then it follows, due to Prior’s result, that both H→ + D and R→ + D
are (non–independent) axiomatizations of T V→ . Moreover,
Proposition 3.2. {I, B, W, K1 , D} is axiomatization of T V→ .
Furthemore,
Proposition 3.3. IBCK1 D ≡ IBCD ≡ BCKD ≡ B 0 KD ≡ BKD.
Let us note that an addition of Linearity Axiom L : ((p → q) → (q →
p)) → (q → p) to B 0 KD gives an axiomatization of implicational fragment
of L
à ukasiewicz’s infinite–valued logic [14].
The question arises: Can we get the desired formula X by a weakening
of formula D? Under a weakening of a formula a we mean a substitution
into a in most general form and without identification of variables. For
example, formula D1 : (((p → q) → r) → r) → ((r → (p → q)) → (p → q))
is an instance of such a substitution into D, and K1 is such instance for
K. Using terminology of Pahi’s work [17] where the interesting results
of interrelationship of implicational logics are presented, formula D is a
“restricted generalization” (r.g.) of D1 . The following result ([17], p.167)
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takes place: R→ + a is equivalent to R→ + a∗ where a is implicational r.g.
of a∗ . Thus, in this way we cannot separate logics R→ + D∗ and H→ + D∗ .
Indeed, we have
Proposition 3.4. R→ + D1 ≡ T V→ .
But really there exists a formula (I denote it U ) which separates R→
from H→ in the correct sense, i.e.
R→ + U 6≡ T V→ ,
but
H→ + U ≡ T V→
where U is ((((p → q) → q) → p) → r) → (((((q → p) → p) → q) →
r) → r). Formula U appeared in [15] where the problem of independent
axiomatization of RM→ (implicational fragment of propositional logic RM
[1]) is solved. From this result it follows that R→ +U 6≡ T V→ . In [2] another
axiomatization of RM→ is given where (6) is one of the axioms:
(6).
((q → q) → (p → p)) → (((p → q) → p) → p).
It is obvious that under axioms I and K1 we get
Proposition 3.5. I, K1 , (6) ` (((p → q) → (p) → p).
Thus, H→ + (6) ≡ T V→ , i.e. H→ + U ≡ T V→ . But unfortunately we
have
Proposition 3.6. IBW K1 + (6) ≡ IBCK1 + (6) ≡ T V→ .
Finally, interrelationship of the most well–known and interesting implicational logics can be represented by properties of the following lattice
which is not even a semimodular, and R→ has not complementation.
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T V→
◦
¡@
¡ @
@◦ RM
→
@
¡@
@
¡
@
¡
@◦¡
@◦ BCIU
BCK ◦¡
R→
@
¡
@
¡
@
¡
@
¡
@
¡
@◦¡
H→ ¡◦¡
BCI
fig. 2
A lattice with such properties can be found already in G. Birkhoff ([4], ch.7),
but on a quite different occasion. It remains to add that an investigation
of algebraic properties of the above lattice, denoted by Q∗ , is given in [12].
4. N –dimensional interpretation of implicational logics1
Main idea of interpretation is that in n–dimensional space every coordinate
axis corresponds to one axiom. Let’s consider the logic H→ .
Let’s choose in the space some point O and draw five mutually perpendicular straight lines through it. The number 5 is determined by the
fact that H→ can be axiomatized with the help of five independent axioms
(Proposition 1.1). However for simplicity, as point O, let us take an implicational logic IB. It enables us to move to the three–dimensional space,
with coordinate axes x, y, z.. On every of these coordinate axes let’s put a
unit vector which we denote by axioms C, W and K1 respectively. On the
ends of vectors, one–dimensional implicational logics BCI, IBW , IBK1
are situated. To compare with the lattice on the fig.1, the following the
coordinate axes will be chosen:
1
The author has reported on this theme for the first time at 9th International Congress
of Logic, Methodology and Philosophy of Science (August 1991, Uppsala).
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BCI
◦
IBW
◦
@
@
@
IBK1
◦
¡
¡
¡
@
@◦¡
¡
IB
fig. 3
Let us consider coordinate planes. Each plane forms a two–dimensional
point which corresponds to implicational logics R→ , BCK or IBW K1 .
R→ ◦@
◦
¡@
@
¡
BCK
@
◦
¡ IBW K1
¡
@·¡
@
·¡
···
·
@
¡
·
···
·
·
·
@◦¡
BCI ·◦·@
IBW ¡◦· IBK1
@
@
¡
¡
@
@◦¡
¡
IB
fig. 4
Now let us construct our figure further to a three–dimensional cube in
the usual way. As a result, we get a cube with the point which in this case
is a three–dimensional logic H→ . This figure we call a three–dimensional
intuitionistic implicational cube:
H·→
◦··
¡
¡ ····@
@
¡ ··· @
···
¡
@
···
¡
@◦
·
◦
◦
R→ @
········ ¡
¡ IBW K1
@
·
BCK
·
···¡
@
···
·
·
·
¡ ···
· @
·
·
··
·
@◦¡
BCI ·◦·@
IBW ¡◦· IBK1
@
@
¡
@
¡
@◦¡
IB
fig. 5
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¡
Let us note that n–dimensionality depends on which logic is chosen
as O. However, it is worth paying attention to the phenomen of space
conversion. In the logical language it means that axiomatization of some
logic can be represented by a single axiom. These results are well–known
for T V→ and H→ . For BCI it was done by C.A. Meredith [13]. Let’s denote
this single axiom as Lil (=Linear implicational logic).
Then our problem of extension H→ to T V→ can be formulated as follows: Is there any three–dimensional classical implicational cube with the
following nodes?
H→
R
T V→
◦··
¡··@
¡
·@
¡ ·· @
··
@
¡
··
¡
@
··
¡
@
··
@◦
·
◦¡
◦
·· ···
¡ BCK + X
@
·
¡
@
·
·· ¡
@ · · R→ + X
··
¡
@
··
··
·
·
··
¡
@
·
·
··
¡
@
·
· ◦·
@◦¡
◦· ·
BCK
BCI + X
¡
@
¡
@
¡
¡
@
@
@
¡
@
@◦¡
¡
BCI
fig. 6
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89
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[11] J. L
à ukasiewicz and A. Tarski, Investigations into the sentential
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à ukasiewicz, Selected Works, Warszawa, Polish Scientific
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à ukasiewicz implication, The
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for implicational calculi , Zeitschrift für Mathematische Logik und
Grundlagen der Mathematik, vol. 18 (1972), pp. 165–167.
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1962.
[19] V.A. Smirnov, Formal Inference and Logical Calculus, Moscow,
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für Logik, Wissenstheorie und Information an der Freien Universität Berlin.
Centre of Logical Investigations
Institute of Philosophy
Russian Academy of Science
119842 Moscow, Volkhonka 14
Russia
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