Caribb. J. Math. Comput. Sci.
13, 2006, 12-15
A WEAK DEDUCTION THEOREM
H.G.R. MILLINGTON
Abstract. For propositional logics without rules of contraction we derive a weakened
form of the deduction theorem.
In what follows, L is an implicational propositional language having implication, →, as its only binary connective, with detachment as its only rule of
deduction, and a countable family of variables, some of which are denoted by
a, . . . , z. Well-formed formulæ (wff ) are constructed according to the rules: if
X is a variable then X is a well-formed formula; if P and Q are well-formed
formulæ then (P → Q) is a well-formed formula. In a recent paper [1] the Deduction Theorem has been studied with a view to finding weakened versions of
it. This note observes that there is already a weak form of the Deduction Theorem appropriate for contraction-less propositional logics. Closely related work
is found in [2].
As is usual, the outermost brackets of well-formed formulæ will be omitted.
Denote by [Q/V ]W the well-formed formula (wff ) obtained by replacing each
occurrence of a variable V in a wff W by a wff Q, and by var W the family of
all variables appearing in W . Let S be a family of wff of L. For each family F
of variables of L, the deductive hull of S relative to F , denoted by hull F (S), is
the smallest family G of wff such that
• each wff of S is in G,
• if P and Q are wff such that P and P → Q are in G, then Q is in G,
• if P is in G, and V is a variable appearing in P , but not in F , then [W/V ]P
is in G for each wff W .
Let Q be a wff. A derivation of Q from S without substitution for variables in
F is a finite sequence W1 , . . . , Wn , such that Wn is Q, W1 ε S, and, for each k
with 1 < k ≤ n, either Wk ε S, or, for some i, j with 1 ≤ i < k, 1 ≤ j < k, Wj
is Wi → Wk , or, for some i < k and variable X appearing in Wi but not in F ,
Wk is [V /X]Wi for some wff V . Note that F may be the set of all variables of
L, i.e. substitution is then not one of the rules of deduction. In that case, we
shall speak of “the deductive hull of S”, and of “a derivation of Q from S”. In a
straightforward manner it may be shown that a wff Q is in the deductive hull of S
relative to F iff there is a derivation of Q from S without substitution for variables
in F . The theorem below asserts that, under the given conditions, deducibility
“S, P ` Q” is equivalent to “there exists a non-negative integer n such that
S ` P n → Q”, where (P 0 → Q) := Q, and (P n+1 → Q) := (P → (P n → Q)).
12
Hugh Millington
A WEAK DEDUCTION THEOREM
13
Theorem 1 (Weak Deduction Theorem). Let S be a family of wff which
contains all instances of the wff (b → c) → ((a → b) → (a → c)), (a → (b →
c)) → (b → (a → c)) and a → a. Let P be any wff of L. A wff Q is in the
deductive hull of {P } ∪ S relative to var P if and only if, for some integer n ≥ 0,
the wff P n → Q is in the deductive hull of S relative to var P .
Corollary 1. Under the hypothesis of the theorem, if S also contains all
substitution instances of the wff (a → (a → b)) → (a → b), then at least one of
Q or P → Q is in the deductive hull of S relative to var P .
The corollary contains as special instances the deduction theorems for the classical and intuitionist propositional logics [5, 8]. The theorem is proved for
L
à ukasiewicz infinite-valued propositional logic in [6]. In the meta language, we
shall take = to be identity of formulæ; in particular, therefore, if V = W , then
V → W , and [V /X]P = [W/X]P , where P is any wff of L, and X is any variable
of L.
Proof. The proof is by induction on the length k of a derivation. Denote
var P by F . Let Q ε hull F ({P } ∪ S). By the earlier remark, we can find a
derivation W1 , . . . , Wk of Q from {P } ∪ S without substitution for variables in
F . If k = 1, the conclusion of the theorem is trivially true. Suppose now that,
for some k ≥ 1, the conclusion of the theorem holds for each wff V having a
derivation of length x ≤ k from {P } ∪ T , without substitution for variables in
F . Let Q be any wff having a derivation W1 , . . . , Wk , Q from {P } ∪ S of length
k + 1, without substitution for variables in F . If Q is obtained from W1 , . . . , Wk
by detachment, then, for some distinct i, j ≤ k, the wff Wj is Wi → Q. Now,
W1 , . . . , Wi and W1 , . . . , Wj are derivations respectively of Wi and Wj , having
lengths less than or equal to k, and without substitution for variables in F .
Hence, for some r and s, the wff P r → Wi and P s → Wj are in hull F (S). Then
P r+s → Q is in hull F (S), by the derivation
(P s → (Wi → Q)) → (Wi → (P s → Q))
P s → (Wi → Q)
Wi → (P s → Q)
(P r → Wi ) → ((Wi → (P s → Q)) → (P r → (P s → Q)))
P r → Wi
(Wi → (P s → Q)) → (P r → (P s → Q))
P r → (P s → Q)
Suppose now that, for some i ≤ k, Q is obtained by substitution of a wff U for
some variable V appearing in Wi but not in P . By the inductive hypothesis, for
some r, the wff P r → Wi is in hull F (S). Thus P r → Q is in hull F (S). Finally,
if Q = P , or Q ε S, then (P → Q) ε S. The theorem follows. 2
For negated well-formed formulæ we can derive a version of Theorem 1 in which
S does not necessarily contain all substitution instances of a → a. We adjoin a
constant 0 to the alphabet of L, and stipulate that 0 is a wff. For each wff Q of
L, denote (Q → 0) by ¬Q.
Theorem 2. Let S be a family of wff which contains all instances of the wff
(b → c) → ((a → b) → (a → c)), (a → (b → c)) → (b → (a → c)) and 0 → a.
A Weak Deduction Theorem
14
H.G.R. MILLINGTON
Let P be any wff of L. A wff ¬Q is in the deductive hull of {P } ∪ S relative to
var P if and only if, for some integer n ≥ 0, the wff P n → ¬Q is in the deductive
hull of S relative to var P .
Proof. We note that the following wff must be in S.
(0 → a) → ((a → 0) → (a → 0)),
¬a → ¬a,
The assertion of the theorem is therefore trivially true for any wff ¬Q having a
derivation of length 1 from {P }∪S. The conclusion now follows by mathematical
induction, as in the previous proof. 2
A converse of the corollary is provided by a result of Herbrand-Tarski [1, 4, 7].
Theorem 3. Let S be a family of wff which is closed under detachment.
Suppose that, for all wff P , Q, and finite family W of wff, if P is in W, and
Q is in the deductive hull of W ∪ S, then (P → Q) is in the deductive hull of
(W \ {P }) ∪ S. Then S contains each substitution instance of the wff ((b → c) →
((a → b) → (a → c))), ((a → (b → c)) → (b → (a → c))), (a → (b → a)),
((a → (a → b)) → (a → b)).
Proof. Let α, β, γ be any wff of L. By repeated detachment it follows that
γ is in the deductive hull of {α, (α → β), (β → γ)} ∪ S. Hence (α → γ) is in the
deductive hull of {(α → β), (β → γ)} ∪ S. Then ((α → β) → (α → γ)) is in the
deductive hull of {(β → γ)} ∪ S, and finally ((β → γ) → ((α → β) → (α → γ)))
is in the deductive hull of S. Thus S contains every substitution instance of the
wff ((b → c) → ((a → b) → (a → c))) . Similarly, by consideration of the families
{α, β, (α → (β → γ))} ∪ S,
{α, β} ∪ S ,
{α, (α → (α → β))} ∪ S,
it follows that S contains all substitution instances of the wff listed in the theorem. 2
It follows that the Deduction Theorem holds if and only if the propositional
logic is stronger than positive implicational logic [3], and therefore cannot hold
in implicational propositional theories which exclude absorption, ((a → (a →
b)) → (a → b)). For the latter, variants of the Deduction Theorem become
significant — for example, Theorems 1 and 2 above. See also [1].
REFERENCES
[1] F. Bou, J.M. Font, and J.L.G. Lapresta, On weakening the deduction theorem and
strengthening modus ponens, Math.Log.Quart., vol. 50 (2004), pp. 303–324.
[2] Petr Cintula, Weakly implicative (fuzzy) logics, Archiv Logic, vol. 7 (2005), pp. 115–
117.
[3] K. Došen, A historical introduction to substructural logics, Substructural Logics, Oxford, 1993, pp. 1–30.
[4] J. Herbrand, Recherches sur la théorie de la démonstration, Travaux Soc. Sci. Lettr.
Varsovie, Cl. III, vol. 33 (1930), pp. 33–160.
[5] E. Mendelson, Introduction to Mathematical Logic, third ed., Wadsworth and
Brooks/Cole, 1987.
Hugh Millington
A WEAK DEDUCTION THEOREM
15
[6] A. Rose and J.B. Rosser, Fragments of many-valued statement calculi, Trans. Amer.
Math. Soc., vol. 87 (1958), pp. 1–53.
[7] A. Tarski, Über einige fundamentale Begriffe der Metamathematik, C.R. Soc. Sci.
Lettr. Varsovie, Cl. III, vol. 23 (1930), pp. 22–29.
[8] D. van Dalen, Logic and Structure, first ed., Springer, 1983.
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