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intuitionistic propositional logic is a
subsystem of classical propositional logic∗
CWoo†
2013-03-22 3:57:19
Suppose logical systems L1 and L2 share the same set L of well-formed
formulas (wff’s). Let F1 be the set of all theorems of L1 , and F2 the set of all
theorems of L2 . We say that L1 is a subsystem of L2 if F1 ⊆ F2 , and a proper
subsystem if F1 ⊂ F2 .
In the subsequent discussion, PLc stands for classical propositional logic,
and PLi for intuitionistic propositional logic. We write PLi ≤ PLc to mean
that PLi is a subsystem of PLc , or < if it is proper.
Proposition 1. PLi < PLc .
Unless otherwise specified, ` A means the wff A is a theorem of PLc .
Before proving this, we need the following facts about PLc :
1. ` A and ` B iff ` A ∧ B.
4. A → B, ¬A → B ` B
2. ` A implies ` B iff ` A → B.
5. if ∆ ` A and Γ, A ` B, then
∆, Γ ` B.
3. A → B, B → C ` A → C
Proof. The first two facts are proved here, and the third is here.
Fact 4 is equivalent to ` (A → B) → ((¬A → B) → B) by the deduction
theorem twice. Use the theorem schemas ` (A → B) ↔ (¬B → ¬A) and
` A ↔ ¬¬A (law of double negation), and the substitution theorem twice, it is
enough to show
` (¬B → ¬A) → ((¬B → A) → ¬¬B).
By the deduction theorem three times, it is enough to show ¬B → ¬A, ¬B →
A, ¬B `⊥, which is provided by the deduction ¬B → ¬A, ¬B → A, ¬B, ¬A, A, ⊥.
For the last fact, given deduction E1 of A from ∆, and deduction E2 of A → B
from Γ, E1 , E2 , B is a deduction of B from ∆ ∪ Γ.
∗ hIntuitionisticPropositionalLogicIsASubsystemOfClassicalPropositionalLogici
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1
Proof. Since in both systems, the only inference rule is modus ponens, and
theoremhood of wff’s is preserved by the inference rule (that is, if ` A and
` A → B, then ` B), all we need to show is that every axiom of PLi is a
theorem of PLc .
1. A → (B → A). This is just an axiom schema for PLc .
2. A → (B → A ∧ B).
A, B, A → ¬B, ¬B, ⊥ leads to A, B, A → ¬B `⊥. Applying the deduction
theorem three times, we get ` A → (B → ((A → ¬B) →⊥)), or ` A →
(B → A ∧ B).
3. A ∧ B → A and A ∧ B → B. See here.
4. A → A ∨ B.
A, ¬A, ⊥, ⊥→ B, B results in A, ¬A ` B, since ⊥→ B is a theorem. By
applying the deduction theorem twice, we get ` A → (¬A → B), or
` A → A ∨ B.
5. B → A ∨ B.
Clearly, B, ¬A ` B. By the deduction theorem, B ` ¬A → B, or B `
A ∨ B. By the deduction theorem again, ` B → A ∨ B.
6. (A → C) → ((B → C) → (A ∨ B → C)).
By fact 3, A → C, B → C, ¬A → B ` ¬A → C. With fact 4: A →
C, ¬A → C ` C, so A → C, B → C, ¬A → B ` C by fact 5. Now apply
the deduction theorem 3 times.
7. (A → B) → ((A → (B → C)) → (A → C)).
From A → B, A → (B → C), A, B → C, B, C, we get A → B, A → (B →
C), A ` C. Applying the deduction theorem three times, we have the
result.
8. (A → B) → ((A → ¬B) → ¬A). This is just 7, where C is ⊥.
9. ¬A → (A → B).
From ¬A, A `⊥ and ⊥` B, we get ¬A, A ` B by fact 5, and the result
follows with two applications of the deduction theorem.
Since A ∨ ¬A is a theorem of PLc and not of PLi (see here), we conclude that
PLi is a proper subsystem of PLc .
Remarks.
1. Strictly speaking, the language Li of PLi under this system is different
from the language Lc of PLc under this system. In Li , the logical connectives consist of →, ¬, ∧, ∨, whereas in Lc , only → is used. The other
connectives are introduced as abbreviational devices: ¬A is A →⊥, A ∨ B
is ¬A → B, and A ∧ B is ¬(A → ¬B). So it doesn’t make much sense to
say that PLi < PLc .
2
2. One way to get around this issue is to re-define what it means for one
logical system to be a subsystem of another. For example, one can define
a mapping τ : Li → Lc such that
• takes propositional variables to propositional variables
• τ preserves the formation rules of wff’s
A mapping satisfying these conditions is called a translation. Then, a
system PLi is a subsystem of PLc if there is a translation τ : Li → Lc
such that
• for any wff A in Li , we have `i A implies `c τ (A).
To further require that PLi be a proper subsystem of PLc , we also need
• there is an wff B in Li such that 6`i B and `c τ (B).
3. Using this definition, define τ : Li → Lc by

A
if A is a propositional variable




 τ (B) → τ (C) if A is B → C
¬τ (B)
if A is ¬B
τ (A) :=


τ
(B)
∧
τ
(C)
if
A is B ∧ C



τ (B) ∨ τ (C) if A is B ∨ C.
where the symbols ¬, ∧, and ∨ in Lc are used as abbreviational tools (see
the first remark). Then we see that the proposition makes sense.
4. Another way of getting around this issue is to come up with another axiom
system for PLc that uses →, ¬, ∧, and ∨ as primitive logical connectives.
Such an axiom system exists. In fact, add ¬¬A → A to the list of axiom
schemas for PLi , and we get an axiom system for PLc .
5. Using the idea in 2 above further, it can be shown that PLc is a subsystem
of PLi (under the right translation)! The existence of such a translation
ν is known as the Glivenko’s theorem. This translation ν has the further
property that for any wff A in Lc ,
`c A
iff
3
`i ν(A).