Deduction theorem

In mathematical logic, the deduction theorem states that if a formula B is deducible from a set $\Delta \cup \{A\}$ then the implication A → B is deducible from $\Delta \,.$ In symbols,

\Delta \cup \{A\}\vdash B

implies

\Delta \vdash A\to B\,.

In particular, if $\Delta \,$ is the empty set, then as a special case we have

A\vdash B

implies

\vdash A\to B

where, as is standard, we have omitted writing the empty set. Moreover, when $\Delta =\{A_{1},\dots ,A_{n}\}\,$ is finite we write

A_{1},\dots ,A_{n}\vdash B\,.

The deduction theorem is a metatheorem: it is used to deduce proofs in a given theory though it is not a theorem of the theory itself.^[1]

The deduction meta-theorem is one of the most important meta-theorems. In some systems of logic, it is taken as a rule of inference, an introduction rule for "→". In other systems, proving it from the axioms is the first major task in proving that the logic is complete. It is usually very hard to prove anything in propositional logic without using the deduction meta-theorem, and it usually becomes quite easy if you can use it.

Examples of deduction

Today many sentential logic languages use three axioms:

(1) P→(Q→P)
(2) (P→(Q→R))→((P→Q)→(P→R))
(3) (~P→~Q)→(Q→P)

Strictly speaking, these are not proved. However, below is a demonstration of a kind of proof, using natural deduction (in which, strictly speaking, there are no axioms).

"Prove" axiom 1:

- P 1. hypothesis
  - Q 2. hypothesis
  - P 3. reiteration of 1
- Q→P 4. deduction from 2 to 3
P→(Q→P) 5. deduction from 1 to 4 QED

"Prove" axiom 2:

- P→(Q→R) 1. hypothesis
  - P→Q 2. hypothesis
    - P 3. hypothesis
    - Q 4. modus ponens 3,2
    - Q→R 5. modus ponens 3,1
    - R 6. modus ponens 4,5
  - P→R 7. deduction from 3 to 6
- (P→Q)→(P→R) 8. deduction from 2 to 7
(P→(Q→R))→((P→Q)→(P→R)) 9. deduction from 1 to 8 QED

Using axiom 1 to show ((P→(Q→P))→R)→R:

- (P→(Q→P))→R 1. hypothesis
- P→(Q→P) 2. axiom 1
- R 3. modus ponens 2,1
((P→(Q→P))→R)→R 4. deduction from 1 to 3 QED

Virtual rules of inference

From the examples, you can see that we have added three virtual (or extra and temporary) rules of inference to our normal axiomatic logic. These are "hypothesis", "reiteration", and "deduction". The normal rules of inference (i.e. "modus ponens" and the various axioms) remain available.

1. Hypothesis is a step where one adds an additional premise to those already available. So, if your previous step S was deduced as:

E_{1},E_{2},...,E_{n-1},E_{n}\vdash S,

then one adds another premise H and gets:

E_{1},E_{2},...,E_{n-1},E_{n},H\vdash H.

This is symbolized by moving from the n-th level of indentation to the n+1-th level and saying

- - - - S previous step
        H hypothesis

2. Reiteration is a step where one re-uses a previous step. In practice, this is only necessary when one wants to take a hypothesis which is not the most recent hypothesis and use it as the final step before a deduction step.

3. Deduction is a step where one removes the most recent hypothesis (still available) and prefixes it to the previous step. This is shown by unindenting one level as follows:

- - - - H hypothesis
        
        ......... (other steps)
        
        C (conclusion drawn from H)
      - H→C deduction

Conversion from proof using the deduction meta-theorem to axiomatic proof

In axiomatic versions of propositional logic, one usually has among the axiom schemas (where P, Q, and R are replaced by any propositions):

Axiom 1 is: P→(Q→P)
Axiom 2 is: (P→(Q→R))→((P→Q)→(P→R))
Modus ponens is: from P and P→Q infer Q

From these one can quickly deduce the theorem schema P→P (see propositional calculus). These axiom schemas are chosen to enable one to derive the deduction theorem from them easily. So it might seem that we are begging the question. However, they can be justified by checking that they are tautologies using truth tables and that modus ponens preserves truth.

Suppose that we have that Γ and H prove C, and we wish to show that Γ proves H→C. For each step S in the deduction which is a premise in Γ (a reiteration step) or an axiom, we can apply modus ponens to the axiom 1, S→(H→S), to get H→S. If the step is H itself (a hypothesis step), we apply the theorem schema to get H→H. If the step is the result of applying modus ponens to A and A→S, we first make sure that these have been converted to H→A and H→(A→S) and then we take the axiom 2, (H→(A→S))→((H→A)→(H→S)), and apply modus ponens to get (H→A)→(H→S) and then again to get H→S. At the end of the proof we will have H→C as required, except that now it only depends on Γ, not on H. So the deduction step will disappear, consolidated into the previous step which was the conclusion derived from H.

To minimize the complexity of the resulting proof, some preprocessing should be done before the conversion. Any steps (other than the conclusion) which do not actually depend on H should be moved up before the hypothesis step and unindented one level. And any other unnecessary steps (which are not used to get the conclusion or can be bypassed), such as reiterations which are not the conclusion, should be eliminated.

During the conversion, it may be useful to put all the applications of modus ponens to axiom 1 at the beginning of the deduction (right after the H→H step).

When converting a modus ponens, if A is outside the scope of H, then it will be necessary to apply axiom 1, A→(H→A), and modus ponens to get H→A. Similarly, if A→S is outside the scope of H, apply axiom 1, (A→S)→(H→(A→S)), and modus ponens to get H→(A→S). It should not be necessary to do both of these, unless the modus ponens step is the conclusion, because if both are outside the scope, then the modus ponens should have been moved up before H and thus be outside the scope also.

Under the Curry-Howard correspondence, the above conversion process for the deduction meta-theorem is analogous to the conversion process from lambda calculus terms to terms of combinatory logic, where axiom 1 corresponds to the K combinator, and axiom 2 corresponds to the S combinator. Note that the I combinator corresponds to the theorem schema P→P.

The deduction theorem in predicate logic

The deduction theorem is also valid in first-order logic in the following form:

If T is a theory and F, G are formulas with F closed, and T∪{F}├G, then T├F→G.

Here, the symbol ├ means "is a syntactical consequence of." We indicate below how the proof of this deduction theorem differs from that of the deduction theorem in propositional calculus.

In the most common versions of the notion of formal proof, there are, in addition to the axiom schemes of propositional calculus (or the understanding that all tautologies of propositional calculus are to be taken as axiom schemes in their own right), quantifier axioms, and in addition to modus ponens, one additional rule of inference, known as the rule of generalization: "From K, infer ∀vK."

In order to convert a proof of G from T∪{F} to one of F→G from T, one deals with steps of the proof of G which are axioms or result from application of modus ponens in the same way as for proofs in propositional logic. Steps which result from application of the rule of generalization are dealt with via the following quantifier axiom (valid whenever the variable v is not free in formula H):

(H→K)→(H→∀vK).

Since in our case F is assumed to be closed, we can take H to be F. This axiom allows one to deduce F→∀vK from F→K, which is just what is needed whenever the rule of generalization is applied to some K in the proof of G.

Example of conversion

To illustrate how one can convert a natural deduction to the axiomatic form of proof, we apply it to the tautology Q→((Q→R)→R). In practice, it is usually enough to know that we could do this. We normally use the natural-deductive form in place of the much longer axiomatic proof.

First, we write a proof using a natural-deduction like method:

- Q 1. hypothesis
  - Q→R 2. hypothesis
  - R 3. modus ponens 1,2
- (Q→R)→R 4. deduction from 2 to 3
Q→((Q→R)→R) 5. deduction from 1 to 4 QED

Second, we convert the inner deduction to an axiomatic proof:

(Q→R)→(Q→R) 1. theorem schema (A→A)
((Q→R)→(Q→R))→(((Q→R)→Q)→((Q→R)→R)) 2. axiom 2
((Q→R)→Q)→((Q→R)→R) 3. modus ponens 1,2
Q→((Q→R)→Q) 4. axiom 1
- Q 5. hypothesis
- (Q→R)→Q 6. modus ponens 5,4
- (Q→R)→R 7. modus ponens 6,3
Q→((Q→R)→R) 8. deduction from 5 to 7 QED

Third, we convert the outer deduction to an axiomatic proof:

(Q→R)→(Q→R) 1. theorem schema (A→A)
((Q→R)→(Q→R))→(((Q→R)→Q)→((Q→R)→R)) 2. axiom 2
((Q→R)→Q)→((Q→R)→R) 3. modus ponens 1,2
Q→((Q→R)→Q) 4. axiom 1
[((Q→R)→Q)→((Q→R)→R)]→[Q→(((Q→R)→Q)→((Q→R)→R))] 5. axiom 1
Q→(((Q→R)→Q)→((Q→R)→R)) 6. modus ponens 3,5
[Q→(((Q→R)→Q)→((Q→R)→R))]→([Q→((Q→R)→Q)]→[Q→((Q→R)→R))]) 7. axiom 2
[Q→((Q→R)→Q)]→[Q→((Q→R)→R))] 8. modus ponens 6,7
Q→((Q→R)→R)) 9. modus ponens 4,8 QED

These three steps can be stated succinctly using the Curry-Howard isomorphism:

first, in lambda calculus, the function f = λa. λb. b a has type q → (q → r) → r
second, by lambda elimination on b, f = λa. s i (k a)
third, by lambda elimination on a, f = s (k (s i)) k

Paraconsistent Deduction Theorem

In general, the classical deduction theorem doesn't hold in paraconsistent logic. However, the following "two-way deduction theorem" does hold in one form of paraconsistent logic:^[2]

\vdash E\rightarrow F

if and only if (

E\vdash F

and

\neg F\vdash \neg E

)

that requires the contrapositive inference to hold in addition to the requirement of the classical deduction theorem.

The resolution theorem

The Resolution theorem is the converse of the deduction theorem. It follows immediately from modus ponens which is the elimination rule for implication.

Notes

^ See Detlovs and Podnieks 1964
^ Hewitt 2008

References

Introduction to Mathematical Logic by Vilnis Detlovs and Karlis Podnieks Podnieks is a comprehensive tutorial. See Section 1.5.
Carl Hewitt (2008). "ORGs for Scalable, Robust, Privacy-Friendly Client Cloud Computing". IEEE Internet Computing. 12 (5): 96. doi:10.1109/MIC.2008.107. September/October 2008

[1] See Detlovs and Podnieks 1964

[2] Hewitt 2008

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