Smale's paradox

In mathematical logic, the deduction theorem is a metatheorem of first-order logic. It is a formalization of the common proof technique in which an implication A → B is proved by assuming A and then deriving B from this assumption conjoined with known results. The deduction theorem explains why proofs of conditional sentences in mathematics are logically correct. Though it has seemed "obvious" to mathematicians literally for centuries that proving B from A conjoined with a set of theorems is sufficient to proving the implication A → B based on those theorems alone, it was left to Herbrand and Tarski to show (independently) this was logically correct in the general case—another instance, perhaps, of modern logic "cleaning up" mathematical practice.

The deduction theorem states that if a formula B is deducible from a set of assumptions $\mathrm{\Delta }\cup \{A\}$, where A is a closed formula, then the implication A → B is deducible from $\mathrm{\Delta }.$ In symbols, $\mathrm{\Delta }\cup \{A\}⊢B$ implies $\mathrm{\Delta }⊢A\to B.$. In the special case where $\mathrm{\Delta }$ is the empty set, the deduction theorem shows that $A⊢B$ implies $⊢A\to B.$

The deduction theorem holds for all first-order theories with the usual deductive systems for first-order logic. However, there are first-order systems in which new inference rules are added for which the deduction theorem fails.

The deduction rule is an important property of Hilbert-style systems because the use of this metatheorem leads to much shorter proofs than would be possible without it. Although the deduction theorem could be taken as primitive rule of inference in such systems, this approach is not generally followed; instead, the deduction theorem is obtained as an admissible rule using the other logical axioms and modus ponens. In other formal proof systems, the deduction theorem is sometimes taken as a primitive rule of inference. For example, in natural deduction, the deduction theorem is recast as an introduction rule for "→".

Examples of deduction

"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:

$E_1,E_2,...,E_{n-1},E_n⊢S,$

then one adds another premise H and gets:

$E_1,E_2,...,E_{n-1},E_n,H⊢H.$

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

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.

From these axiom schemas one can quickly deduce the theorem schema P→P (reflexivity of implication) which is used below:

1. (P→((Q→P)→P))→((P→(Q→P))→(P→P)) from axiom schema 2 with P, (Q→P), P
2. P→((Q→P)→P) from axiom schema 1 with P, (Q→P)
3. (P→(Q→P))→(P→P) from modus ponens applied to step 2 and step 1
4. P→(Q→P) from axiom schema 1 with P, Q
5. P→P from modus ponens applied to step 4 and step 3

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 correspondence:

• 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:

$⊢E\to F$ if and only if ($E⊢F$ and $\mathrm{\neg }F⊢\mathrm{\neg }E$)

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.

See also

• Conditional proof
• Propositional calculus
• Peirce's law

References

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 September/October 2008
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

External links

• Introduction to Mathematical Logic by Vilnis Detlovs and Karlis Podnieks Podnieks is a comprehensive tutorial. See Section 1.5.