P řednáška 12
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Proof calculi, Hilbert 1
Principles of proof calculi
Hilbert’s programme and Hilbert calculus
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Proof calculi, Hilbert 2
1921: Hilbert’s Program of Formalisation of Mathematics Reasoning with infinites paradoxes (Zeno, infinitesimals in the
17th century, Russell, …) Hilbert: ‘finitary’ methods of axiomatisation and reasoning in
mathematics; Kant: We obviously cannot experience infinitely many events or
move about infinitely far in space. (actual infinityactual infinity) However, there is no upper bound on the number of steps we
execute, we can always move a step further. (potential infinity)(potential infinity) But at any point we will have acquired only a finite amount of finite amount of
experienceexperience and have taken only a finite number of steps. Thus, for a Kantian like Hilbert, the only legitimate infinity is a
potential infinity, rather thanrather than the actual infinity the actual infinity. “mathematics is about symbols” (?), mathematical reasoning -
Syntactic laws of symbol manipulation (?); consistency proof
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Statements of Hilbert’s program The main goal of Hilbert's program was to provide secure secure
foundations for all mathematicsfoundations for all mathematics. In particular this should include: A formalization of all mathematicsformalization of all mathematics; in other words all
mathematical statements should be written in a precise formal language, and manipulated according to well defined rules.
CompletenessCompleteness: a proof that all true mathematical statements can be proved in the formalism.
ConsistencyConsistency: a proof that no contradiction can be obtained in the formalism of mathematics. This consistency proof should preferably use only "finitistic" reasoning about finite mathematical objects.
ConservationConservation: a proof that any result about "real objects" obtained using reasoning about "ideal objects" (such as uncountable sets) can be proved without using ideal objects.
DecidabilityDecidability: there should be an algorithm for deciding the truth or falsity of any mathematical statement.
Proof calculi, Hilbert 3
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Gödel's incompleteness theorems Kurt Gödel showed that most of the goals of Hilbert's program were impossible
to achieve, at least if interpreted in the most obvious way. Gödel's second incompleteness theorem shows that any consistent theory powerful enough to encode addition and multiplication of integers cannot prove its own consistency. This presents a challenge to Hilbert's program:
It is not possible to formalize all of mathematics within a formal system, as any attempt at such a formalism will omit some true mathematical statements. There is no complete, consistent extension of even Peano arithmetic based on a recursively enumerable set of axioms.
A theory such as Peano arithmetic cannot even prove its own consistency, so a restricted "finitistic" subset of it certainly cannot prove the consistency of more powerful theories such as set theory.
There is no algorithm to decide the truth (or provability) of statements in any consistent extension of Peano arithmetic. Strictly speaking, this negative solution to the Entscheidungsproblem appeared a few years after Gödel's theorem, because at the time the notion of an algorithm had not been precisely defined.
Proof calculi, Hilbert 4
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Kurt Gödel, Albert Einstein
5
Kurt Gödel with an unknown local farmer
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Step No. 1; proof calculus, the goals
Recall CompletenessCompleteness: a proof that all true mathematical statements can be proved in the formalism.
So that we start with proving all true logical statements
To this end we build up a formal system proving all logically valid sentences
and/or proving all logically valid arguments which are two sides of the same coin due to
PP11, …,, …, PPnn |= C|= C iff |= (|= (PP11 … … PPnn) ) CC
Proof calculi, Hilbert 6
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Proof calculi, Hilbert 7
Formal systems, Proof calculi A proof calculus (of a theory) is given by:A. a languageB. a set of axiomsC. a set of deduction rulesad A. The definition of a language of the
system consists of: an alphabet (a non-empty set of symbols), and a grammar (defines in an inductive way a set of
well-formed formulas - WFF)
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Proof calculi, Hilbert 8
Hilbert-like calculus.Language: restricted FOPLAlphabet: 1. logical symbols:
(countable set of) individual variables x, y, z, …connectives , quantifiers
2. special symbols (of arity n)predicate symbols Pn, Qn, Rn, …functional symbols fn, gn, hn, …constants a, b, c, – functional symbols of arity 0
3. auxiliary symbols (, ), [, ], …Grammar:1. terms
each constant and each variable is an atomic termif t1, …, tn are terms, fn a functional symbol, then fn(t1, …, tn) is a (functional) term
2. atomic formulasif t1, …, tn are terms, Pn predicate symbol, then Pn(t1, …, tn) is an atomic (well-formed) formula
3. composed formulasLet A, B be well-formed formulas. Then A, (AB), are well-formed formulas.Let A be a well-formed formula, x a variable. Then xA is a well-formed formula.
4. Nothing is a WFF unless it so follows from 1.-3.
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Proof calculi, Hilbert 9
Hilbert calculusAd B. The set of axioms is a chosen subset of the set of
WFF. The set of axioms has to be decidable: axiom schemes:1. A (B A) 2. (A (B C)) ((A B) (A C)) 3. (B A) (A B)4. x A(x) A(x/t) Term t substitutable for x in A 5. x [A B(x)] A x B(x), x is not free in
A
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Proof calculi, Hilbert 10
Hilbert calculusAd C. The deduction rules are of a form:
A1,…,Am |– B1,…,Bn enable us to prove theorems (provable
formulas) of the calculus. We say that each Bi is derived (inferred) from the set of assumptions A1,…,Am.Rule schemas:MP: A, A B |– B (modus ponens)G: A |– x A (generalization)
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Proof calculi, Hilbert 11
Hilbert calculusNotes:1. A, B are not formulas, but meta-symbols denoting any formula.
Each axiom schema denotes an infinite class of formulas of a given form. If axioms were specified by concrete formulas, like
1. p (q p)2. (p (q r)) ((p q) (p r))3. (q p) (p q)
we would have to extend the set of rules with the rule of substitution:Substituting in a proved formula for each propositional logic symbol another formula, then the obtained formula is proved as well.
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Proof calculi, Hilbert 12
Hilbert calculus2. The axiomatic system defined in this way works only with the
symbols of connectives , , and quantifier . Other symbols of the other connectives and existential quantifier can be introduced as abbreviations ex definicione:
A B =df (A B)A B =df (A B)A B =df ((A B) (B A))xA =df x A
The symbols , , and do not belong to the alphabet of the language of the calculus.
3. In Hilbert calculus we do not use the indirect proof.
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Proof calculi, Hilbert 13
Hilbert calculus4. Hilbert calculus defined in this way is sound
(semantically consistent). a) All the axioms are logically valid formulas. b) The modus ponens rule is truth-preserving. The only problem – as you can easily see – is the
generalisation rule. This rule is obviously not truth preserving:
formula P(x) x P(x) is not logically valid. However, this rule is preserving the truth in an interpretation:
If the formula P(x) at the left-hand side is true in an interpretation, then x P(x) is true in this interpretation as well.
Since the axioms of the calculus are logically valid, the rule can be applied in a correct way.
After all, this is a common way of proving in mathematics. To prove that something holds for all the triangles, we prove that for any triangle.
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Proof calculi, Hilbert 14
A sound calculus: if | A (provable) then |= A (True)
WFF
|– ATheorems
Axioms
|= ALVF
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Proof calculi, Hilbert 15
Proof in a calculus A proof of a formula A (from logical axioms of the
given calculus) is a sequence of formulas (proof steps) B1,…, Bn such that:
A = Bn (the proved formula A is the last step) each Bi (i=1,…,n) is either
an axiom or Bi is derived from the previous Bj (j=1,…,i-1) using a
deduction rule of the calculus. A formula A is provable by the calculus, denoted
|– A, if there is a proof of A in the calculus. A provable formula is called a theorem.
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Proof calculi, Hilbert 16
Hilbert calculus Note that any axiom is a theorem as well. Its
proof is a trivial one step proof. To make the proof more comprehensive, you
can use in the proof sequence also previously proved formulas (theorems).
Therefore, we will first prove the rules of natural deduction, transforming thus Hilbert Calculus into the natural deduction system.
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Proof calculi, Hilbert 17
A Proof from AssumptionsA (direct) proof of a formula A from assumptions A1,
…,Am is a sequence of formulas (proof steps) B1,…Bn such that:
A = Bn (the proved formula A is the last step) each Bi (i=1,…,n) is either
an axiom, or an assumptionassumption Ak (1 k m), or Bi is derived from the previous Bj (j=1,…,i-1) using a rule of the
calculus.
A formula A is provable from A1, …, Am, denoted A1,…,Am |– A, if there is a proof of A from A1,…,Am.
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Proof calculi, Hilbert 18
Examples of proofsProof of a formula schema A A:
1. (A ((A A) A)) ((A (A A)) (A A)) axiom A2: B/A A, C/A
2. A ((A A) A)axiom A1: B/A A
3. (A (A A)) (A A) MP:2,14. A (A A) axiom A1: B/A5. A A MP:4,3 Q.E.D.
Hence: |– A A .
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Proof calculi, Hilbert 19
Examples of proofsProof of: A B, B C | A C
(transitivity of implication TI):1. A B assumption2. B C assumption3. (A (B C)) ((A B) (A C)) axiom A2 4. (B C) (A (B C)) axiom A1 A/(B C), B/A5. A (B C) MP:2,46. (A B) (A C) MP:5,37. A C MP:1,6 Q.E.D.Hence: A B, B C |– A C .
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Proof calculi, Hilbert 20
Examples of proofs|– Ax/t xAx
(the ND rule – existential generalisation)Proof:1. x Ax Ax/t axiom A42. x Ax x Ax theorem of type C C (see below)3. x Ax Ax/t C D, D E |– C E: 2, 1 TI4. x Ax = xAx Intr. acc. (by definition)5. xAx Ax/t substitution: 4 into 3 6. [xAx Ax/t] [Ax/t xAx] axiom A37. Ax/t xAx MP: 5, 6 Q.E.D.
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Proof calculi, Hilbert 21
Examples of proofs
A Bx |– A xBx (x is not free in A)Proof:
1. A Bx assumption2. x[A Bx] Generalisation:13. x[A Bx] [A xBx] axiom
A54. A xBx MP: 2,3 Q.E.D.
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Proof calculi, Hilbert 22
Theorem of Deduction Let A be a closed formula, B any formula. Then: A1, A2,...,Ak |– A B if and only if
A1, A2,...,Ak, A |– B.
Remark: The statement a) if |– A B, then A |– B
is valid universally, not only for A being a closed formula (the proof is obvious – modus ponens).
On the other hand, the other statement b) If A |– B, then |– A B
is not valid for an open formula A (with at least one free variable). Example: Let A = A(x), B = xA(x).
Then A(x) |– xA(x) is valid according to the generalisation rule.But the formula Ax xAx is generally not logically valid, and therefore not provable in a sound calculus.
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Proof calculi, Hilbert 23
The Theorem of Deduction Proof (we will prove the Deduction Theorem only for
the propositional logic):1. Let A1, A2,...,Ak |– A B.
Then there is a sequence B1, B2,...,Bn, which is the proof of A B from assumptions A1,A2,...,Ak.
The proof of B from A1, A2,...,Ak, A is then the sequence of formulas B1, B2,...,Bn, A, B, where Bn = A B and B is the result of applying modus ponens to formulas Bn and A.
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Proof calculi, Hilbert 24
The Theorem of Deduction2. Let A1, A2,...,Ak, A |– B.
Then there is a sequence of formulas C1,C2,...,Cr = B, which is the proof of B from A1,A2,...,Ak, A. We will prove by induction that the formula A Ci (for all i = 1, 2,...,r) is provable from A1, A2,...,Ak. Then also A Cr will be proved.
a) Base of the induction: If the length of the proof is = 1, then there are possibilities:1. C1 is an assumption Ai, or axiom, then:2. C1 (A C1) axiom A13. A C1 MP: 1,2 Or, In the third case C1 = A, and we are to prove A A (see example 1).b) Induction step: we prove that on the assumption of A Cn being proved for n = 1, 2, ..., i-1 the formula
A Cn can be proved also for n = i. For Ci there are four cases: 1. Ci is an assumption of Ai, 2. Ci is an axiom, 3. Ci is the formula A, 4. Ci is an immediate consequence of the formulas Cj and Ck = (Cj Ci), where j, k < i. In the first three cases the proof is analogical to a). In the last case the proof of the formula A Ci is the sequence of formulas:
1. A Cj induction assumption2. A (Cj Ci) induction assumption 3. (A (Cj Ci)) ((A Cj) (A Ci)) A24. (A Cj) (A Ci) MP: 2,35. (A Ci) MP: 1,4 Q.E.D
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Proof calculi, Hilbert 25
Semantics A semantically correct (sound) logical calculus
serves for proving logically valid formulas (tautologies). In this case the
axioms have to be logically valid formulas (true in every interpretation), and the
deduction rules have to make it possible to prove logically valid formulas. For that reason the rules are preserving truth in an interpretation, i.e., A1,…,Am |– B1,…,Bn can be read as follows: if all the formulas A1,…,Am are true in an interpretation I,
then B1,…,Bn are true in this interpretation as well.
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Proof calculi, Hilbert 26
Theorem on Soundness (semantic consistence) Each provable formula in the Hilbert calculus is also
logically valid formula: If |– A, then |= A.Proof (outline): Each formula of the form of an axiom schema of
A1 – A5 is logically valid (i.e. true in every interpretation structure I for any valuation v of free variables).
Obviously, MP (modus ponens) is a truth preserving rule.
Generalisation rule: Ax |– xAx ?
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Proof calculi, Hilbert 27
Theorem on Soundness (semantic consistence) Generalisation rule Ax |– xAx is preserving
truth in an interpretation: Let us assume that A(x) is a proof step such that in
the sequence preceding A(x) the generalisation rule has not been used as yet.
Hence |= A(x) (since it has been obtained from logically valid formulas by using at most the truth preserving modus ponens rule).
It means that in any interpretation I the formula A(x) is true for any valuation v of x. Which, by definition, means that |= xA(x) (is logically valid as well).
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Proof calculi, Hilbert 28
Hilbert & natural deduction According to the Deduction Theorem each
theorem of the implication form corresponds to a deduction rule(s), and vice versa. For example:
Theorem Rule(s)|– A ((A B) B) A, A B |– B (MP rule) |– A (B A) ax. A1 A |– B A; A, B |– A |– A A A |– A |– (A B) ((B C) (A C)) A B |– (B C) (A C);
A B, B C |– A C /rule TI/
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Proof calculi, Hilbert 29
Example: a few simple theorems and the corresponding (natural deduction) rules:
1. |– A (A B); |– A (A B) A, A |– B
2. |– A A B; |– B A B A |– A B; B |– A B ID
3. |– A A A |– A EN
4. |– A A A |– A IN
5. |– (A B) (B A) A B |– B A TR
6. |– A B A; |– A B B A B |– A, B EC
7. |– A (B A B); |– B (A A B) A, B |– A B IC
8. |– A (B C) (A B C) A (B C) |– A B C
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Proof calculi, Hilbert 30
Some proofsAd 1. |– A (A B); i.e.: A, A |– B.
Proof: (from a contradiction |-- anything)1. A assumption2. A assumption 3. (B A) (A B) A34. A (B A) A15. B A MP: 2,46. A B MP: 5,37. B MP: 1,6 Q.E.D.
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Proof calculi, Hilbert 31
Some proofs
Ad 2. |– A A B, i.e.: A |– A B. (the rule ID of the natural deduction)Using the definition abbreviation A B =df A B, we are to prove the theorem:|– A (A B), i.e. the rule A, A |– B, which has been already proved.
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Proof calculi, Hilbert 32
Some proofsAd 3. |– A A; i.e.: A |– A.
Proof:1. A assumption 2. (A A) (A A) axiom A33. A (A A) theorem ad 1. 4. A A MP: 1,35. A A MP: 4,26. A MP: 1,5 Q.E.D.
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Proof calculi, Hilbert 33
Some proofs
Ad 4. |– A A; i.e.: A |– A.Proof:1. A assumption2. (A A) (A A)axiom A33. A A theorem ad 3. 4. A A MP: 3,2Q.E.D.
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Proof calculi, Hilbert 34
Some proofsAd 5. |– (A B) (B A), i.e.: (A B) |– (B A).
Proof: 1. A B assumption2. A A theorem ad 3. 3. A B TI: 2,14. B B theorem ad 4. 5. A B TI: 1,4 6. A B TI: 2,57. (A B) (B A) axiom A38. B A MP: 6,7Q.E.D.
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Proof calculi, Hilbert 35
Some proofsAd 6. |– (A B) A, i.e.: A B |– A.
(The rule EC of the natural deduction)Using definition abbreviation A B =df (A B) we are to prove
|– (A B) A, i.e.: (A B) |– A.Proof:1. (A B) assumption2. (A (A B)) ((A B) A) theorem ad 5. 3. A (A B) theorem ad 1. 4. (A B) A MP: 3,25. A MP: 1,46. A A theorem ad 3. 7. A MP: 5,6 Q.E.D.
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36
Some meta-rulesLet T is any finite set of formulas: T = {A1, A2,..,An}. Then(a)if T, A |– B and |– A, then T |– B.
It is not necessary to state theorems in the assumptions.(b)if A |– B, then T, A |– B. (Monotonicity of proving)(c) if T |– A and T, A |– B, then T |– B.(d)if T |– A and A |– B, then T |– B.(e)if T |– A; T |– B; A, B |– C then T |– C.(f) if T |– A and T |– B, then T |– A B.
(Consequences can be composed in a conjunctive way.)(g)T |– A (B C) if and only if T |– B (A C).
(The order of assumptions is not important.)(h)T, A B |– C if and only if both T, A |– C and T, B |– C.
(Split the proof whenever there is a disjunction in the sequence – meta-rule of the natural deduction)
(i) if T, A |– B and if T, A |– B, then T |– B.Proof calculi, Hilbert
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Proof calculi, Hilbert 37
Proofs of meta-rules (a sequence of rules)
Ad (h) :Let T, A B |– C, we prove that: T, A |– C; T, B |– C.Proof:1. A |– A B the rule ID2. T, A |– A Bmeta-rule (b): 13. T, A B |– Cassumption4. T, A |– C meta-rule (d): 2,3
Q.E.D.5. T, B |– C analogically to 4.
Q.E.D.
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Proof calculi, Hilbert 38
Proofs of meta-rules (a sequence of rules)
Ad (h) :Let T, A |– C; T, B |– C, we prove that T, A B |– C.
Proof:1. T, A |– C assumption2. T |– A C deduction Theorem:1 3. T |– C A meta-rule (d): 2, (the rule TR of natural deduction)4. T, C |– A deduction Theorem: 35. T, C |– B analogical to 4.6. T, C |– A B meta-rule (f): 4,57. A B |– (A B) de Morgan rule (prove it!)8. T, C |– (A B)meta-rule (d): 6,7 9. T |– C (A B) deduction theorem: 810. T |– A B C meta-rule (d): 9. (the rule TR)11. T, A B |– C deduction theorem: 10
Q.E.D.
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Proof calculi, Hilbert 39
Proofs of meta-rules (a sequence of rules)
Ad (i):Let T, A |– B; T, A |– B, we prove T |– B.Proof:
1. T, A |– B assumption2. T, A |– B assumption 3. T, A A |– B meta-rule (h): 1,24. T |– B meta-rule (a): 3
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Proof calculi, Hilbert 40
A Complete Calculus: if |= A then | A Each logically valid formula is provable
in the calculus The set of theorems = the set of logically
valid formulas (the red sector of the previous slide is empty)
Sound (semantic consistent) and complete calculus: |= A iff | A Provability and logical validity coincide in FOPL
(1st-order predicate logic) Hilbert calculus is sound and complete
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Proof calculi, Hilbert 41
Sound calculus: if | A (provable) then |= A (True)
WFF
|– ATheorems
Axioms
|= ALVF
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Proof calculi, Hilbert 42
Sound and complete calculus: | A (theorem) iff |= A (tautology)
WFF
|– ATheorems
Axioms
|= ALVF
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Proof calculi, Hilbert 43
Properties of a calculus: deduction rules, consistency
The set of deduction rules enables us to perform proofs mechanically, considering just the symbols, abstracting of their semantics. Proving in a calculus is a syntactic method.
A natural demand is a syntactic consistency of the calculus. A calculus is consistent iff there is a WFF such that is not
provable (in an inconsistent calculus everything is provable). This definition is equivalent to the following one: a calculus is
consistent iff a formula of the form A A, or (A A), is not provable.
A calculus is syntactically consistent iff it is sound (semantically consistent).
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Proof calculi, Hilbert 44
Sound and Complete Calculus: |= A iff | A Soundness
(an outline of the proof has been done) In 1928 Hilbert and Ackermann published a concise small book
Grundzüge der theoretischen LogikGrundzüge der theoretischen Logik, in which they arrived at exactly this point: they had defined axioms and derivation rules of predicate logic (slightly distinct from the above), and formulated the problem of completeness. They raised a question whether such a proof calculus is complete in the sense that each logical truth is provable within the calculus; in other words, whether the calculus proves exactly all the logically valid FOPL formulas.
Completeness Proof: Stronger version: if T |= , then T |– . Kurt Gödel, 1930 A theory T is consistent iff there is a formula which is not
provable in T: not T |– .
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Proof calculi, Hilbert 45
Strong Completeness of Hilbert Calculus: if T |= , then T |– The proof of the Completeness theorem is based on
the following Lemma:Each consistent theory has a model. if T |= , then T |– iff if not T |– , then not T |= {T } does not prove as well
( does not contradict T) {T } is consistent, it has a model M M is a model of T in which is not true is not entailed by T: T |=
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Proof calculi, Hilbert 46
Properties of a calculus: Hilbert calculus is not decidable
There is another property of calculi. To illustrate it, let’s raise a question: having a formula , does the calculus decide ?
In other words, is there an algorithm that would answer Yes or No, having as input and answering the question whether is logically valid or no? If there is such an algorithm, then the calculus is decidable.
If the calculus is complete, then it proves all the logically valid formulas, and the proofs can be described in an algorithmic way.
However, in case the input formula is not logically valid, the algorithm does not have to answer (in a final number of steps).
Indeed, there are no decidable 1st order predicate logic calculi, i.e., the problem of logical validity is not decidable in the FOPL.
(the consequence of Gödel’s Incompleteness Theorems)
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Proof calculi, Hilbert 47
Provable = logically true?Provable from … = logically entailed by …?
The relation of provability (A1,...,An |– A) and the relation of logical entailment (A1,...,An |= A) are distinct relations.
Similarly, the set of theorems |– A (of a calculus) is generally not identical to the set of logically valid formulas |= A.
The former is syntactic notion defined within a calculus, the latter notion is independent of a calculus, it is semantic.
In a sound calculus the set of theorems is a subset of the set of logically valid formulas.
In a sound and complete calculus the set of theorems is identical with the set of logically valid formulas.
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Proof calculi, Hilbert 48
Hilbert Calculus
WFF
|– ATheorems
Axioms
|= ALVF
???