Principles of proof calculi Hilberts programme and Hilbert calculus Proof calculi, Hilbert 1 1921: Hilberts 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 infinity) infinity However, there is no upper bound on the number of steps we execute, we can always move a step further. (potential infinity) But at any point we will have acquired only a finite amount of experience and have taken only a finite number of steps. Thus, for a Kantian like Hilbert, the only legitimate infinity is a potential infinity, rather than the actual infinity. infinity mathematics is about symbols (?), mathematical reasoning Syntactic laws of symbol manipulation (?); consistency proof Proof calculi, Hilbert 2 Statements of Hilberts program

The main goal of Hilbert's program was to provide secure foundations for all mathematics. mathematics In particular this should include: A formalization of all mathematics; mathematics in other words all mathematical statements should be written in a precise formal language, and manipulated according to well defined rules. Completeness: Completeness a proof that all true mathematical statements can be proved in the formalism. Consistency: Consistency 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. Conservation: Conservation 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. Decidability: Decidability there should be an algorithm for deciding the truth or falsity of any mathematical statement. Proof calculi, Hilbert

3 Gdel's incompleteness theorems Kurt Gdel showed that most of the goals of Hilbert's program were impossible to achieve, at least if interpreted in the most obvious way. Gdel'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 Gdel's theorem, because at the time the notion of an algorithm had not been precisely defined. Proof calculi, Hilbert 4 Kurt Gdel, Albert Einstein 5 Kurt Gdel with an unknown local farmer Step No. 1; proof calculus, the goals Recall Completeness: Completeness 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 P1, , Pn |= C iff |= (P1 Pn) C Proof calculi, Hilbert 6 Formal systems, Proof calculi A proof calculus (of a theory) is given by: A. a language B. a set of axioms C. a set of deduction rules ad 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) Proof calculi, Hilbert 7 Hilbert-like calculus. Language: restricted FOPL Alphabet: 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 term if t1, , tn are terms, fn a functional symbol, then fn(t1, , tn) is a (functional) term 2. atomic formulas if t1, , tn are terms, Pn predicate symbol, then Pn(t1, , tn) is an atomic (well-formed)

formula 3. composed formulas Let 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. Proof calculi, Hilbert 8 Hilbert calculus Ad 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 Proof calculi, Hilbert 9

Hilbert calculus Ad 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) Proof calculi, Hilbert 10 Hilbert calculus Notes: 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. Proof calculi, Hilbert 11 Hilbert calculus 2. 3. 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. In Hilbert calculus we do not use the indirect proof. Proof calculi, Hilbert 12 Hilbert calculus 4. 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. Proof calculi, Hilbert 13 A sound calculus: if | A (provable) then |= A (True) WFF |= A LVF Axioms | A Theorems

Proof calculi, Hilbert 14 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. Proof calculi, Hilbert 15 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.

Proof calculi, Hilbert 16 A Proof from Assumptions A (direct) proof of a formula A from assumptions A1,,Am is a sequence of formulas (proof steps) B1, Bn such that: A=B (the proved formula A is the last step) n each B (i=1,,n) is either i an axiom, or an assumption 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.

Proof calculi, Hilbert 17 Examples of proofs Proof 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:2,1 4. A (A A) axiom A1: B/A 5. A A MP:4,3 Q.E.D. Hence: | A A . Proof calculi, Hilbert 18 Examples of proofs

Proof of: A B, B C | A C (transitivity of implication TI): 1. A B assumption 2. B C assumption 3. (A (B C)) ((A B) (A C)) axiom A2 4. (B C) (A (B C)) axiom A1 A/(B C), B/A 5. A (B C) MP:2,4 6. (A B) (A C) MP:5,3 7. A C MP:1,6 Q.E.D. Hence: A B, B C | A C . Proof calculi, Hilbert 19 Examples of proofs | Ax/t xAx

(the ND rule existential generalisation) Proof: 1. x Ax Ax/t axiom A4 2. x Ax x Ax theorem of type C C (see below) 3. x Ax Ax/t C D, D E | C E: 2, 1 TI 4. x Ax = xAx Intr. acc. (by definition) 5. xAx Ax/t substitution: 4 into 3 6. [xAx Ax/t] [Ax/t xAx] axiom A3 7. Ax/t xAx MP: 5, 6 Q.E.D. Proof calculi, Hilbert 20 Examples of proofs

A Bx | A xBx (x is not free in A) Proof: 1. A Bx assumption 2. x[A Bx] Generalisation:1 3. x[A Bx] [A xBx] axiom A5 4. A xBx MP: 2,3 Q.E.D. Proof calculi, Hilbert 21 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. Proof calculi, Hilbert 22 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. Proof calculi, Hilbert 23 The Theorem of Deduction 2. 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 A1 3. 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 assumption 2. A (Cj Ci) induction assumption 3. (A (Cj Ci)) ((A Cj) (A Ci)) A2 4. (A Cj) (A Ci) MP: 2,3 5. (A Ci)MP: 1,4 Q.E.D Proof calculi, Hilbert 24 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. Proof calculi, Hilbert 25 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 ? Proof calculi, Hilbert 26 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). Proof calculi, Hilbert 27 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

| A ((A B) B) | A (B A) ax. A1 | A A Rule(s) A, A B | B (MP rule) A | B A; A, B | A A | A | (A B) ((B C) (A C)) A B | (B C) (A C); A B, B C | A C /rule TI/ Proof calculi, Hilbert 28 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 Proof calculi, Hilbert 29 Some proofs Ad 1. | A (A B); i.e.: A, A | B. Proof: (from a contradiction |-- anything) 1. A assumption 2. A assumption 3. (B A) (A B) A3 4. A (B A) A1 5. B A MP: 2,4 6. A B MP: 5,3 7. B

MP: 1,6 Q.E.D. Proof calculi, Hilbert 30 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. Proof calculi, Hilbert 31 Some proofs Ad 3. | A A; i.e.: A | A. Proof: 1. A 2. (A A) (A A)

3. A (A A) 4. A A 5. A A 6. A Q.E.D. Proof calculi, Hilbert assumption axiom A3 theorem ad 1. MP: 1,3 MP: 4,2 MP: 1,5 32 Some proofs Ad 4. | A A; i.e.: A | A. Proof: 1. A assumption 2. (A A) (A A)axiom A3 3. A A theorem ad 3. 4. A A

MP: 3,2 Q.E.D. Proof calculi, Hilbert 33 Some proofs Ad 5. | (A B) (B A), i.e.: (A B) | (B A). Proof: 1. A B assumption 2. A A theorem ad 3. 3. A B TI: 2,1 4. B B theorem ad 4. 5. A B TI: 1,4 6. A B TI: 2,5 7. (A B) (B A) axiom A3 8. B A

MP: 6,7 Q.E.D. Proof calculi, Hilbert 34 Some proofs Ad 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) assumption 2. (A (A B)) ((A B) A) theorem ad 5. 3. A (A B) theorem ad 1. 4. (A B) A MP: 3,2 5. A MP: 1,4 6. A A

theorem ad 3. 7. A MP: 5,6 Q.E.D. Proof calculi, Hilbert 35 Some meta-rules Let T is any finite set of formulas: T = {A 1, 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) 36 Hilbert then T | B. (i) if T, A | B and if T, Proof Acalculi, | B, 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 ID 2. T, A | A B meta-rule (b): 1 3. T, A B | Cassumption 4. T, A | C meta-rule (d): 2,3 Q.E.D. 5. T, B | C analogically to 4. Q.E.D. Proof calculi, Hilbert 37 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 assumption 2. 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: 3 5. T, C | B analogical to 4. 6. T, C | A B meta-rule (f): 4,5 7. 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: 8 10. T | A B C meta-rule (d): 9. (the rule TR) 11. T, A B | C deduction theorem: 10 Q.E.D.

Proof calculi, Hilbert 38 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 assumption 2. T, A | B assumption 3. T, A A | B meta-rule (h): 1,2 4. T | B meta-rule (a): 3 Proof calculi, Hilbert 39

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 Proof calculi, Hilbert

40 Sound calculus: if | A (provable) then |= A (True) WFF |= A LVF Axioms | A Theorems Proof calculi, Hilbert 41 Sound and complete calculus: | A (theorem) iff |= A (tautology) WFF |= A LVF

Axioms | A Theorems Proof calculi, Hilbert 42 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). Proof calculi, Hilbert 43 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 Grundzge der theoretischen Logik, 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 Gdel, 1930 A theory T is consistent iff there is a formula which is not provable in T: not T | . Proof calculi, Hilbert 44 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 |= Proof calculi, Hilbert 45 Properties of a calculus: Hilbert calculus is not decidable

There is another property of calculi. To illustrate it, lets 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 Gdels Incompleteness Theorems) Proof calculi, Hilbert 46 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. Proof calculi, Hilbert 47 Hilbert Calculus ???

WFF A |= F LV Axioms | A Theorems Proof calculi, Hilbert 48