1 / 23

Proof calculi

Proof calculi. Lesson 11. Ancient antinomies, paradoxes. Zeno’s paradoxes Achilles & tortoise Achilles is unable to catch up with, not to mention overtake a tortoise. The quick runner Achilles is engaged in a race with a slow tortoise who has been granted a head start of 100 m.

smcduff
Télécharger la présentation

Proof calculi

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Proof calculi Lesson 11

  2. Ancient antinomies, paradoxes • Zeno’s paradoxes • Achilles & tortoise • Achilles is unable to catch up with, not to mention overtake a tortoise. • The quick runner Achilles is engaged in a race with a slow tortoise who has been granted a head start of 100 m. • Though Achilles runs 10 times faster than the tortoise, Achilles will never overtake the tortoise if both run at a constant speed. • While Achilles is covering the gap of 100 m between himself and the tortoise, the tortoise creates a new gap of 10 m. • To Achilles’s great frustration, while he is covering this new gap, the slow but steady animal is creating a new one of 1 m; and so on, ad infinitum. • No matter how quickly Achilles closes each new gap, the slow but steady tortoise always opens a new one. • Hence, Achilles can never catch up with or overtake the tortoise.

  3. Ancient antinomies, paradoxes • Zeno’s ‘motion’ paradoxes • The arrow • the flying arrow is at rest, which result follows from the assumption that time is composed of moments …. • he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always in a now, the flying arrow is therefore motionless. • (Aristotle Physics, 239b30) • Zeno abolishes motion, saying “What is in motion moves neither in the place it is nor in one in which it is not”. • (Diogenes Laertius Lives of Famous Philosophers, ix.72) • This argument against motion explicitly turns on a particular kind of assumption of plurality: that time is composed of moments (or ‘nows’) and nothing else. • Consider an arrow, apparently in motion, at any instant. First, Zeno assumes that it travels no distance during that moment—‘it occupies an equal space’ for the whole instant, because at that instant it has no time to move. • But the entire period of its motion contains only instants, all of which contain an arrow at rest, and so, Zeno concludes, the arrow cannot be moving.

  4. Mathematics, late 19th century, the beginning of 20th century • Those ancient paradoxes began to crop up again !!! • Georg Cantor, 1874 • Naïve theory of sets • Bertrand Russell – paradox! • LetN be the set ofallnormal sets: • N = {M | M  M} • Question: isN N ? • These paradoxes arise from our involving actual infinity into the reasoning; yet, they undermine our belief in rational reasoning • But, we don’t want to give up the beautiful paradise of Cantor’s theory of sets

  5. Historical background • The reason why proof calculi have been developed can be traced back to the end of 19th century. At that time formalization methods had been developed and various paradoxes arose. All those paradoxes arose from the assumption on the existence of actual infinities. • To avoid paradoxes, David Hilbert (a significant German mathematician) proclaimed the program of formalisation of mathematics (logistic).The idea was simple: to avoid paradoxes we will use only finitist methods: • First: • start with a decidable set of certainly (logically) true sentences, • use truth-preserving rules of deduction, and • infer all the logical truths. • Second, • begin with sentences true in the area of interest (interpretation domain), • use truth-preserving rules of deduction, and • infer all the truths about this area under scrutiny. • In particular, he intended to axiomatisemathematics in this way, in order to avoid paradoxes.

  6. ‘pre-Hilbert’ formalists • „Mathematics is a game with symbols“ • A simple system S: Constants: , Predicates:  Axioms of S: (1) x (xx) (2) x x (3)  Inference rules: MP (modus ponens), E (general quantifier elimination), I (existential quantifier insertion) Theorem: Proof: 1.  axiom 3 2. xx I, 1. 3. x x axiom 2 4.  MP, 2,3 5. x (xx) axiom 1 6. E, 5 7.  MP, 4,6 It is impossible to develop mathematics in such a purely formalist way. We must take into account the meaning of symbols, otherwise, we cannot use it Again, itis a consequenceofGödel’s Incompleteness theorem

  7. Historical background • Hilbert believed that these goals can be met. • Kurt Gödel(the greatest logician of the 20th century) proved the completeness of the 1st order predicate calculus, which was expected. He even proved the strong completeness: if SA |= T then SA |– T (SA – a set of assumptions). That is, the first goal had been met. • But Hilbert wanted more: he believed that all the truths of mathematics can be proved in this mechanic finite way (the second goal). • That is, that a theory of arithmetic (e.g. Peano) is complete in the following sense: each formula is in the theory decidable, i.e., the theory proves either the formula or its negation, which means that all the formulas true in the intended interpretation over the set of natural numbers are provable in the theory: • Gödel’s theorems on incompleteness give a surprising result: there are true but not provable sentences of natural numbers arithmetic. • Hence Hilbert program is not fully realizable (the second goal cannot be met).

  8. Formal systems; Proof calculi • are build up to meet the first Hilbert’s goal; to prove all the logical truths, and all the logically valid arguments • second Hilbert’s goal logical theories of arithmetics (Robinson, Peano) built up within a proof calculus A proof calculus (of a theory) is given by: • a language • a set of axioms • 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 an infinite set of well-formed formulas - WFF)

  9. Proof calculi; Example of a language: FOPL Alphabet: 1. logical symbols: (countable set of) individual variablesx, 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. termseach constant and each variable is an atomic termif t1, …, tnare terms, fn a functional symbol, then fn(t1, …, tn) is a (functional) term 2. atomic formulasif t1, …, tnare 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, (AB), (AB), (AB), (AB), are well-formed formulas.Let A be a well-formed formula, x a variable. Then xA, xA are well-formed formulas. 4. Nothing is a WFF unless it so follows from 1.-3.

  10. Proof calculi Ad b)The set ofaxiomsis a chosen subset of the set of WFF. • The axioms are chosen as the basic (analytically true) formulas that are not being proved, or rather prove themselves. Example: {pp, pp}. Ad c)Thededuction rulesare 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. Example: pq, p |– q (modus ponens) pq |– p, q (conjunction elimination) pq, q |– p (modus tollens)

  11. Proof calculi • 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.

  12. A Proof from Assumptions A (direct) proof of a formula Afrom assumptions A1,…,Amis 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 assumptionAk (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 provablefrom A1,…,Am, denoted A1,…,Am|– A, if there is a proof of A from A1,…,Am.

  13. A Proof from Assumptions An indirect proof of a formula Afrom assumptions A1,…,Am is a sequence of formulas (proof steps) B1,…Bn such that: • each Bi (i=1,…,n) is either • an axiom, or • an assumptionAk (1 km), or • an assumptionA of theindirect proof (formula A that is to be proved is negated) • Bi is derived from the previous Bj (j=1,…,i-1) using a rule of the calculus. • Some Bk contradicts to Bl, i.e., Bk = Bl(k {1,...,n}, l {1,...,n})

  14. Why do we build up a proof calculus? • To prove • Logically valid formulas (tautologies), recall the definition of a proof of a formula A • Logically valid arguments, recall the definition of a proof of a formula A from assumptions A1, …, An.

  15. Sound calculus: if | A (theorem) then |= A (logically true) WFF |= A LVF Axioms |– A Theorems

  16. 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 and complete calculus: • |= A iff | A • Provability and logical validity coincide in FOPL (1st-order predicate logic) • There are sound and complete calculi for the FOPL, e.g.: Hilbert-likecalculi, Gentzensequent calculi, natural deduction, resolutionmethod, … • But, provability in a calculusis a completelydifferentnotionfromlogical validity !

  17. 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 under every interpretation), and the • deduction rules have to make it possible to prove logically valid formulas. For this reason the proofs must be truth-preserving in the following sense: • A1,…,Am|– B1,…,Bn • if all the formulas A1,…,Amare true in an interpretation I, then B1,…,Bn are also true in this interpretationI.

  18. The Theorem of Deduction • A sound proof calculus should meet the following Theorem of Deduction. Theorem of deduction. A formula  is provable from assumptions A1,…,Am, iff the formula Am is provable from A1,…,Am-1. Symbolically: A1,…,Am |–  iff A1,…,Am-1 |– (Am). • In a sound calculus meeting the Deduction Theorem the following equivalence holds: If A1,…,Am |– if and only ifA1,…,Am |= . • If the calculus is sound and complete, then provability coincides with logical entailment: • A1,…,Am |– iffA1,…,Am |= .

  19. Properties of a calculus: axioms • The set of axiomsof a calculus is non-empty and decidablein the set of WFFs (otherwise the calculus would not be reasonable, for we couldn’t execute proofs if we did not know which formulas are axioms). • It means that there is an algorithm that for any WFF  given as its input answers in a finite number of steps an output Yes or NO on the question whether  is an axiom or not. • A finite set is trivially decidable. The set of axioms can be infinite. In such a case we define the set either by an algorithm or (usually) as a finite set of axiom schemata. • The set of axioms should be minimal, i.e., each axiom is independent of the other axioms (not provable from them).

  20. Properties of a calculus: deduction rules, consistency • The set of deduction rulesenables us to make the proofs mechanically, considering just the symbols, abstracting of their semantics. Proving in a calculus is a syntactic method. • A natural demand is a syntactic consistencyof the calculus. • A calculus is (syntactically) consistentiff there is a WFF  such that  is not provable (in an inconsistent calculus everything is provable). • This definition is equivalent to the following, more natural 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).

  21. Properties of a calculus: (un)decidability • There is another desirable property of a calculus. 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 / No the question whether anywell-formed formula is logically valid? • If there is such an algorithm, then the calculus is decidable. • Calculi of propositional logic are decidable. • However,there are no decidable 1st-order predicate logic calculi, i.e.,the problem of logical validity is not decidable in the FOPL. Now consider this. • If the calculus is complete, then it proves all the logically valid formulas, and the proofs can be described in an algorithmic way. • Hence, the algorithm answers YES, the formula is logically valid. • How come that there are no decidable calculi for FOPL? • The problem is this. If the given input formula  is just satisfiable rather than logically valid, the algorithm does not have to answer in a final number of steps. It may get captured in a loop. • (the consequence of Gödel’s Incompleteness Theorems)

  22. 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 set of theorems is a syntactic matter and defined within a calculus, while the set of logically valid formulas is a semantic matter, independent of provability in a calculus. • 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 formulas.

  23. A sound calculus: if | A (theorem) then |= A (logically true)complete calculus: | A (theorem) iff |= A (logically true) the pink area is empty WFF |= A LVF Axioms |– A Theorems

More Related