Mathematical Logic

Mathematical Logic Marie Duží marie.duzi@vsb.cz Mathematical Logic

Texts to study: http://www.cs.vsb.cz/duzi Courses Mathematical Logic(English pages) Presentation of lectures Book: Gamut L.T.F., Logic, Language and Meaning, Chicago Press 1991, Vol. 1. Chapter 1, Chapter 2 (except of 2.7), Chapter 3, and Chapter 4 – Sections: 4.1, 4.2, 4.4. Mathematical Logic

Requirements to pass the course • Accreditation: • Written test • Min. 15 grades, max. 40 grades • No repetitions for the tests! • Requirement: obtaining at least 15 grades. • Exam: Written test (min.21 grades, max. 40 grades) • Oral exam (max. 20 grades) • Total: • at least 51 grades – good (3), • at least 66 grades – very good (2), • at least 86 grades – excellent (1) Mathematical Logic

1. Introduction What is logic about? What is the subject of logic? Logic is the science of correct, valid reasoning, or, in other words, the art of a valid argumentation What is an argument? Argument: On the assumption of true premises P1,...,Pn it is possible to reason that the conclusion Z is true as well: P1, ..., Pn⊢Z Example: On the assumption that it is Wednesday I belief that the lecture on Mathematical logic takes place today:Wednesday⊢Lecture on Logic Mathematical Logic

Introduction: valid arguments In this course we deal only with deductively valid arguments. Notation:P1,...,Pn |= Z The conclusion Z logically follows from the premises P1,..., Pn. Definition 1: The conclusion Z logically follows from the premises P1,...,Pn, denoted P1,...,Pn |= Z, iffunderno circumstancesit might happen that the premises were true and the conclusion false. In other words, the conclusion Z logically follows from the premises P1,..., Pn if and only if assuming that premises are true and the conclusion false yields a contradiction. Mathematical Logic

Introduction: valid arguments Example: As it is Wednesday today I believe that the lecture on ‘Mathematical Logic’ takes place: It is Wednesday  invalid Lecture on Math-Logic takes place Is it a deductively valid argument? No, it is not: It might happen that Duzi were sick and the lecture didn’t take place though it is Wednesday; what is missing in the argument for it be valid? A premise is missing, for instance that each Wednesday the lecture takes place. Each Wednesday the lecture on Logic takes place. It is Wednesday today  valid Today the lecture on Logic takes place. Mathematical Logic

(Deductively) invalid arguments: generalization (induction), abduction We will not deal with arguments that are not deductively valid, like: generalization (induction), abduction, and other –ductions  the subject of Artificial Intelligence (non-monotonic reasoning) Examples: Till now logic always took place on Tuesdays.  induction, invalid (Therefore) Logic will take place also this Tuesday All swans that I have seen till now are white.  induction, invalid (Therefore) All swans are white Mathematical Logic

Deduction, generalization (induction), abduction Examples: All rabbits in the hat are white. These rabbits are taken from the hat.  These rabbits are white. Deduction, valid These rabbits are taken from the hat. These rabbits are white.  (Probably) All rabbits in the hat are white. Generalization, Induction, invalid All rabbits in the hat are white. These rabbits are white.  (Probably because) These rabbits are from the hat. Abduction, invalid Seeking premises, causes of events, diagnosis of malfunctions Mathematical Logic

Examples of deductively valid arguments • He is at home or he has gone to a pub. If he is at home then he plays a piano. But he does not play a piano. ------------------------------------------------ Hence He has gone to the pub. Sometimes the arguments are so obvious that it seems as if we did not need any logic. Well, if he did not play a piano (3. premise), then he was not at home (2. premise), and according to the first premise he must have gone to the pub. We all use logic in our everyday life, we wouldn’t survive without logic: • All agarics (mushrooms) have a strong toxic effect. The mushroom I have picked up is an agaric. ---------------------------------------------------------------------- The mushroom I have picked up has a strong toxic effect. Will you examine the mushroom by tasting it, or will you rely on logic? Mathematical Logic

Examples of deductively valid arguments All agarics (mushrooms) have a strong toxic effect. This apple is an agaric. ---------------------------------------------------------------------- Hence  This apple has a strong toxic effect. The argument is valid. But the conclusion is evidently not true (false). Hence, at least one premise is false (obviously the second). Circumstances according to Definition 1 are particular interpretations (depending on the expressive power of the logical system). Logical connectives (‘and’, ‘or’, ‘if …then …’) and quantifiers (‘all’, ‘some’, ‘every’, …) have a fixed interpretation; we interpret elementary propositions and/or their parts, i.e. predicates and descriptions In our example, if ‘this apple’ and ‘agarics’ were interpreted in such a way that the second premise were true, the truth of the conclusion is guaranteed. We also say that the argument has a valid logical form. Mathematical Logic

Deductively valid arguments Logicis atoolthat helps us to discoverthe relation of logical entailment, to answer questions like „What follows from these assumptions“?, etc. • If the course is good then it is useful. • The lecturer is sharply demanding studiousness or the course is not useful. • But the lecturer is not demanding. -------------------------------------------------------------------------- Hence • The course is not good. • It helps our intuition that can sometimes fail. • The assumptions can be complicated, “enmeshed in negations and other connectives”, so that the relation of entailment is not obvious at first sight. • Similarly as all the mother-tongue speakers use intuitively rules of grammar without knowing the grammar explicitly (often not being able to formulate the rules). • But sometimes it is useful to consult the grammar book or a dictionary (in particular when taking part in a TV competition). Mathematical Logic

Examples of valid arguments • All men like football and beer. • Some beer-lovers do not like football. • Xaver likes only those who like football and beer. –––––––––––––––––––––––––––––––––––– • Xaver does not like some women. Necessarily, if premises are true the conclusion has to be true as well. Is this argument valid? Certainly, if Xaver likes only those who like football and beer (premise 3), then he does not like some beer-lovers (namely those who do not like football – according to the premise 2). Hence, (according to 1) he does not like some “non-men”, i.e., women. But according to the Definition 1 the argument is not valid: the argument is valid if necessarily, i.e., in all the circumstances (under all interpretations) in which the premises are true the conclusion is true as well. But: in our case those individuals that are not men would not have to be interpreted as women. A premise is missing, viz. the premise “who is not a man is a woman”. Moreover, to be precise, we should also specify that “who is a lover of something he likes that”. Mathematical Logic

Examples of valid arguments Hence: We have to state all the premises necessary for deriving the conclusion. • All men like football and beer. • Some beer-lovers do not like football. • Xaver likes only those who like football and beer. • Who is not a man is a woman. • Who is a lover of something he likes it. –––––––––––––––––––––––––––––––––––– • Xaver does not like some women. Now the argument is logically valid, it has a valid logical form. The conclusion is logically entailed by (follows from) the premises. We also say that the conclusion is informationally (deductively) contained in the premises. Mathematical Logic

Valid arguments in mathematics Argument A: No prime number is divisible by three. The number 9 is divisible by three. ––––––––––––––––––––––––––– valid Thenumber 9 is not a prime. Argument B: No prime number is divisible by six. The number eight is not a prime. ––––––––––––––––––––––––––– invalid The number eight is not divisible by six. Though in the case of argument B it cannot happen that the premises were true and the conclusion false, the argument is invalid. The conclusion is not logically entailed by the premises. If the term ‘eight’ were interpreted as the number 12, the premises would be true and the conclusion false. (The conclusion is not deductively contained in the premises) Mathematical Logic

Theorem of deduction; semantic variant The argument P1,...,Pn|= Z is valid if and only if the proposition of the form“ifP1and...andPnthen Z” P1 &...& Pn Z is analytically (necessarily) true. Notation: |=P1 ... Pn Z. Hence: P1,...,Pn|= Z (if and only if) P1,...,Pn-1|= (PnZ) P1,...,Pn-2|= ((Pn-1  Pn) Z) P1,...,Pn-3|= ((Pn-2 Pn-1 Pn) Z) … |= (P1...Pn)Z Mathematical Logic

Logical analysis of language Validness of an argument is determined by the meaning (interpretation) of particular sentences that are analyzed (formalized) in a less or more fine-grained way according to the expressive power of a logical system: • Propositional logic: makes it possible to analyze only the way in which a molecular sentence is composed from atomic sentences. The structure of atomic sentences is not examined, they contribute only by its truth value: True – 1, False – 0 (an algebra of truth values) • 1st-order Predicate logic: to some degree it makes it possible to analyze also the structure of elementary atomic sentences; namely the way in which properties and/or relations are ascribed to (tuples of) individuals. • 2nd-order Predicate logic: in addition, it makes it possible to analyze also properties of properties, propertied of functions and relations between them. • Modal logics (analyze “necessary” and “possible”), epistemic logics (knowledge), doxastic logics (of hypotheses) deontic logics (of commands),... • Transparent intensional logic (perhaps the most powerful system) – see the course “Natural Language Processing“. Mathematical Logic

Properties of valid arguments • A valid argument may have a false conclusion: • All primes are odd • The number 2 is not odd •  The number 2 is not a prime But then at least one premise has to be false In such a case we also say that the argument is not sound. A valid argument that is not sound may also be useful  a proof ad absurdum. If you want to show that your boss is not right, it is not diplomatic to say to him “You are not right”. Instead, you may argue by way of the proof ad absurdum: “Well, you say P – interesting, but P entails Q, and Q entails R, which is obviously false.” (Hence, P must have been false as well.) • Monotonicity: if an argument is valid then enriching the set of assumptions by another premise does not cancel the validity of the argument. Mathematical Logic

Properties of valid arguments • From contradictory (inconsistent) assumptions (such that it can never happen that all of them were simultanously true) any conclusion follows. • If I study hard then I’ll pass the exam. • I haven’t passed the exam though I studied hard.-------------------------------------------------------------------- •  (e.g.) My dog is playing a piano right now • Reflexivity: If A is one of the assumptions P1,...,Pn, thenP1,...,Pn|= A. • Transitivity: If P1, …, Pn |= Z and Q1, …, Qm, Z |= Z’, then P1, …, Pn, Q1, …, Qm |= Z’ . Break Mathematical Logic

Naïve theory of sets (George Cantor) • What is it a set? • A set is a collection of elements, and it is determined just by its elements;a set consisting of elements a, b, c is denoted: {a, b, c} • An element of a set can be again a set, a set may consist of no elements, it may beempty(denoted by ) ! • Examples: , {a, b}, {b, a}, {a, b, a}, {{a, b}}, {a, {b, a}}, {, {}, {{}}} • Sets are identical if and only if (iff) they have exactly the same elements (the principle of extensionality) • Notation: x  M– „x is an element ofM“ • a  {a, b}, a {{a, b}}, {a, b}  {{a, b}},  {, {}, {{}}},  {, {}}, but: x for anyx (i.e., for nox holds x  ). • {a, b} = {b, a} = {a, b, a}, but: {a, b}  {{a, b}}  {a, {b, a}} Mathematical Logic

Set-theoretical operations (create new sets from sets) • Union: A  B = {x | x  A orx  B} read: „The set of allx such that x is an element of A orx is an element of B.“ • {a, b, c}  {a, d} = {a, b, c, d} • {odd numbers}  {even numbers} = {natural numbers} – denotedNat • UiI Ai = {x| x  Aiforsomei  I} • LetAi = {x| x = 2.i for some i  Nat} • UiNatAi = the set of all even numbers Mathematical Logic

Set-theoretical operations (create new sets from sets) • Intersection: A  B = {x | x  A andx  B} read: „The set of allx such that x is an element of A andx is an element of B as well.“ • {a, b, c}  {a, d} = {a} • {even numbers}  {odd numbers} =  • iI Ai = {x| x  Aiforalli  I} • LetAi = {x| x  Nat, x  i} • TheniNatAi =  Mathematical Logic

Relations between sets • a setAis asubsetof a setB, denoted A B, iff each element of Ais also an element ofB. • a setAis aproper subset of a setB, denotedA B, iff each element of Ais also an element ofBbut not vice versa. {a} {a}  {a, b}  {{a, b}} !!! • It holds: A  B, iffA  B andA  B • It holds: A  B, iffA  B = B, iffA  B = A Mathematical Logic

Some other set-theoretical operations • Difference: A \ B = {x | x  A andx  B} • {a, b, c} \ {a, b} = {c} • Complement: Let A  M. The complement of A with respect to M is the setA’ = M \ A • Cartesian product: A  B = {a,b|aA, bB} • wherea,b je an ordered couple (the ordering is important: a is the first, b is the second) • It holds: a,b = c,diffa = c, b = d • But: a,b  b,a, though{a,b} = {b,a} !!! • generalization: A  …  Athe set of n-tuples, denoted also by An Mathematical Logic

Some other set-theoretical operations • Potential set: 2A = {B | B  A}, denoted also by P(A) 2{a,b} = {, {a}, {b}, {a,b}} 2{a,b,c} = {, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}} How many elements are there in 2A? If |A| is the number of elements (cardinality) of a set A, then 2Ahas 2|A| elements (hence the notation: 2A) 2{a,b} {a}= {, {a,a}, {b,a}, {a,a, b,a}} Mathematical Logic

Grafical picturing (in a universe U): S A: S\(PM) = (S\P)(S\M) S(x)  (P(x)  M(x)) S(x)  P(x)  M(x) B: P\(SM) = (P\S)(P\M) P(x)  (S(x)  M(x))  P(x)  S(x)  M(x) C: (S P) \ M S(x)  P(x) M(x) D: S  P  M S(x)  P(x)  M(x) E: (S  M) \ P S(x)  M(x) P(x) F: (P  M) \ S P(x)  M(x) S(x) G: M\(PS) = (M\P)(M\S) M(x)  (P(x)  S(x))  M(x)  P(x)  S(x) H: U \ (S  P  M) = (U \ S  U \ P  U \ S) (S(x)  P(x)  M(x))  S(x)  P(x)  M(x) A E C D G B F P M H Mathematical Logic

Russell’s paradox • Is it true that any collection of elements (i.e., a collection defined in an arbitrary way) can be considered to be a set? • It is normal that a set and its elements are entities of different types. Hence a “normal set” is not an element of itself. • Let N be a set of all normal sets: N = {M | M  M}. • Question: Is N N ? In other words, is N itself normalor not? • Yes? But then according to the definition of N it holds that N is normal, i.e., NN. • No? But then NN, hence N is normal, and therefore it belongs to N, i.e., NN. • Both the answers yield a contradiction; N is not well defined. The definition does not determine a collection of elements that could be considered to be a set. Mathematical Logic

Mathematical Logic