a first course in logic an introduction to model theory proof theory computability and complexity sep 2004

[...]... A means x is not an element of A, ∅ denotes the unique set containing no elements, A ⊂ B means every element of set A is also an element of set B, A ∪ B denotes the union of sets A and B, A ∩ B denotes the intersection of sets A and B, and A × B denotes the Cartesian product of sets A and B Recall that the union A ∪ B of A and B is the set of elements that are in A or B (including those in both A and. .. Cantonese, a logic is an artificial language having a precisely defined syntax One purpose for such artificial languages is to avoid the ambiguities and paradoxes that arise in natural languages Consider the following English sentence Let n be the smallest natural number that cannot be defined in fewer than 20 words Since this sentence itself contains fewer than 20 words, it is paradoxical A logic avoids such... an English word, on the 2 Propositional logic other hand, always depends on the context For example, ∧ represents a concept that is similar but not identical to and. ” For atomic formulas A and B, A ∧ B always means the same as B ∧ A This is not always true of the word and. ” The sentence She became violently sick and she went to the doctor does not have the same meaning as She went to the doctor and. .. relations (such as < for less than) and functions (such as + for addition) Depending on what our interests are, we may consider these sets with any number of various functions and relations The interplay between mathematical structures and formal languages is the subject of model theory First- order logic, containing various relations and functions, is the primary language of model theory We study model. .. are summarized by Tables 1.1 and 1.2 and the definitions of the symbols ∨, →, and ↔ The semantics and the rules for deduction that follow from the semantics are implicit in the words “not,” and, ” “or,” “implies,” and “if and only if” (although this correspondence is not exact) For example, if (A ∧ B) is true (truth value 1), then we can deduce that both A and B are true And if A → B and A both have truth... of A to S ∪ S0 has the same value for F0 , then we define A( F0 ) to be this value Example 1.10 Let A and B be atomic formulas Let A be the assignment of {A, B} defined by A( A) = 1 and A( B) = 0 Then A( A ∧ B) = 0, A( A ∨ B) = 1, A( A ∧ (C ∨ ¬C)) = 1, and A( B ∨ (C ∧ ¬C)) = 0 8 Propositional logic The reason A( A ∧ (C ∨ ¬C)) = 1 is that A( A) = 1 and, no matter what truth value we assign to C, (C ∨ ¬C) has... both A ⊂ B and B ⊂ A In particular, the order and repetition of elements within a set do not matter For example, A = {α, β, γ} = {γ, β, α} = {β, β, α, γ} = {γ, α, β, β, α} Note that A ⊂ B includes the possibility that A = B We say that A is a proper subset of B if A ⊂ B and A = B and A = ∅ A set is essentially a database that has no structure For an example of a database, suppose that we have a phone... whereas the intersection A ∩ B is the set of only those elements that are in both A and B The Cartesian product A × B of A and B is the set of ordered pairs (a, b) with a ∈ A and b ∈ B We simply write A2 for A × A Likewise, for n > 2, An denotes the Cartesian product of An 1 and A This is the set of n-tuples (a1 , a2 , , an ) with each ai ∈ A For convenience, A1 (the set of 1-tuples) is an alternative... as Fagin’s theorem) and o state them precisely in due time Let us now end our preliminary ramblings and begin our study of logic 1 Propositional logic 1.1 What is propositional logic? In propositional logic, atomic formulas are propositions Any assertion will do For example, A = “Aristotle is dead,” B = “Barcelona is on the Seine,” and C = “Courtney Love is tall” are atomic formulas Atomic formulas... relationship between complexity and logic on its head We show that, in a certain setting (namely, graph theory) the complexity classes of P and NP (and others) can be defined as logics For example, Fagin’s Theorem states that (for graphs) NP contains precisely those decision problems that can be expressed in second-order existential logic So the P = NP problem and related questions can be rephrased as . Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey.