Logic and discrete mathematics (HKGAB4) What is logic? - PDF document

Logic, Lecture I Introduction to logics Logic, Lecture I Introduction to logics Logic and discrete mathematics (HKGAB4) What is logic? http://www.ida.liu.se/ ∼ HKGAB4/ “All rational inquiry depends on logic, on the ability The first approximation: logic is the science of correct of people to reason correctly most of the time, and, when reasoning, i.e., reasoning based on correct (sound) arguments. they fail to reason correctly, on the ability of others to point A correct ( sound ) argument is one in which anyone who out the gaps in their reasoning” (Barwise & Etchemendy) accepts its premises should also accept its conclusions. To see whether an argument is correct, one looks at the connec- Logic: contents tion between the premisses and the conclusion. One does not judge whether there are good reasons for accepting the premisses, but 1. Logic: informal introduction, syntactic and semantic per- whether person who accepted the premisses, for whatever reasons, spective. Meta properties (soundness, completeness). good or bad, ought also accept the conclusion. 2. Logical connectives and 0-1 reasoning. Examples 3. Introduction to formal reasoning: Fitch format (notation). 1. Correct arguments: 4. Fitch rules for propositional connectives. • if x is a parent of y , and y is a parent of z , 5. Quantifiers. then x is a grandparent of z 6. Fitch rules for quantifiers. • if A and B is true, then A is true. 7. Normal forms for formulas and reasoning by resolution. 2. Incorrect arguments: 8. Logic as a database querying language. • if A implies B then B implies A • if A or B is true, then A is true. 9. Modal logics. 3. are the following arguments correct? • if A implies B then not B implies not A • if A is true, then A or B is true. � A. Sza� � A. Sza� c las - 1 - c las - 2 - Logic, Lecture I Introduction to logics Logic, Lecture I Introduction to logics What is logic? – continued What is logic? – continued Logical formalisms are applied in many areas of science as a Traditionally, there are two methodologies to introduce basis for clarifying and formalizing reasoning. a logic: Intuitively, a logic is defined by the (family of) language(s) it • syntactically , via a notion of a proof and proof system uses and by its underlying reasoning machinery. • semantically , via a notion of a model, satisfiability and truth. The intensive use of formal reasoning techniques resulted in defining hundreds, if not thousands, of logics that fit nicely to particular application areas. Both methodologies first require to chose a language that suits best a particular application. For example, We then first need to clarify what do we mean by a logic. 1. talking about politics we use terms “political party”, “prime min- In order to make any reasoning fruitful, we have ister”, “parliament”, “statement”, etc. etc. 1. to decide what is the subject of reasoning or, in other words, what 2. talking about computer science phenomena we use terms “soft- are we going to talk about and what language is to be used ware”, “program execution”, “statement”, etc. etc. 2. to associate a precise meaning to basic notions of the language, in order to avoid ambiguities and misunderstandings Of course we use different vocabulary talking about different areas. 3. to state clearly what kind of opinions (sentences) can be formu- lated in the language we deal with and, moreover, which of those opinions are true (valid), and which are false (invalid). Logical language is defined by means of basic concepts, formulas and logical connectives or operators. Connectives and operators have a fixed meaning. Vocabularies reflecting particular application domains Now we can investigate the subject of reasoning are flexible. via the validity of expressed opinions. Such an abstraction defines a specific logic. � A. Sza� � A. Sza� c las - 3 - c las - 4 -

Logic, Lecture I Introduction to logics Logic, Lecture I Introduction to logics What is a logical language? Elements of logical language Language of a logic is a formal language (as defined in the Building blocks of logical languages are usually among the discrete math part of the course), reflecting natural language following: phenomena and allowing one to formulate sentences (called formulas ) about a particular domain of interest. • individual constants ( constants , for short), representing particular individuals , i.e., elements of the underlying doamin – examples: 0 , 1 , John Example • variables , representing a range of individuals, – examples: x, y, m, n Language of arithmetics of natural numbers consists of: • function symbols , representing functions, • constants, e.g., 0 , 1 , 2 , 3 , . . . – examples: + , ∗ , father () • variables, e.g., l, m, n, k, . . . • relation symbols , representing relations, – examples: = , ≤ , � • function symbols, e.g., addition (+) or multiplication ( ∗ ) • logical constants : True , False , sometimes also other, • relation symbols, e.g., = or ≤ . – examples: Unknown , Inconsistent Examples of formulas: • connectives and operators , allowing one to form more n ≤ 2 ∗ n + 1 complex formulas from simpler formulas, n ∗ ( k + 1) = n ∗ k + n – examples of connectives: “and”, “or”, “implies”, n ∗ n ∗ n + 2 ∗ n ∗ n + 3 = 20 – examples of operators: “for all”, “exists”, “is necessary”, “always” However, n 3 + 2 ∗ n 2 + 3 = 20 is a formula only if n 3 and n 2 are • auxiliary symbols , making notation easier to understand operations allowed in the language. – examples: “(”, “)”, “[”, “]”. � A. Sza� � A. Sza� c las - 5 - c las - 6 - Logic, Lecture I Introduction to logics Logic, Lecture I Introduction to logics Why “function/relation symbols” Terms (expressions) instead of “functions/relations”? Expressions formed using individual constants, variables and In natural language names are not individuals they denote! function symbols, are called terms . Intuitively, a term represents a value of the underlying domain. A ground term is a term without variables. Examples 1. Name “John” is not a person named “John”. Examples 2. An individual can have many names, 1. In arithmetics the following expressions are terms: – e.g., “John” and “father of Jack” can denote the same person. 1 + 2 + 3 + 4 ∗ 2 – a ground term 3. An individual may have no name – e.g., we do not give separate ( n + 2 ∗ k ) ∗ 5 + n ∗ n – a term (but not ground) names for any particle in the universe. but the following are not: 4. Many different individuals may have the same name, – e.g., “John” denotes many persons. 1 + 2 + 3 + . . . + 10 – “ . . . ” is not a symbol in arithmetics 5. Some names do not denote any real-world individuals, – e.g., “Pegasus”. ( n + 2 ∗ k ) ∗ 5 ≤ 20 – “ ≤ ” is a relation symbol. 2. If father and mother are function symbols and John is a con- stant then the following expressions are (ground) terms: In logic symbols correspond to names. father ( John ) A function/relation symbol is not a function/relation, mother ( father ( John )) but its name. father ( father ( father ( John ))) However, comparing to natural language, usually mother ( father ( mother ( John ))) . a symbol denotes a unique individual . � A. Sza� � A. Sza� c las - 7 - c las - 8 -