Logic Preliminaries¶
2 Further preliminaries
2.1 Use and mention
2.1.1 Use : name (identifier)
2.1.2 Mention : use the name for reference (object which identifier indicates)
2.2 Sense and denotation
2.2.1 Sense (an expression) : meaning
2.2.2 Denotation (an expression) : objects which it refers
2.2.3 Condition of sense and denotation
2.2.3.1 The denotation (a complex expression) is a function of the denotations of its parts
2.2.3.2 The sense (a complex expression) is a function of the senses of its parts
2.2.4 Ordinary vs Oblique sense/denotation
2.2.4.1 Oblique denotation = Ordinary sense
2.2.4.2 Direct occurrence (an occurrence of a name or description in a expression) : in the context it has ordinary sense
2.3 Variables : expression of generalization, must specify substituents
2.3.1 Substituents : a set of values which could substitute a variable
2.3.2 Values : names
2.4 Sentence forms
2.4.1 Replacement of a direct occurrence of a sentence with the expression with same ordinary denotation does not change its truth value
2.4.2 Replacement of an indirect occurrence of a sentence with the expression with same ordinary denotation can change its truth value
2.4.3 Sentence form : it is an expression that is a sentence or is obtainable from a sentence by replacing some or all direct occurrences of names by variables
2.4.4 Quantifier : often asserted in sentence forms
2.4.4.1 Universal quantifier : For every x, ..
2.4.4.2 Existential quantifier : There is an x such that …
2.4.5 Variables
2.4.5.1 Free occurrences : needs additional quantifiers or replacements to obtain a sentence
2.4.5.2 Bound : already has those
2.5 Description forms
2.5.1 Replacement of a direct occurrence of a description with the expression with same ordinary denotation does not change its truth value
2.6 Set ~ Class
2.6.1 Elements : objects constituting a set ~ members
2.6.1.1 Sets having same members are identical
2.6.2 Empty set : no members
2.6.3 Universal set : set of all objects satisfying (x is identical with x)
2.6.4 원소나열법 : notation for any finite number of occurrences of variables, names, or descriptions
2.6.5 A subset of B : every element of a set A is also an element of a set B ~ included in
2.6.5.1 Φ⊂A
2.6.5.2 A⊂A
2.6.5.3 A⊂(Universal set)
2.6.5.4 If A⊂B and B⊂A, then A = B
2.6.5.5 If A⊂B and B⊂C, then A⊂C
2.6.6 The Union of A and B : a set of all members belonging to A or to B
2.6.7 The Intersection of A and B : a set of all members belonging to A and to B
2.6.8 The Complement of A : a set of all members not belonging to A
2.6.9 Russell’s Antinomy : K = {x | x is not an element of x} K ∈K or K not∈ K ?
2.6.9.1 Give up Assumption that all sets can themselves be members of sets
2.6.10 Relation : any set of ordered n-tuples of objects is an n-ary relation
2.6.10.1 Binary Relation : x R y = <x,y> ∈ R
2.6.10.2 Domain : a set of all objects x s.t. for some y , x R y
2.6.10.3 Converse domain : a set of all objects y s.t. for some x, x R y
2.6.10.4 Field (binary relation) : (Domain) U (Converse domain)
2.6.10.5 Converse (binary relation R) : for all objects x, y , x R y iff y S x
2.6.10.6 Function (a binary relation R) : for all objects x, y, z , if x R y and x R z , then y = z
2.6.10.7 1-1 relation (binary relation R) : R and its converse are both function
2.6.10.8 N-ary operation (n+1-ary relation R) : with respect to set D, for each n-tuple <x1, x2, … , xn> of objects in D there uniquely exists an object y in D s.t. <x1,x2,..,xn,y> ∈ R
2.7 Object-language and Metalanguage
2.7.1 Explain one language by using another, then former is Object-language and latter is Metalanguage (ex. artificial language explained by English)
2.7.2 Metalinguistic variables : variables of the metalanguage, Greek letters