Content

THE DEFINITE LANGUAGE 1

Content

Background

  • What is syntax

    In logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them. Syntax is concerned with the rules used for constructing, or transforming the symbols and words of a language, as contrasted with the semantics of a language which is concerned with its meaning

    The symbols, formulas, systems, theorems, proofs, and interpretations expressed in formal languages are syntactic entities whose properties may be studied without regard to any meaning they may be given, and, in fact, need not be given any.

Preface

  • The purpose of present work is to give a systematic exposition of such a mthod, namely, of the method of “logical syntax”.
  • The aim of logical syntax is to provide a system of cocncepts, a language, by the help of which the results of logical analysis will be exactly formulable. phylosophy is to be replaced by he logical syntax of the language of science.
  • Language I covers a narrow field of concepts, lanuage II is richer in modes of expression.

Introduction

What is logical syntax

  • The logical syntax mean the formal theory of the linguistic forms of a language-the systematic statement of the formal rules which govern it together with the development of consequences which follow from these rules.

  • The syntax of a language is supposed to lay down rules according to which the linguistic structures (e.g. the sentences) are to be built up from the elements (e.g. words). The chief task of logic is supposed to be that of formulating rules according to which judgemets may be inferred from other judgements; in other words, according to which conclusions may be drawn from premisses.

  • The logic can be studied with any degree of accuracy when it is based not on judgements (thoughts, or the content of thoughts) but rather on linguistic expressions. Sentence is the most important. The rules of logic are formal, logical characteristics of sentences and the logical relations are dependent on syntactic structure of the sentences. -> logic will become a part of syntax. -> difference: formation rules & transformation rules. -> designate as logical syntax the system which comprises the rules of formation and transformation

  • natural word-language (such as German & Latin) is unsystematic and logically imperfect structured. The same arises in the artificial word-languages.

  • Rules of formation and transformation permit of being formally apprehended.

  • -> Consider the syntax of two artificially constructed formal symbolic language (employ formal symbols instead of words).

  • The sentences, definitions, and rules of the syntax of a language are concerned with the forms of that language.

    1. how are them correctly expressed?
    2. is a kind of super-language necessary?
    3. a third language to explain the super-language?
    4. is it possible to formulate the syntax of a language with in the language itself?
    • -> it is possible to express the syntax of a language in the language itself.
    • start by constructing the syntax, then procced to formalize its concepts and thereby determine its logical character.

  • object-language, syntax-language

Language as calculi

  1. A system of conventions or rules is understood by a calculus.
  2. The rules are concerned with symbols about the nature, and relations of the nature are distributed in various classes.
  3. Any finite series of these symbols is called an expression of the calculus in question.
  • The rules of the calculus determine
    1. the condition under which an expression can be said to belong to a certain category of expressions. -> formation = syntactic rules
    2. under what conditions the transformation of one or more expressions into another or others are allowed. -> transformation = logical laws of deduction.
    • a system of a language is a calculus
    • every well-determined mathematical discipline is a calculus.
  • logical syntax is the same thing as the construction and manipulation of a calculus. Languages are the most important examples of calculi.
  • The syntax is only concerned with the formal properties of expressions, thw design of the individual symbols is indifferent.
  • Any series of any things will equally well serve as terms or expressions in a calculus, or in a language.
  • pure syntax and descriptive syntax ~ mathmetical geometry and physical geometry
    1. pure syntax concerned with the forms of sentences.

THE DEFINITE LANGUAGE 1

Rules of formation for language 1

Predicates and functors

  • previous knowledge:
    1. objects: proper names. e.g. house name, color names. name-language
    2. sysematic positional co-ordinates (symbols show the place of the objects in the system, and their positions in relation to one another). e.g. house number, color figures/letters. co-ordinate-language
  • In language 1, natural numbers as co-ordinates.
  • predicates: to express a property of an object, or of a position, or a relation between several objects or positions.
  • predicates are proper names for the properties of positions.
  • descripptive predicates: express empirical properties or relations
  • logical predicates:
  • functors: express properties or relations of position by means of numbers.
  • descriptive functors, logical functors.
  • numerical expression: an expression which in any way designates a number. e.g. “te(3), sum(3,4)”
  • sentence: an expression which corresponds to a propositional sentence of a word-language. e.g.“te(3)=5, sum(3,4)=7, blue(3)”

Syntactical gothic symbols

  • two expressions are equal when their corresponding symbols are equal symbols. If two symbols or two expressions are equal, they have same syntactical design.
  • An exprssion of I consists of an ordered series of symbols of I.
  • By a syntactic form we understand any kind or category of expressions which is syntactically determined.
  • Five kinds of symbols:
    1. eleven indivisual symbols:
    2. variables
    3. constant
    4. predicates
    5. fuctors

The junction symbols

  • The one-term or two-term junction symols are used to costruct a new sentences outof one or two sentences respectively.