Hardegree, Metalogic, Formal Languages – 2 page 1 of 28 2 Formal Languages – 2 1. Introduction ......................................................................................................................................3 2. The Meta-Language ..........................................................................................................................3 3. Prefix, Infix, and Postfix...................................................................................................................3 1. Prefix Format........................................................................................................................3 2. Postfix Format......................................................................................................................4 3. Infix Format..........................................................................................................................4 4. Prefix (Polish) Formatted ZOL’s......................................................................................................4 1. General Vocabulary..............................................................................................................4 2. General Rules Of Formation.................................................................................................4 5. Abstract Example of a Polish Formatted ZOL..................................................................................4 1. Vocabulary (lexicon)............................................................................................................4 2. Rules of Formation...............................................................................................................5 6. Concrete Example of a Polish Formatted ZOL.................................................................................5 1. vocabulary............................................................................................................................5 2. rules of formation: ................................................................................................................5 7. Postfix Formatted ZOL’s ..................................................................................................................5 8. Infix (Algebraic) Formatted ZOL’s ..................................................................................................6 1. Vocabulary...........................................................................................................................6 2. Rules of formation................................................................................................................6 9. Abstract Example of an Algebraically Formatted ZOL ....................................................................6 1. vocabulary (lexicon) ............................................................................................................6 2. rules of formation.................................................................................................................6 10. Concrete Example of an Algebraically Formatted ZOL....................................................................6 1. vocabulary............................................................................................................................6 2. rules of formation.................................................................................................................7 11. Introduction ......................................................................................................................................8 12. First-Order Languages......................................................................................................................8 13. The Rules of Formation for First-Order Languages..........................................................................9 14. First-Order Theories ........................................................................................................................9 15. Example 1: Peano Arithmetic........................................................................................................10 16. Example 2: Pure First-Order Logic...............................................................................................11 17. Example 3: Elementary Group Theory ..........................................................................................11 18. Example 4: Elementary Boolean Algebra......................................................................................12 19. Example 5: Classical Set Theory ..................................................................................................13 20. Example 6: Modern Set Theory.....................................................................................................13 21. Introduction ....................................................................................................................................14 22. The Syntax of Simple Second-Order Languages.............................................................................15 1. Logical Vocabulary............................................................................................................15 2. Non-Logical Vocabulary....................................................................................................16 3. Rules of Formation.............................................................................................................16 23. Example 1: Second-Order Arithmetic ............................................................................................17 24. Example 2: Second-Order Set Theory............................................................................................17 Hardegree, Metalogic, Formal Languages – 2 page 2 of 28 1. Classical Set Theory ..........................................................................................................18 2. Modern Set Theory.............................................................................................................18 25. Introduction ....................................................................................................................................18 26. The Fundamental Categorial Rule ..................................................................................................18 27. Sentential Modal Logic ..................................................................................................................20 28. Quantified Sentential Logic ............................................................................................................20 29. An Abstract Example......................................................................................................................21 30. Generalized First-Order Languages................................................................................................21 31. The Syntax of Generalized First-Order Languages.........................................................................23 1. Logical Vocabulary: ...........................................................................................................23 2. Non-Logical Vocabulary: ...................................................................................................23 3. Rules of Formation.............................................................................................................23 32. An Example of A Generalized First-Order Theory ........................................................................24 33. Zero-Order Languages....................................................................................................................24 34. Converting between Prefix Notation and Infix Notation.................................................................25 35. First-Order Languages....................................................................................................................25 36. Categorial Languages .....................................................................................................................26 37. Answers to Selected Exercises ......................................................................................................26 Hardegree, Metalogic, Formal Languages – 2 page 3 of 28 Part 1 - Zero-Order Languages 1. Introduction In the current part we develop a general account of a very simple class of formal languages – namely, zero-order languages. For these purposes, a zero-order language (ZOL) is a formal language in which every grammatical expression is either a sentence (category S) or a sentential connective (category Sk¢S). See Appendix A3 for a general description of categorial grammar. 2. The Meta-Language In the current part, the meta-language (English+) is specified informally as follows. (1) Metalinguistic variables: lower case Greek letters (with or without subscripts) are constants and variables that range over strings of symbols. (2) Quote-Notation: the official name of a string of symbols of the object language is obtained by enclosing that string in single quotes. (3) String concatenation: this operation is explicitly indicated – by the plus sign ‘+’. Algebraically speaking, the plus operation is associative, so parentheses are dropped [e.g., ‘a+b+g’]. (4) Metaphysics: As it turns out, our notation cannot distinguish between the letter ‘a’ and the word ‘a’. Furthermore, our metaphysics supports this “confusion” — we treat strings mereologically; in particular, we regard