9.1: Introduction

This chapter is about more complicated translations and some tools for understanding equivalences and contradictions.

The formal syntax of MQL, the language of MQT, will be presented.  

We put statements involving quantifiers into classes according to an ancient system and there is a standard translation into MQL for them.

A    All S are P                             ("x) (Sx É Px)

E     No S are P                             ("x) (Sx É ~Px)    [ NB: same as  ~($x) (Sx & Px) ]

I      Some S are P                         ($x) (Sx & ~Px)

O    Some S are not P                  ($x) (Sx & ~Px)

Propositions in this form are in categical form

Recall the traditional square of opposition from chapter 1:

                 A      E                 

       All S are P       No S are P 

      Some S are P       Some S are not P 

                                              I                    O

The statements that are diagonally opposed are contradictories. So the following are logical equivalences:

A º ~O, E º ~I, I º ~E, O º ~A

There is also the Simpler Square of opposition:

           Half-A      Half-E

   Everything is S       Nothing is S

             ("x) Sx       ("x) ~Sx

    Something is S       Something is not S

           Half-I      Half-O

             ($x) Sx       ($x) ~Sx

Using MQL and the resources of quantifiers and propositional logic connectives we can formally treat many more complex sentences.

Consider these examples using the following translation dictionary:

       a = Arthur

       e = Excalibur

       c = Clare

       E = has a sharp edge

       I = is made of iron

       K = is a king

       P = is a person

       S = is a sword

       W = is made of wood

       Kings are all people

       ("x) (Kx É Px)

       Every sword which is made of iron has a sharp edge

       ("x) ((Sx & Ix) É Ex)

       There is a sword with a sharp edge which is not made of iron

       ($x) ((Sx & Ex) &  ~Ix)

       No sword is made of either wood or iron

       ("x) (Sx É ~(Wx v Ix))

The rules for wffs for MQL are as follows:

A. Define a set of elements

1. Take a defined set of propositional constants: P, Q, ...

2. Take a defined set of propositional variables: p, q, ...

3. Take the eight defined logical symbols (or incomplete symbols) of PL : ~, &, v, x, É, º, (, ).

4. Take a defined set of individual constants:  a, b, c, ...

5. Take a defined set of individual variables: x, y, z, ...

Individual constants and individual variables are the individual letters.

6. Take a defined set of predicate letters: F, G, H, ...

7. Take the quantifier symbols: $, ".

B. Set up a recursive definition for the wff (well-formed formula of QML.)

     Use these symbols for the rules:

        F - for predicate letters

        c - for individual constant

        x - for individual variable

        k - for individual letter

1. basis clauses:

    B1.       Any propositional letter is a wff.

    B2M.   Fk is a wff.

    These define atomic formulae.

2. recursive clauses.

    R~.       If X is a wff then ~X is a wff;

    R".      If X is a wff then ("x)X is a wff. X is then said to be the scope of ("x).

    R$.      If X is a wff then ($x)X is a wff. X is then said to be the scope of ($x).

    R*.       If X and Y are wffs, and * is one of &, v, º, É, then (X * Y) is a wff.

3. terminal clause: Nothing is a wff unless it is a wff because of the preceding clauses.

If the variable x occurs in the scope of the quantifier ("x) or ($x) it is said to be a bound occurrence of that variable, and x in the scope is a bound variable or just bound.

If a variable is not bound then it is free. We speak of free occurrences and free variables.

A quantifier binds the variable beside it and all free occurrences of the same variable in its scope.

For example: in ("x)(Px É Qx) all the xs are bound.

                       in ("x)(Px É Qy) only the xs are bound, the y is free.

                       in (("x)Px É Qx) only the first two xs are bound, the last x is free.

A wff which contains free occurrences of a variable is an open formula.

Otherwise it's a closed formula.

Main operators of wff are defined just the same as for PL.

The universe of discourse indicates the range of the quantifier. It may be entirely unrestricted in which case ("x)Px means absolutely everything is P. Or it may be restricted; for example if the UD is people, then ("x)Px means everybody is P.

A universe of discourse can be declared in the dictionary for a translation.

An open formula can be closed by placing an existential or a universal quantifier for every free variable to the left of the wff.

If an open formula is closed by using just existential quantifiers it is an existential closure.

Similarly we can have universal closures.