12.1: Introduction

To the Monadic Predicate Logic we add predicates for relations to get General Predicate Logic.

Then we can handle statements like 'Ann is taller than Bill'. Previously we could only say things like 'Anne is tall and Bill is tall.'

12.2: Relations

The proposition 'Anne is taller than Bill' affirms that there is a relation of 'being taller than' that holds between Anne and Bill. That relation is predicated of Anne and Bill. Similarly in 'Arthur struck Mordred' the relation of 'striking' is predicated of Arthur and Mordred.

The order of the things of which the relation is predicated is important:

The statement that

            'Anne is taller than Bill'

is different from the statement that

            'Bill is taller than Anne.'

Sometimes, when the relationship is named with a verb, the order in English can be reversed by putting the active verb into the passive voice (or vice versa). So

            'Arthur struck Mordred'

means the same thing as

            'Mordred was struck by Arthur.'

These are two-place relations, but we can have relations with any number of places:

            1 place = monadic, which we introduced in the previous chapter as properties;

            2 place = dyadic;

            3 place =  triadic, e.g.  'Jane stands between Anne and Bill';

            ...

            n place = n-adic;

            many places =  polyadic.

You can see why we call the number of places for a relation its adicity.

We introduce letters for all these sorts of relations.

In best practice we mark the number of places that a relation has by a superscript on the corresponding predicate letter, thus

            L²xy     for     x loves y

            B³xyz   for     x stands between y and z.

But we don't really need them in the normal run of events so we won't be using them again.

Note that for convenience we often write Lxy as xLy.

12.3: Translations to Predicate Logic

Two major difficulties:

1.  Translation from ordinary language into QT is the hardest thing about learningPredicate Logic.

2.  There is no finite limit on counterexamples, but the truth-trees don't change

Here are some general rules for translating from English to QT:

1.   Move quantifier phrases to the left, where possible.

           E.g. 'Jupiter attracts everything' becomes 'Everything is such that Jupiter attracts it.'

2.   After transposing we can translate into QT.

           'Everything is such that Jupiter attracts it'

            j = Jupiter

           xAy = x attracts y

           ("y) jAy

           Nothing is not attracted by Jupiter

            j = Jupiter

           xAy = x attracts y

           ~($y) ~jAy

3.   If there are multiple quantifications it may be possible to apply that process repeatedly.

           'Everything attracts something'

           xAy = x attracts y

           Partially translated becomes:

                ("x) x attracts something

           Do a quantifier move:

                ("x) something is the case that x attracts it

           So the full translation is:

                ("x)($y) xAy

4.    More complex cases require consideration of whether the proposition is in A, E, I, O form.

           'John loves everybody'

           j = John

           Px = x is a person

           xLy = x is loved by y

           Do a quantifier move:

                Everybody is such that John loves him.

           In our dictionary that becomes:

                Every person is such that John loves him.

           This is an A proposition, and partially translates as:

                ("x)(Px É John loves x)

           Fully translated it is:

                ("x)(Px É jLx)

       More complex cases also arise. Practise them.

           'Everyone who loves something is happy'

           In our dictionary that becomes:

                'Every person who loves something is happy'

           Which looks like an A proposition if we group it like this:

                'Every (person who loves something) is happy'

           Partially translated it is:

                ("x)((Px & x loves something) É Hx)

           Do a quantifier move on the untranslated part:

                ("x)((Px & something is such that x loves it) É Hx)

           Further partially translated it is:

                ("x)((Px & ($y) x loves y) É Hx)

           And finally:

                ("x)((Px & ($y) xLy) É Hx)

12.4: Translation to English

In order to see what our calculations mean we have to be able to translate back into English. The best way is to reverse the process that was dmonstrated above.

           ($x)("y) xAy

           Replace the predicate letters

                ($x)("y) x attracts y

           Transpose to get y closer to its quantifier

                ($x)("y) y is attracted by x

           Partial translation

                ($x) Everything is attracted by x

           Transpose x

                ($x) x attracts everything

           Final translation

                Something attracts everything

12.5: Syntax for QL

The language QL is an extension of MQL. The only real difference is the addition of the predicate letters for polyadic relations.

The rules for wffs for QL 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 n-place predicate letters: Fⁿ, Gⁿ, Hⁿ, ...

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:

        Fⁿ - 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.   If k₁, ..., k_n are n individual letters then Fⁿ k₁, ..., k_n 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.

12.6: Semantics and Finite Worlds for QT

The semantics for QT is an extension of that for MQT. We only need to extend the tables we used for monadic predicates so that they can deal with polyadic predicates.

For convenience we'll only do tables for dyadic relations.

We can describe relations in a PW by means of tables. Suppose we have a PW with 3 items a, b, c and a relation S, such that aSb, aSc, bSc. Then this is described by the table

          S  a  b  c

          a  0  1  1

          b  0  0  1

          c  0  0  0

Note1: read it vertical-relation-horizontal

Note2: if the vertical and the horizontal lists of elenments are in the same order then the truth value for xSy and the truth value for ySx will be opposite each other across the diagonal line from top left to bottom right. And the value xSx will be on that diagonal.

In order to apply the values in finite world tables we must expand the formulae of QT just as we did for MQT.

1.   Begin with innermost quantifiers and work your way out.

2.   a.   Replace universally quantified formulae with conjunctions

      b.   Replace existentially quantified formulae with disjunctions