Chapter Six, Tutorial Three
We have been using PL now for some time. But we have yet to say exactly
what counts as a sentence of this language. It is time to spell
this out. Let's begin with a few preliminary considerations. These
will give the basic idea of PL sentence construction.
- We cannot define a PL sentence as we did before in SL,
we cannot start with the atomic sentence and use connectives
to build molecular ones. Why not?
- Answer: the "atoms" of PL are not
themselves sentences. For example, take the sentence
We will build this sentence by starting with the atoms 'Ax'
and 'Bx', combining these with the horseshoe to give 'Ax>Bx'
and finally adding the quantifier.
- But 'Ax', 'Bx', and 'Ax>Bx'
are not sentences! Because they contain a variable
'x' instead of a name, they do not express complete thoughts.
For instance, 'Ax' says that "x has property A"
which is a little like saying "_____ has mumps".
It is not a complete sentence.
- So, 'Ax', 'Bx', and 'Ax>Bx'
are incomplete sentences. We will call them "formulas"
- To make a formula like these into sentences, we must either
replace variables with a name or add a quantifier.
- Thus we will make the sentence '(^x)(Ax>Bx)'
by building it up from atoms ('Ax' and 'Bx') using a truth
functional connective and a quantifier. When each variable
has a quantifier, the formula is also a sentence.
- So, we construct sentences from formulas:
- First, we take , 'Ax', 'Da', 'Rxa', 'Bxyz', etc. as our
- Then we build more complex formulas adding truth functional
connectives or quantifiers. For example, we may build '~Ax',
'(Da&Bxyz)', '(%x)Rxa', etc.
These are new formulas.
- We may keep on building by adding more truth functional
connectives and quantifiers to formulas already constructed.
For example: '~Ax>(%x)Rxa'
or '(^y)(Da&Bxyz)' count as
more complex formulas of PL.
- Only when a PL formula has a quantifier for each instance
of a variable does it count as a sentence of PL. Of our examples
just above, only '(%x)Rxa' is a
- This is all a bit vague. We will make these syntactical
definitions more precise beginning on the next page.
But first, try your hand at the following. Which
of these count as formulas of PL? (We drop outside parentheses as with
And which of the following are sentences of