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Please keep your Handout because we will use it again next class.
Today we finished the section 7.5 on PL syntax and we practiced translation from
English to PL and the other way round. We finished 7.6 and started 7.7.
Section 7.6 introduces basic notions of Aristotelian logic. Aristotle focused on four types
of sentences, which medieval logicians designated as A-, E-, I-, O- sentences. Remember
that E and O sentences can be conveniently translated in two ways:
No As are B
as
x(Ax  ~Bx) or ~x(Ax & Bx)
Some As are not B
as
x(Ax & ~ Bx) or ~x(Ax  Bx)
Here is a proof that x(Ax  ~Bx) and ~x(Ax & Bx) are equivalent. The proof uses the
rules Impl, DeM, double negation (listed on the front cover of the book under the SD+
rules) and the definition of the universal quantifier in terms of existential quantifier and
negation: xAx  ~x~Ax
1
2
3
4
5
x(Ax  ~Bx)
~x~(Ax  ~Bx)
~x~(~Ax  ~Bx)
~x(~~Ax & ~~Bx)
~x(Ax & Bx)
def of 
Impl
DeM
DN
This proves that 1 implies 5. To prove that 5 implies 1 just read the proof backwards,
since any two lines are equivalent.
We talked about the distinction between the object-language and the metalanguage. There
are lots of reasons to make that distinction. We mentioned one of them: the distinction
prevents the so-called Liar Paradox and various versions of it to be formulated in our
language PL. The distinction is explained in the section 2.4, but you are not yet required
to study that section for your assignments. However, you should know some basic
notions from the metalanguage, because lots of definitions are written in metalanguage.
Bold capital letters are so called meta-variables that stand for PL formulae; bold letter x
is a meta-variable that stands for PL variables (x, y, z ...); a stands for PL individual
constants (a, b, c, ...).
For practice:
Prove that x(Ax & ~ Bx) is equivalent to ~x(Ax  Bx)
7.6E
What to read for the next class: sections 7.7