Up through the midterm we worked with a
special teaching language that used postfix literal notation. A special
advantage of it was how easily it lends itself to programming truth table
relationships, which are the key to understanding valid argument forms.
The transition to conventional notation should be very easy (although for
someone who has started with conventional infix notation it often is difficult
to adjust to PLN).
Let's look at how we do things with conventional symbolism.
For the connectives (what we called signals before) we have one less than in PLN.
negation: ~P
(PN)
conjunction: P
& Q (PQA)
disjunction: P
v Q (PQO)
implication: P
-> Q (PQC)
biconditional (mutual implication):
P
<-> Q (PQB)
For mutual exclusion (exclusive disjunction,
or the sense of "or" when only one choice is possible), we need to use
a more roundabout expression: ~(P <->
Q) or (P v Q)
& ~(P & Q).
With postfix notation grouping was handled automatically by the placement of the signals. For infix notation we have new punctiation with parentheses:
PQRAO would become P v (Q & R) while PQARO is (P & Q) v R.
Because there are no longer letters for signals, we do not need to restrict the letters used for variables, and we do not need to set up a code as we do in PLN. Normally, we will symbolize by using a letter that goes with the English words, as in the example below.
The test is not easy but the students are ready. ~T & S or maybe ~E & R
With quantifier logic we no longer have the relative simplicity of PLN for shorter expressions. However, to allow for easier use of the keyboard, we are going to avoid the usual practice of indicating quantifiers by inverting the letters A and E as you will see it done in textbooks (such as the Schaum book for this course). To avoid confusion, since the same letters can be used as variables, we will always use parentheses following a quantifier.
universal quantification: Ax(Fx)
and Ax(Fx -> Gx)
for FV
and FGD
existential quantification: Ex(Fx)
and Ex(Fx & Gx)
for FW
and FGE
Please note that for convenience we are using the normal rather than the inverted or reversed versions of the letters "A" and "E" for quantifiers, and accordingly we always use a parenthesis for the quantified expression. In most texts you will see these forms:
universal quantification: ∀xFx and ∀x(Fx->Gx) [or sometimes (x)Fx and (x)(Fx->Gx)]
existential quantification: ∃xFx and ∃x(Fx&Gx)
To review symbolization and practice translations, go to these pages: propositional variables, predicate variables, quantifiers, and monadic predicates. Take the time to do the practice exercises before going on to the assignment for this lesson, which involves replacing PLN expressions with wff's that use conventional notation.