In symbolic logic we also have some use for x and y as variables representing unknown or changing individuals. This use is much less in PLN than in conventional notation, however. But we also use capital letters as variables, either without restrictions so that they are the first letters of actual words, or according to a particular code that is established. In PLN we use capital letters both for variables and for signals about the logical relationships among these variables. We use small letters according to a code that names individuals or to represent stand-in names when we want to give instances of some generalization.
We will use the letters from P through U to represent complete basic ideas according to a code that expresses these ideas positively and in the present tense. For instance:
P: Logic is easy.
Q: Logic is fun.
R: Lee studies.
S: Lee does well.
We have these five signals:
A for conjunction
(P and Q)
O for disjunction
(P or Q -- but with the understanding that this includes the possibility
of both being true)
C for implication
(If P then Q, P only if Q)
B for equivalence
(P if and only if Q)
M for mutual
exclusion (P or Q, but not both)
N for negation
(not P)
A sentence such as “Lee studies if logic is not easy” is symbolized
as PNRC. Notice that with conditionals
the English word order may not correspond to the required logical order.
PNRC
would also “translate” these sentences:
We’ll know logic is not easy only if Lee is studying.
Lee was studying if logic was not easy.
If logic is not easy then Lee will study.
If a difference in time matters, then this should be reflected in the code. Otherwise, the English statements are reduced to purely truth-functional relationships so that the difference in time or whether something happens continuously or is a single event is not reflected in the symbolization.
You should note that PLN requires a signal to follow the elements it links, as in PN and PQA. Those elements can themselves be signals, as in PQARC in which C links the elements A and R. To keep track of the elements we can use the count technique.
We also use the letters F though K to represent predicates--the things we might say about someone. Again we set up a code:
Fx: x is rich
Gx: x is famous
Hx: x is lucky
a: Ted
b: Jane
FaFbA would stand for "Ted and
Jane are both rich."
FaGaA would stand for "Ted is
both rich and famous."
We also have additional signals to express quantification--the idea
of whether we are speaking of everyone in a group or just about some.
V and D
are universal quantifiers, W and E
are particular or existential quantifiers.
“Everyone is lucky” would be expressed by HV.
“Anyone rich is lucky” would be expressed by FHD.
“Someone is famous” or “There are those who are famous”
would be GW.
“There are rich people who are unlucky” would be FHNE.
“Nobody’s famous” would be GNV.
“Nobody famous is lucky” would be GHND.
“Anyone not rich is lucky” would be FNHD.
“All those who are rich and famous are lucky” would be FGAHD.
“Only the rich are famous” would be GFD.
(Note this last example and think why again the English word order can
be misleading.)
Although our variables are assigned by a code, we also use them to represent patterns with no definite code in mind. PQC, for instance, expresses the conditional relationship of P and Q regardless of what P and Q stand for. It is then understood that these actual letters could be replaced with any other letter or module made up of meaningful strings. For instance, PQARC expresses the same relationship between the module PQA and the variable R that we find in PQC. We could then say that PQARC is a substitution instance of PQC.
For additional practice, run the interactive program Introducing
PLN available from the files in our SymbolicLogic group at Yahoogroups.com
Next: learn about working with conditionals.
Go back to the starting page.