Welcome to the course. There are several ideas that I want to begin with.
Symbolic logic was designed originally to represent the relationships in our thoughts, but since then it has come to be used to represent what happens with electronic circuits. The key link is between the fact that a current can either be going through a line or not, and we can talk about many of the things we say as statements which are either true or not.
To emphasize what happens here I am going to do the first half of the course entirely in PLN (Postfix Literal Notation), which allows me to show you some things with computer graphics that may make some of these ideas more clear. In the second half we will make the transition to conventional notation.
It helps, I think, if we approach symbolic logic first off as a game. Like chess, it has fixed pieces (symbols) with precise rules for how they can be moved. And just as we record the progress of a game in a fixed manner, we can show what goes on with a logical relationship in what we call a proof or a derivation. But even more important, we realize that there arfe certain properties or characteristics we expect in a game that also apply here.
Right now, please click on and read about the properties of effective games. (To really get into the spirit of things, see how you might answer the questions at the bottom).
Two things should stand out. First, we need to work with just the right symbols (letters in the game above) put together in just the right away. This is what we mean by a well-formed formula--a wff.
Next, we need to have fixed rules for making any changes in what we start with. In symbolic logic we see these as introduction and elimination rules, and we also have rules for exchanging one wff for another (equivalence rules).
Now, to get started, I am going to ask you
to go to the SLIPS pages and learn about postfix literal
notation and the variables and our
signals used for PLN. One difference between working with PLN
and working with the conventional notation found in most texts is that
we set up a code with a very limited range of letters. Now, since
we are just getting started, do not worry about absorbing everything on
the pages. Let's keep it simple by talking only about the propositional
variables (the letters in the alphabet from P to U) and these six signals
expressing our key logical relationships:
A (think of the
"a" in "and") to express conjunction
B to express
a biconditional or equivalence relationship
C to express
implication or a conditional relationship
M to express
the opposite of equivalence, which is a mutually exclusive relationship
N to express
negation
O (think of the
"o" in "or") to express disjunction
In a later lesson we will be concerned with how to express English sentences, but right now I want you to think about what it takes to have a wff here. The signals always look backwards (this is what we mean by postfix notation), so that PQC means we are setting up a conditional relationship between P and Q. We could express this as "If P is true then Q is true" or as "P is true only if Q is true." What matters here is that this is a tight signal in that the order has to be correct.
We can also negate any part of this.
Let's set up a first code so that P represents "Logic is easy" and Q represents
"Logic is fun." These are different strings that we can have with
"translations" for each:
PNQC:
If logic is not easy then it is fun.
PQNC:
If logic is easy then it is not fun.
PQCN:
It's false to say that logic is easy only if it is fun.
Now, let's bring in two rules. One allows us to eliminate the C, the other to exchange one string or part of a string for another.
This is the first. If we have any conditional
and we have the part to the left (the antecedent) by itself, we can also
have the part to the right (the consequent) by itself. We represent
the rule in a general manner this way:
PQC, P |- Q C elim (the sign
|- represents that whenever we have what is the left the thing to the right
will follow)
This is the second. We can switch the
antecedent and the consequent if we negate both.
PQC
= QNPNC Contra (for contraposition)
What we will be working with very soon are
derivations, not too different from what you saw in the letter game.
Let's look at a sample:
PQC, QN |- PN
1. PQC Ass
2. QN Ass
3. QNPNC
1 Contra
4. PN
2,3 C elim
After you have reviewed this page, go on to
the
first
assignment.