We will here introduce and
learn how to use a deductive system for sentential or propositional logic. The
goal of our system is to enable us to move from a set of premises to any
conclusion that does indeed follow truth-functionally from those premises. A
system that in principle enables us to do that is said to be
complete
.
But there is another virtue that a system should have. As well as being
complete it should be
sound
. That is, if you follow the rules of the system you will not be able to reach
a conclusion that does not follow truth-functionally from the set of premises
with which you are working.
By the end of these tutorials we will have developed a complete system that
will,
with modest effort, be able to use. The tutorial utilizes a proof-builder with
which you will construct derivations,
but you should be able to transfer the skills you will gain to pencil and
paper. There will remain one significant difference between the use of the
proof-builder and use of pencil and paper. The proof-builder will, in
effect, not allow you to make mistakes. If you attempt to do something not
allowed in the system you will receive an error message. However when you turn
to pencil and
paper you may make mistakes. But study with the proof-builder should enable
you to minimize the number of mistakes you make when you move to pencil and
paper. And you can always use the proof-builder to check derivations you have
done on paper.
Certain arguments are valid,
certain others are invalid. Suppose you are presented with an argument that you
do not know to be valid. Can you find out whether the argument is valid or
invalid by using the deductive system? The answer is yes and no. It is yes in
the sense that if you do reach the conclusion by using the deductive system
correctly you know, since the system is a sound one, that the argument in
question is a valid one. But what if you are unable to reach the conclusion?
Have you established that the argument is invalid? The answer to this is no.
You may simply have overlooked a way to reach the conclusion. Of course in
certain cases you will, by noting why it is that you are unable to reach a
conclusion, be able to "see" that an argument is invalid.
Why do we have or need
deductive systems? After all, there are other ways to determine whether an
argument is valid or invalid. We could use truth tables, for example. But there
are various reasons why deductive systems are of value. First, as we shall see
when we study arguments that look at arguments that involve looking at more
than sentential structure, there are cases in which there is no full-fledged
analog of truth-tables, that is no "mechanical" means of always
determining in a finite number of steps whether or not an argument is valid.
When we come to study such arguments we will rely upon a deductive system.
Second, in much of our actual reasoning we often do move from premises to a
conclusion by the use of rules of inference. A system of the sort we are
developing is sometimes called a system of "natural deduction". It is
not of course true that it is completely "natural" in the sense of
exactly mirroring our ordinary reasoning. But it is nonetheless related to the
reasoning that we do use in ordinary life. By mastering the deductive system we
can hopefully develop the ability to reason in a more disciplined fashion. We
may make fewer mistakes and may be better able to see that certain reasoning
does involve mistakes.
When using our system we will
typically proceed in the following way. An argument begins with certain
premises. We will first list those premises. Each one will be entered on a
separate numbered line. For example, if our premises are
P v Q and ~P
the opening of what we shall speak of as a
derivation
will look like this:
|
1.
|
P v Q
|
Premise
|
If an argument is valid, we should be able to move from the premises to the
conclusion in a sequence of steps. Each step, each additional line, will be one
that the rules of our deductive system will allow us to add or, something we will discuss later, an assumption. If we reach a
conclusion in this way, we shall say that we have a
derivation
or
proof
of the conclusion from those premises, or, to be slightly more formal, a
derivation of our conclusion from that set of premises. In order to keep track
of where a line comes from, we shall enter on the right an account—we
shall call it a
justification
—that tells us how the line was derived, where it came from. In this case
the two lines we have are justified in the sense that they are premises.
Frequent use will be made of
the symbol 'I-', a symbol that we shall call the turnstile. This indicates that
what is on the right of it is derivable from the set of sentences specified on
the left.
Let us now proceed to the
development of what we shall call the basic rules of our deductive system.
After developing them we will introduce three more rules that are conceptually
different from the basic rules.