We will here introduce and
learn how to use the rules that apply to conjunctions, to 'and' sentences. The
initial lessons will, as well as introducing the rules, instruct you in how to
use the proof-builder. Our first rule is one that we call &Elimination
(&E). &E allows us to move from a conjunction p & q to whichever
of p, q we want. You can enter our 'and" by using the ampersand. On a standard
keyboard it is found above the 7. Our 'not' is the ~ found on the left above
the tab key. In both cases pushing that key while holding shift will enter the
symbol. You should also note that you can often save time by copying and
pasting. On a PC this can be accomplished by dragging the mouse along the text
you desire while holding down the left mouse button. Then right click on that
and choose copy. One quick way of pasting what you have copied is to move your
mouse cursor to the place to which you wish to copy and then pushing control+v.
Note that the browsers have edit functions. You should experiment and find the
method of copying and pasting that you find most convenient. The one just
mentioned is simply the one which the author of this tutorial finds convenient.
Note that you cannot copy text from the program window, only from the pale
yellow window. Very little typing is required. And the builder is very tolerant
so far as spacing and capitalization are concerned. For example:
p&Q
P & Q
p & q
will all be entered as:
P & Q
Just be careful not to introduce any extraneous symbols and be careful about
parentheses (recall that we are always eliminating the outermost pair if there
is such. Should you make a mistake you need not always rewrite everything. You
can edit out, for example, an extraneous symbol. Note that the ~ key is, if
unshifted, `. I frequently forget the shift and have a ` in my line. But by
using your mouse or arrow keys you can edit it out, replacing it with the ~.
You should now:
(Note that all lessons open in a new window. Maximize the window when opened. When you have finished a lesson close that window to return here and continue the tutorial.)
You will recall that from, for
example, '(P & Q) & R', 'Q' does indeed follow. But we cannot get there
in one step. Our rule only enables us to obtain a sentence that is the left or
the right conjunct of the line to which we appeal. 'P' is not either the left
or right conjunct. You may have noticed when you did the preceding exercise
that a link called 'tactics' opens in the upper left corner of your screen.
This link will open a new window, one that can be resized, maximized or
minimized. It provides suggestions as to how to use and obtain various kinds of
sentences. You should try using it (even though you will not need the help it
provides) when you do this lesson.
You should now:
Let us now introduce the rule
&Introduction (&I) Recall that this rule enables us, for example, to construct
the following derivation:
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1.
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P
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Premise
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Here we have shown that P, Q I- P & Q
All that remains is to try a
derivation that involves applying both rules. Here is one sample.
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1.
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(P & R) & Q
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Premise
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Notice that you could have derived the conclusion by applying the rules in a
different order. It is never true that there is only one way of deriving a
given conclusion from a given set of premises. In cases where there is more
than one premise, please enter the premises in the order specified. Otherwise
the builder will not be able to inform you that you have completed your task
successfully.
You should now:
From any premise p it follows that p & p. To see how to prove any instance
of this in the builder:
Conjunctions, like additions
and multiplications in arithmetic, are both commutative and associative. That
is:
Commutative- any sentence p & q is logically equivalent to the sentence q
& p. Our system will not allow you to make such a move in one step. However
all that you need to do is to apply &E twice and then apply &I,
choosing the order that you desire.
Associative- any sentence p & (q & r) is logically equivalent to (p & q) & r. Again our system will not allow you to make such a move in one step. However to see one case of this: