Introduction to Assumptions

         Our current system is not complete. For example, it follows from p -› q and q -› r that p -› r. But we have as of yet no means of showing this. Similarly ~(p v q) I- ~p & ~q, but, again, we have as yet no way to show it. We will here introduce what we shall call modes of proof. These modes of proof will involve utilizing assumptions. From the standpoint of the builder it is easy enough to introduce an assumption. Simply enter some sentence in R1 and click on the assumption button A. When you do so you will notice that the assumption is indented. It is justified as an assumption. If you add another assumption it will indent further. Should you wish to see this:

Go to Lesson 1

When you are working on paper you should follow the practice of indenting assumptions. You will find that it will be of considerable help in avoiding errors. We will speak of the unindented column as the primary column. This is of course the column in which we have been entering any premises that we have. When you enter an assumption we will say that you have initiated a subderivation.

So much for the initial mechanics. Let us now look at a common argument which involves utilizing assumptions. You have said the following:

If the switch is turned to the left when it is in the middle, the light dims. If the light dims, then the display can't be seen. So, if the switch is turned to the left when it is in the middle, then the display can't be seen.

Someone challenges your conclusion. One way in which you might defend it is as follows:

Look you are granting that (1) if the switch is turned to the left when it is in the middle, the light dims. You are also granting that (2) if the light dims, then the display can't be seen. Now suppose that the switch is turned to the left when it is in the middle. It follows, given (1) that the light dims. Since the light dims it follows, given (2) that the display can't be seen. So if the switich is turned to the left when it is in the middle, then the display can't be seen.

Here you are using a mode of proof that we shall call conditional introduction (-›I). (It is sometimes called 'conditional proof'.) You have made an assumption (assumptions are often introduced by 'suppose') and then you make a claim that you have reached a conclusion that depends only upon the premises. Note that you do not (and could not correctly) claim that the light dims. This claim was reached via an appeal to your assumption as well as to one of your premises. We will study conditional introduction in more detail in the subsequent tutorial. But you now might be able to see that our indentation enables us to mark off claims that will typically depend on the assumption that precedes them. The assumption itself is the intial line of what we will call a subderivation. In the system which we are developing you can, "underneath" an assumption, apply the previously introduced rules to the assumption or to lines that are, as we shall say, "accessible". We will say more about accessibility below. For now we can note that all previous sentences in the primary column are accessible. The following is, for example, allowed:

1. 2. 3. 4.

P -› Q Q -› R      P      Q

Premise Premise A 1, 3 -›E

The subderivation can be continued as follows:

1. 2. 3. 4. 5.

P -› Q Q -› R      P      Q      R

Premise Premise A 1, 3 -›E 2, 4 -›E

Notice that the last line here is the consequent of the conditional that we wish to obtain. Once you obtain that you are in a position to apply the -›I rule. Highlight the assumption and the last line and click the button. You will now see the following:

1. 2. 3. 4. 5. 6.

P -› Q Q -› R       P       Q       R P -› R

Premise Premise * A * 1, 3 -›E * 2, 4 -›E 3-5 -›I

Note that line 6 is no longer indented. At line 6 the assumption has been discharged. The particular subderivation, lines 3-5 is terminated. We will occasionally say, for example, that the subderivation is terminated at line 5. No line within a terminated subderivation may be appealed to at any later point in the derivation. Such lines are, as we shall say inaccessible. The builder will not allow you to use inaccessible lines. And, as a helpful visual cue, it will place an * just before the justification once a line has become inaccessible. Note that we did that here. But when you are writing out a derivation you will have to take care yourself not to appeal to inaccessible lines. That line 6 is not indented indicates that it does follow from the premises that you have- notice that it is in the same column as those premises. Notice that the justification for line 6 says 3-5 rather than 3, 5. This indicates that you are appealing to the subderivation as a whole. If you wish to build this proof yourself:

Go to Lesson 2

Note that when you apply -›I you the consequent of the conditional you are obtaining must be the last sentence of the subderivation in question. The following is erroneous:

1. 2. 3. 4.

P & Q Q      R R -› Q

Premise &E A 2-3 -›I

Erroneous

To enable us to do derivations such as this we have introduced a rule called reit -short for 'reiteration'. Reit allows you to introduce at the point you are any accessible sentence. So this derivation could be completed in this way:

1. 2. 3. 4. 5.

P & Q Q       R       Q R -› Q

Premise &E * A * 2 Reit 3-4 -›I

There is an alternative way of completing this proof that does not involve reiteration:

1. 2. 3. 4.

P & Q       R       Q R -› Q

Premise * A * 1 &E 2-3 -›I

        Consider the following argument:

It's not the case that either John will come or Mary will come. So John will not come.

Someone again (implausibly I admit) challenges your conclusion. You might respond as follows:

You grant that (1) it's not the case that either John will come or Mary will come. Now suppose that it is true that John will come. Then it is true that either John will come or Mary will come. But (1) tells us that that is false. So the supposition that John will come is also false. That is, John will not come.

Here you are using a mode of proof that we shall call negation introduction (~I). (It is sometimes called 'indirect proof' or reductio ad adsurdam .) The core idea is that if an assumption, given your premises, leads to a contradiction then the assumption must, given your premises, be false. But if the assumption, given your premises, must be false, the negation of the assumption, given your premises, must be true. In our system you can only apply negation introduction if you have, as the last line in a subderivation, a line that is an instance of either p & ~p or ~p & p, that is if we have an explicit contradiction. Here is a way to show the above argument valid in our system:

1. 2. 3. 4. 5.

~(P v Q)      P      P v Q      (P v Q) & ~(P v Q) ~P

Premise * A * 2 vI * 1, 3 &I 2- 4 ~I

Notice that the line 5 here is an explicit contradiction. Any application of ~I will give us only the negation of the assumption with which we began. If you have assumed ~p and derive an explicit contradiction, you may apply negation elimination (~E) and obtain p. So, if you wish to establish some sentence p that is not a negation assume ~p and then use ~E to obtain p. You can also use ~I to establish some sentence p that is not itself a negation. To do this you would simply assume the negation ~p of p. If we can discharge this assumption we will obtain ~~p. We could then obtain p by ~~E. You should now:

Go to Lesson 3



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