We will here introduce and
learn how to use certain rules that apply to disjunctions, to 'or' sentences. Our 'or' is simply a lower case v. Recall that our 'or' sentences are designed to reconstruct what is often called an "inclusive" or, that is, our or sentences still count as true even if both disjuncts are true. Our first rule, one that we shall call disjunctive syllogism (DS) allows us, when given any disjunction p v q and the negation of one of the disjuncts, to add the other disjunct as a line. Here are illustrations of both:
1. |
(P v R) v Q |
Premise |
1. |
(P v R) v (Q -› S) |
Premise |
And do keep in mind that if there are multiple premises you should enter them in the specified order. This is necessary if the builder is to recognize that you have completed your task successfully.
You should now:
The important point to remember is that the one of the lines to which this rule applies must be a disjunction. The other must be the negation of one of the disjuncts. Here, as elsewhere, you must pay attention to the form of the lines. We are not here appealing to our intuitions. Here is an example of a defective derivation.
1. |
(P v Q) & R |
Premise |
|
In this case you can derive the conclusion. However to do so within the system you need to obtain 'P v Q' as a line by applying &E. Consider
1. |
(P v Q) -› R |
Premise |
|
This cannot be repaired. The conclusion does not follow from the premises.
We will now introduce a rule, vIntroduction (vI) that causes no end of confusion. It allows us to move from any sentence p to p v q or to q v p where q can be any sentence you want. Suppose you know that if either the power goes off or the cord is implugged then the television will not work. That is, you know that '(P v C) -› ~T' is true. You notice that the cord is unplugged. That is, you know that 'C' is true. Clearly this is a case where there is an -› in the offing. But note that you do not have as an accessible line 'P v C'. To get it you use vI. Here is a derivation within our system:
1. |
(P v C) -› ~T |
Premise |
Using this rule in the builder requires that you highlight the line to which you wish you apply the rule, then enter in R1 what you wish to add (this rule is sometimes called 'addition') and then click on vI. In R2 you will be asked to choose whether you wish to add it as a left or a right disjunct.
You should now:
When should you apply this rule (it is after all one that can always be applied)? The short and uninformative answer is when you need to. What disjunct should be added? The short and uninformative answer here is that you should add something that will be useful. The initial example exhibited one case of need. We will study this elsewhere in more detail, but consider the following. vI can be used to introduce sentences that have not hitherto occured. Should the conclusion that you wish to reach include sentences that do not occur elsewhere you might have to use vI. It is one of the ways (no other ones have yet been introduced though they will be) in which we can introduce sentences that include sentences do not occur as subsentences in any preceding accessible line. Let us look at one particular case, one often found puzzling. Given that you have accessible a sentence p and its negation ~p you can obtain any conclusion r that you like. Here is one particular example that illustrates the tactic used to obtain what you like.
1. |
(P & Q) & ~(P & Q) |
Premise |
In this way you can always show that p & ~p I- q. This is as it should be. There is no case in which the premise is true, hence there is no case in which both the premise is true and the conclusion is false.
You should now:
Our or elimination rule (vE) is one form of a rule that is sometimes called 'constructive dilemma'. (We will later show how you can reconstruct other kinds of constructive dilemmas.) Consider the follow argument in English:
If team A wins, then my (C)lub team will be out of the race. If team B wins, then my (C)lub team will be out of the race. But either team A or team B will win. So, my team will be out of the race.
Clearly this is a valid argument, that conclusion does follow from the premises. But at this point we do not have the capacity to show that it is valid. So we will introduce a rule that enables us to move from a disjunction p v q, a conditional p -› r and a conditional q -› to r. This is our vE rule. Note that it is the first rule that appeals to three preceding lines. Here is a reconstruction of the preceding argument:
1. |
A -› C |
Premise |
You should now:
You will recall that conjunctions are both commutative and associative. So too are disjunctions. However demonstrating this using the rules we have so far is not possible. We will return to this question when the development of our system has been completed.