Working with Conditionals

        We will here introduce and learn how to use certain rules that apply to conditionals, to 'if...then...' sentences. These tutorials are cumulative, that is, in each tutorial we will on occasion utilize material introduced in previous tutorials. Our conditional is officially entered by entering two symbols, first the hyphen (no shift required- located to the right of the 0) and then the '›' above the period (a shift is required). However when you are entering something you may simply type '> '. The constructor will then enter '-›' for you. Our first rule is the one that we call -›Elimination (-›E). -›E allows us to move from p and a conditional p -› q to q. This rule is often called 'modus ponens'. This rule appeals to two lines- one must be a conditional and the other must be the antecedent of the conditional. If you wish to apply this rule but you do not already have the antecedent as a line you might see if there is a way in which you can obtain that antecedent from the lines which are accessible. Here is a simple example of this rule.

1. 2. 3.

P -› Q P Q

Premise Premise 1, 2 -›E

Please recall 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:

Go to Lesson 1

        Later on we will introduce a rule that enables us to introduce conditionals. But the use of that rule involves more advanced techniques than we are concerned with at this point. Instead we will introduce a rule that we shall call 'modus tollens' (our mt rule). It enables us to move from lines p -› q and ~q to the line ~p.

1. 2. 3.

P -› (Q & R) ~(Q & R) ~P

Premise Premise 1, 2 MT

It does not matter how complex the sentences you appeal to are. But one must be a conditional. And the other must be the negation of the consequent of that conditional. Otherwise this rule simply does not apply.

You should now:

Go to Lesson 2

All that remains is to try a derivation that involves applying both of our new rules. Here is one sample.

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

(P -› R) ~R & (~P -› (S & A)) ~R ~P -› (S & A) ~P S & A

Premise Premise 2 &E 2 &E 1, 3 MT 4, 5 -›E

Notice again 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.

You should now:

Go to Lesson 3

Next Section- on Disjunctions      Conditional Exercises      Main Menu