Chapter 5, Tutorial 2
Trusting Rules: Ampersand Introduction and Elimination
So far we have only three rules of deduction. You will have noticed that we did not add AC or DA.
Why not?
Of course: AC and DA are not valid forms. They are our two named fallacies! We will only add valid forms as "rules of inference".
So, for example, if we have any sentence P on one line of our derivation and the sentence of form P>Q on another, then we can trust the inference to Q; this inference is just our old friend MP. Example:
Here P = 'J' and Q = '~(TvS)'.
We've always seen it as a valid form.
Now we take this form as a rule of derivation.
When we need to write down this rule, we'll do it this way:
| MP | |
|---|---|
| input 1: input 2:  output: |
P>Q P  Q |
Remember that P and Q are metavariables. You may prefer to think about this in terms of our bigger metavariables, the box and oval.
| MP | |
|---|---|
| input 1: input 2: output: |
 > |
This helps make it clear that our metavariables are just placeholders. You need to fit the form...just like back in chapter one!
Most important:
For any three sentences fitting this form, we have a valid argument with output as the conclusion from the input as premises.
Put another way, there is no truth value assignment (i.e., no row) making the input true and the output false.
...we can put trust in MP: So long as the "inputs" are trustworthy, then Q (or , if you prefer) must be true.
Now, let's write down are other two valid forms which are now also rules of inference:
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Notice that there are two forms for DS.
Again, to make thinking about these rules a little clearer, here are the box and oval versions.
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|---|---|---|---|---|---|
| input 1: input 2: output: |
v ~ |
or | v ~ |
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|---|---|---|---|
| input 1: input 2: output: |
> ~ ~ |
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Keep in mind that these mean the same thing as the P, Q versions. All of box, oval, P, and Q are metavariables...that is, they are stand-ins for absolutely any sentence of SL.
Now, let's
think about a new rule.
Now, let's look at a new rule and try to figure out what it's output might be.
| New Rule: | |
|---|---|
| input: output: |
P&Q ??? |
We can fill the "???" with what? That is, click on all the forms just below that would provide a valid replacement for the question marks to complete our "new" rule.
(Think of P as box and Q as oval if that helps...they are meant to be any sentence. The idea of the "new rule" is simple. Suppose you have a conjuncton. What follows from it? Conjuct one? Conjuct two? The rest?)