T6: 6 of 12

Part II: moving toward rules for = and v

Remember that a derivation rule can be written this way:

&E
input:

output:
P&Q P
or P&Q Q

The idea is that if a sentence of a certain form -- a conjunction P&Q -- is given on a derivation, then on any succeding line, one can write either P or Q. This is justified because the input line entails the output; in other words, the inference from P&Q to P or to Q is a valid one.

There are a number of other rules like this. Let's see just a few more for now.

We've already seen &I in action. Here's the statement of the rule:

&I
input 1:
input 2: output:
P
Q P&Q

This just says that if you've derived two statements, P and Q, then you are also justified in concluding P&Q.

From time to time, you'll find it best just to repeat a statement from an earlier line. You can do it by the rule called "reiteration":

R
input: output:
P P

And here are a few more rules of inference you may want to use. Make sure you see that these are valid.

DS
(Disjunctive Syllogism)
input 1:
input 2: output:
PvQ
~P Q
or PvQ
~Q P
MT
(Modus
Tolens )
input 1:
input 2: output:
P>Q
~Q ~P
HS
(Hypothetical Syllogism)
input 1:
input 2: output:
P>Q
Q>R P>R

Rules for the triple-bar

Next consider two rules for the triple bar. Make sure you see why both make good sense:

=E
input 1:
input 2: output:
P=Q
P Q
or P=Q
Q P
=I
input 1:
input 2: output:
P>Q
Q>P P=Q

And, consider their use in this example:

 Premise 1 A>(C&D) Premise 2 (C&D)>A Premise 3 C Premise 4 D 1,2 =I 5 A=(C&D) 3,4 &I 6 C&D 5,6 =E 7 A

This isn't the most efficient way to derive 'A' from the four premises, but it uses both rules. Notice that each step is a valid one...to see why focus on what the triple bar means.

Next consider the v-rules:

Rules for 'v'

Let's start with the introduction rule for the wedge. This rule is a very simple one but may seem strange at first.

vI
input:

output:
P PvQ
or Q PvQ

This rule seems to present a problem because it allows one to pull Q out of the blue. That is, if we have P on a derivation, then we can just write PvQ without knowing anything about Q. Why?

We need to make sure this rule makes sense, that is, that it corresponds to valid argument. So, first remember that 'v' is an inclusive "or". That is to say, an SL disjunction PvQ is true just in case at least one of its disjuncts is true. Thus, given that one disjunct is true, it follows that the whole disjunction is true. So, if P is true, then of course PvQ is too. This is so no matter what Q is or from where we've "pulled it".

The elimination rule for the wedge is like a long winded horseshoe elimination. It's easiest to understand by way of example. So, think about the following.

Aconcagua, the highest peak in the Americas, is either in Peru or Argentina.
But, if Aconcaqua is in Peru, then it's in South America.
Likewise, if this great peak is in Argentina, it's in South America.
__________________________
Thus, either way, Aconcaqua is in South America.

If we symbolize this thinking, we'll see how our rule of vE will work:

PvA
P>S
A>S
S

Thus, we have an argument with three premises. This corresponds to three inputs lines for our rule:

vE
input 1:
input 2:
input 3: output:
PvQ
P>R
Q>R R

The idea here is simple: there are two possibilities (input 1) both of which lead to R (inputs 2 and 3), thus (whichever possibility is right) the output R must hold.

Now, using just our two wedge rules, vI and vE, which sentences can be derived on line 4 of the following?

1.
2.
3.
4.

Click on all the following which could correctly be placed as line 4.