4.5e
Symbolization and Prep for Chapter 5

This exercise is meant to do two things.

1. Clarify some very complicated logic in English via symbolization into SL, and
2. Prepare us for material in the next chapter, especically in 5.3.

So, let's begin.

We'll continue to symbolization with names and predicates. But all the points made about alternative symbolizations hold true when just using sentence letters.

a: Argument A, a particular argument I have in mind, b: Argument B, another argument.

S_:  ___ is sound.
M_:  ___ is Modus Tolens.
T_:  ___ has only true premises.
W_:  ___ is valid. (Note: we can't use 'V' in our langauage; the computer confuses it with wedge.)

You must use upper and lower case letters here: You WILL need the shift-key!

The first two of these are very easy. You might as well just try for the "expected answer". Remember: you must use "Wa" not "Va".

1. Argument A is valid.
2. Argument A is invalid.

Now, let's try one that is a little harder.

If I say that an argument is invalid, we symbolize that with a negation: the argument is not valid. But what if I say that an argument is not invalid.

This "it's not invalid" looks like a double negative! Well, it is. It's often funny English to use a double negative, and your high school English teacher wouldn't like the style. But it's still as meaningful as a double negation in mathematics.

So, for 3, try to answer in more than one way. Try to give both the "expected answer" and then come back (Shft-TAB a couple of times) and redo it in a shorter way.

1. Argument A is not invalid. (hint)

DN:

So, the idea of 3 is that the double negations cancel out. We'll even give this principle a name in the next chapter: "double negation" or DN for short.

Now let's think about "Neither...nor..." and "not...both...".

I might say that neither argument A nor argument B are valid. There are at least two good ways to symboize that.

On the other hand, I might say that they are not both valid. Let's try to think these through, and do this by coming up with more than one good symbolization.

1. Neither argument A nor argument B is valid.
2. Not both of A and B are valid.

Now, try a couple more. You could give more than one good symbolization for these too.

1. Both arguments are invalid
2. Either A is invalid or B is invalid.

You should have noticed in all this that a tilda outside parentheses is very different from one inside. The general idea here is call De Morgan's Principle. For short we'll call it DM.

DM:

Anything of the form ~( P v Q ) is logically equivalent to ~P & ~ Q. And anything of the form ~( P & Q ) is logically equivlalent to ~P v ~Q.

We'll see this plays a big role in the next chapter. And you might remember that you've already proved these sorts of thing are logically equivalent with truth tables. Here's one done for you. You could always do the other on paper. Good practice for the exam. Very good! (Hint, hint.)

Here's another example. You might look at a difficult argument and say:

Hmm, it's invalid unless it's Modus Tolens.

There are a few ways to symbolize this one. One is with the wedge. That's the one we've learned. It's the easiest to memorize. (hint) But you can also do this next one with the horseshoe. Try to do 8 in three different ways. (One with 'v' as the main connective, and two different ones with '>' (hint).)

1. Argument A is invalid unless it's Modus Tolens. (hint)

IM:

One principle here with 8 is called IM: ~P v Q is the same as P > Q.

Keep this in mind and let's move on to one last sort of symbolization.

One may say that an argument like argument A is sound if and only if argument A is both valid and has all true premiese. But "if and only if" seems like two conditionals. One to go with the "if" part and one with the "only if". See if this makes sense:

EQ:

The usual way to symbolize "P if and only if Q" is as "P = Q" But here are two more ways.

1. "P if and only if Q" = "P if Q and P only if Q" = "(Q>P) & (P>Q)", i.e.,

So, "if and only if" is called the biconditional. It amounts to two conditionals and can be symbolized that way: "if one component is true then so is the other".

2. But, then, yet another way to say the same thing is that there is no way for one component to be true while the other is false: "P & Q or not-P and not-Q" = "(P&Q) v (~P&~Q)"

These different but logically equivalent ways of symbolizing "if and only if" also apply to "just in case". And we call them all EQ, meaning they are equivalent.

1. An argument A is sound if and only if it is both valid and has only true premises. (hint)