T6.1: 4 of 5

It is time to add quantifiers. Instead of just symbolize that Halpin is a faker (or whatever you might want to say) we will symbolize things like "everyone is a faker" or "someone is a faker".

For these we will need quantifiers. Jumping the gun a bit, we'll symbolize the word "every" (or equivalently "all", "any", etc.) with an upside down 'A': . And we'll use a backwards-E for "some" and it's synonyms: .

And soon we will want to get to the more useful "someone at O.U. is an employee and not a professor". Or, "everyone at OU is a professor or a student". (Of course, only the first of these two sentences is true.)

Note: You'll need The Logic Font to display our symbols. Use the '%' key for the backwards-E and the '^' for the upside down 'A'.

OK?

We start very simple. And with a very precise example. For purposes of this example will be discussing only sound arguments. We will ignore everything else. Again: for now the subject matter is the collection of all sound arguments. (We could talk instead about all cats, say. But we just need to restrict ourselves in some specific way for the example.)

The subject matter for this example, the universe of discourse, is just the collection of all sound arguments.

Here's one, a disjunctive syllogism (DS) with only true premises:

(a) Halpin (whoever this guy is) is either a professor or a faker.
But (I'm telling the truth, really!) he is no faker.
So, Halpin is a professor

We call this one 'a'. Of course, there are lots of others. The collection of sound arguments is infinite in principle: you can make longer and longer true sentences making longer and longer arguments fitting DS or other valid forms.

Let's think about all the sound arguments in existence. First we say:

Universe of Discourse = all sound arguments

W_: __ is valid.

a: the sound argument above in gray.

Now, it's easy to say of our one argument, a above, that it is valid:

Wa

But we need to say more. That a is representative in this way of any sound argument. We do this with variables: we use 'x' and later ('w', 'y', and 'z') as placeholders for anything in the universe of discourse. They are a little like the blank above in "__ is valid". In fact, hereafter when we say 'W' means "is valid" we will write this as 'Wx' means "x is valid". Let's make the change right now:

Universe of Discourse = all sound arguments

Wx: x is valid.

a: the sound argument above in gray.

Now, we say that some argument (in our universe of discourse) is valid with our backwards-E:

(%x)Wx

(read this "there is an x such that x is valid" or as "there is an x making 'Wx' true.")

So, '(%x)Wx' means that some x, some member of the universe of discourse, is a valid argument. There is a way to "fill the blank" making 'W_' true.

Of course, we already knew that some arguments are valid. Because argument a above is sound, it's an example of a valid argument.

More importantly, because we are only speaking about sound arguments, we can say that they all are valid. We symbolize that with the upside down 'A':

(^x)Wx

(read this as "every x is such that it is valid" or as "all x make 'Wx' true".)

So, '(^x)Wx' means that every x, every member of the universe of discourse, is a valid argument: Any way of filling in the blank of 'W_' is true (so long as we fill in with names for members of our universe of discourse: sound arguments).

Here's the .pdf Wrap-up...and time to start symbolizing.

Universe of Discourse = all sound arguments

Wx: x is valid.

a: the sound argument, argument a, above in gray.

Important Typing Short-cuts:

You may enter '3x' instead of '(%x)' and a 'Vx' instead of '(^x)'. The computer will clean this up for you.

You WILL need the logic font though: remember '%' for the backwards-E and '^' key for the upside-down 'A'.

Feel free to use the Help! feature if you don't see how to begin. (Enter a question mark instead of an answer.)

1. Some arguments are valid.
2. Some arguments are valid and, in fact, argument a is valid.
3. Argument a is valid and, in fact, every argument (in our universe of discourse) is valid.
4. Not only is argument a valid, but all sound arguments are valid. (hint)

Next...

Let's make these problems a little more interesting, and add one more "predicate" to talk about a particular sort of argument, those that are disjunctive syllogisms. For example, argument a above is a disjunctive syllogism. But other sound arguments have different forms (some are MP, some are valid but have forms without a name). We'll use 'Dx' to mean "x is a disjunctive syllogism".

Universe of Discourse = all sound arguments

Wx: x is valid.
Dx: x is a disjunctive syllogism.
a: the sound argument above in gray.

Break for a syntax lesson: Some of these have more than on quantifier, something like '(%x)Wx & (%x)Dx'. This is built up by...

1. Start with 'Wx' and 'Dx'. These are "open" sentences; they are like any English sentence with a blank: e.g., "___ is a valid argument".
2. Build up from these by attaching a quantifier on the left of each to get: '(%x)Wx' and '(%x)Dx'. Quantifiers -- along with their parentheses and the variable 'x' -- work just like a tilde as far as the correct construction of a sentence goes.
3. Finish by connecting the two sentences just formed with an ampersand. This gives '(%x)Wx & (%x)Dx' with main connective ampersand (the last connective used) .

OK, try some more.