Friday, June 08, 2012

Premiseless Argument

We often think of premises as necessary for inferences, but this depends to some extent on what one counts as premises. Here is a simple example, although since it uses a system that most people don't know, Peirce's existential graphs, it needs a bit of explaining.

Peirce's existential graphs are a purely diagrammatic way of doing logic, in which logical steps consists primarily of making and erasing shaded ovals on a blank sheet of paper according to some very, very basic rules. Obviously it would be a pain to reproduce the shaded and unshaded ovals here, so we'll do a work around using brackets; you can imagine a pair of brackets as a large oval on a page. Whether it is shaded or not depends on what surrounds it. Thus this is a shaded oval:

[ ]

This is an unshaded oval in a shaded oval:

[ [ ] ]

This is a shaded oval within an unshaded oval within a shaded oval:

[ [ [ ] ] ]

So if you count pairs of brackets, the first or outermost pair is always shaded, and every even pair represents an unshaded oval, every odd pair represents a shaded oval.

The basic way the system works is this. You take a blank sheet of paper, which Peirce calls the Sheet of Assertion, and this is your universe of discourse -- the whole universe of things that are relevant to whatever you'll be talking about. You draw your premises on the sheet with pencil. A shaded area represents a negative. An unshaded area represents the positive. A line to the left, called the line of assertion indicates that something is definitely in the universe of discourse (colloquially, that some exists). This allows you to say, "There is something that is not a phoenix":

-[-phoenix]

And also "There is nothing that is a phoenix":

[-phoenix]

And things like "There is a phoenix who rises" and "There is a phoenix who does not rise" and "There is no phoenix who rises":

-phoenix-rise
-phoneix-[-rise]
[-phoenix-rise]

There are three things you can do once you have any premises drawn, which Peirce calls Permissions. They are, roughly:

(1) You may erase any graph-instance on an unshaded area as you please, and you may insert a graph-instance on any shaded area that already exists. (Shaded areas themselves are not considered graph-instances.)

(2) Any graph-instance may be repeated in the same area, or in any area enclosed within that area, provided that any lines of assertion have the same features each time; and for any graph-instance already repeated in this way, the innermost instance (or either if they are in the same area) may be erased.

(3) Any vacant ring-shaped area may be collapsed; any vacant ring-shaped area may be created by shading and erasure. (An area is not vacant if crossed by a line of assertion, even if nothing else is in it.)

And that's all; with this you can do nearly everything you would learn in an undergraduate logic class. It's a bit tricky to use, at times, because it wasn't designed to be easy to use but to break down reasoning to its very bare essentials. But you can do predicate calculus with it (and it can be extended even further to do modal logic). And as it happens, we don't need the full system here, because we won't need lines of assertion. And in this context we can see that you can have an argument that does not start with any premises.

(1) We start with the blank sheet.

(2) Third Permission allows us to draw

[ [ ] ]

(3) Then Third Permission allows us again to put in more ovals:

[ [ [ [ ] ] ] ]

(4) Then First Permission allows us to add a letter representing a proposition, any proposition you please, in a shaded area:

[ a [ [ [ ] ] ] ]

(5) Then Second Permission allows us to repeat the letter in another area:

[ a [ [ [ a ] ] ] ]

(6) Then First Permission allows us to add another letter:

[ a [ [ b [ a ] ] ] ]

And this, as it happens, is logically equivalent to "If a, then if b, a" or as we would usually represent it: a -> (b -> a). And we started with a blank sheet empty of premises. (If you want to see how this looks in the real graph format, see Sowa's commentary about halfway down.)

Of course, it's true that you can't get conclusions if you don't have anything at all to start with. Besides premises there are two other sources of information: the universe of discourse itself and the rules of inference. In this case we've drawn out a limit of possibility for the three Permissions -- no matter what universe of discourse you are in, whatever result you get using the Permissions will be consistent with a -> (b -> a). So we can call the universe of discourse and the rules of inference principles of argument, and say that every argument requires principles. But not every argument requires premises, as we see here. The only possible alternative is to claim, Tortoise-like, that (1) positing a universe of discourse and (2) every rule of inference are premises, in which case, also Tortoise-like, you are really claiming that all arguments are infinitely dense -- between every premise there are infinitely many premises, whether we explicitly name them or not -- and infinitely long -- whenever we identify a premise, there are infinitely many premises already on the table, namely, all the rules of logic and their every possible combination. You can have some arguments with no premises, or every argument with infinitely many; you can take your pick, but you are stuck with one of the two.

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