Re: Connectivity of a graph in FO

On Thu, 29 Jun 2006, un student wrote:

un student wrote:
William Elliot wrote:
That particular FO sentence with two free variables is a binary relation.

Maybe the problem is here... Which sentences of FO with two free
variables are binary relations? All of them?

Now that I thinked of this I have understood models and logics are as:

A model M is a triplet M=(D,F,S) where D is universe, ie elements with
no inner structure, S is alphabet, ie the symbols we use for relations,
functions and constants and F is the interpretation function which
gives the symbols their interpretation on this particular model. Now
the interpretation function is the one which defines actually all
properties, expect the cardinality of D, of model. It gives function
symbols their meanings (for example the natural meaning of "+" etc) and
so on.

Now, logic L is a triplet L=(s, O, R) where s is the alphabet, O is the
set of rules by which we are allowed to create wffs and R is the
function which gives the truth values for the wffs. Now s is subset of

R is not part of the logic. The rules of inference are.

S, that is L and M share the alphabet in the sense that logic uses the
symbols of M in defining the language. Actually logic is model
dependent because it uses the models alphabet in defining the wffs.

Baloney.

When we create a sentence with two free variables in our logic L and
interpreted is as a binary relation, doesn't that mean that we are
defining a new model M' which satisfies the sentence we created? Or
actually a set of, possibly isomorphic, models?

No, it means you interpreted a sentence with two free variable as a binary
relation within the model which you used to interpreted the sentence.

I still don't quite get why we are allowed to change arbitrary binary
relation in a sentence with arbitrary sentence with two free variables.

You aren't. See other post. You had a model. Within that model, you
created a binary relation P. The model interpreted the language with a
binary relation symbol E as S. Then you created P. Finally you want
to validate that the same model, except for interpreting E as S, is
another model for the language when E is assigned P.

Let's say you've building plans for a house. Now you want to interpret
those plans for a housewife. You choose color of walls, type of windows,
select appliances and their colors, type and finish of kitchen counter,
etc. So I show her a finished virtual model, complete with furniture.
She likes the color of the sofa and wants the color of the rug to be the
same.

Now you tell me I can't color the rug with the same color? No problem.
Now I've created another different model home interior.
.

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