Re: truth/falsity of sentences in first-order logic

H. J. Sander Bruggink wrote:
Charlie-Boo wrote:

What does the definition of how to determine if 1+1=3 is true or not
have to do with models? How would you determine if 1+1=3 and how would
you use models to do that?

1. Find the object in the model which is denoted by the
term "1";

I said 1+1=2 not f2(f1(a1),f1(a1))=f1(f1(f1(a1)))

1 is the object in the system. a1 would be a term.

2. find the binary function in the model which is denoted by
the function symbol "+";

+ is a binary function not a function symbol. f2 would be a function
symbol.

3. in this function, find the tuple where the object of step
(1) is in the first two positions (because it's a function,
there is exactly one such tuple);
4. look at the object in the third position of the tuple;
5. compare this object to the object in the model denoted by
the term "3";
6. if the two objects are equal, output "true", else output
"false".

This is the only necessary step. The rest is just needless (in fact,
it is a hinderance) playing with variables instead of getting directly
to the point of what the expression is, mathematically (the functions
and relations and net result - the truth value.)

If the model is a model for any conventional arithmetical
theory, then this procedure will actually output "false".

That's because Models are Backwards. You are looking up something and
you already know what it has to be in order for your system to work!
You've lost the freedom of making the axioms dependent on the actual
functions and relations - which they are dependent on!

I wouldn't call it one of the most basic concepts

The concept is explained in any first year logic course.

(quantified: What
portion of books or articles that teach logic exclude it?

Depending on what you mean by "teach logic", I'd say 0%.

One example and you're wrong?

I could
survey if need be.), but what is the relevance of whether it is basic
or not? It always confounds me that some people like to talk about how
certain things are "basic" or "fundamental" or "elementary"
- what's the point of that?

Suppose I claimed I was an expert in cooking, and that I,
in fact, revolutionized haute cuisine. But then, after
some time, it turns out that I can't even boil an egg.
Would you believe me?

If boiling an egg is required to revolutionize haute cuisine then logic
says the statement is false. But what does that have to do with the
price of tea in China? We're talking about verifying mathematical
assertions, not people's bragging rights. What's the point of that?

groente
-- Sander

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