# Prover9 and Mace4

Just found two fantastic programs and a GUI for exploring first-order classical models and also automated proof, Prover9 and Mace4. There are many other theorem provers and model checkers out there. This one is special as it comes as a self-contained and easy to use package for Windows and Macs.

There are many impressive examples built in which you can play with. To start easy, I gave it a little syllogism:

all B are A

no B are C

with existential presupposition, which is expressed simply:

exists x a(x).

exists x b(x).

exists x c(x).

all x (b(x) -> a(x)).

all x (b(x) -> -c(x)).

and asked it to find a model. Out popped a model with two individuals, named 0 and 1:

a(0).

– a(1).b(0).

– b(1).– c(0).

c(1).

So individual 0 is an *A*, a *B*, but not a *C*. Individual 1 is not an *A*, nor a *B*, but is a *C*.

Then I requested a counterexample to the conclusion *no C are A*:

a(0).

a(1).b(0).

– b(1).– c(0).

c(1).

The premises are true in this model, but the conclusion is false.

Finally, does the conclusion *some A are not C* follow from the premises?

2 (exists x b(x)) [assumption].

4 (all x (b(x) -> a(x))) [assumption].

5 (all x (b(x) -> -c(x))) [assumption].

6 (exists x (a(x) & -c(x))) [goal].

7 -a(x) | c(x). [deny(6)].

9 -b(x) | a(x). [clausify(4)].

10 -b(x) | -c(x). [clausify(5)].

11 b(c2). [clausify(2)].

12 c(x) | -b(x). [resolve(7,a,9,b)].

13 -c(c2). [resolve(10,a,11,a)].

16 c(c2). [resolve(12,b,11,a)].

17 $F. [resolve(16,a,13,a)].

Indeed it does. Unfortunately the proofs aren’t very pretty as everything is rewritten in normal forms. One thing I want to play with is how non-classical logics may be embedded in this system.