You should now Launch Deriver and do the 3 exercises of Predicate Exercise 3 (Predex3).
This is going to be a little more involved to set up. The Interpretation in Predex3 shows an Interpretation similar to this
and it certainly would be useful if your running Deriver also displayed this same Interpretation.
There is an issue here, though. (For the geeks among you it is that there is no easy way to get an intelligent image, displayed in a Web Browser, into a separate running program, namely Deriver.)
There are a few different workarounds to achieve the desired end:
Use the Interpretation Tab to show drawing or interpretation. Consider a) whether the terms ('names') in the following formulas refer properly to entities in the Universe, and b) whether each formula is true or whether it is false. Select the formula and click True? (under the Semantics Menu) to see what the program thinks.
a) Fa
b) Ga
c) Fb
d) Gb
e) Fa≡Gb
f) Ga∧Fb
h) Fb⊃Gb
f) Gc
g) Ha
h) Fa ∨ Fc (*see Note*)
(*Note. There are different ways of handling a case like this. On the one hand you can say that Fa is true, therefore the disjunct is true; or you can say that Fc is not entirely proper, therefore it is inappropriate to ask about the truth value of the disjunct. We follow the latter here; but there are occasions when it would be better to follow the former.*)
This exercise is similar to the first, but this time we are interested in lists of formulas and the 'simultaneous truth' of all the formulas in a list. A list of formulas is just some formulas separated by commas. And a list of formulas is true if and only if all the formulas in the list are true. (So, if at least one formula in a list is false, the whole list is taken to be false.)
Use the Interpretation Tab to show drawing. Consider whether each list of formulas is true or whether it is false. Select the list of formulas and click True? to see what the program thinks.
a) Fa, Gb
b) Fa, Fa⊃Gb,~Gb
c) Ga∧Fa, Ha
d) Gb∨Ha, Fa
e) Ra∧Rb,Fa
When we are trying to establish that an argument is invalid, we look for a counter-example to it. A counter-example is a world or scenario in which all the premises are true and the conclusion false.
For example, consider the argument
Therefore
Bert is famous or Arthur is a gambler.
which symbolizes to
(Fb∨Ga)⊃(Fa∧Gb)
Gb
Fa
∴
Fb∨Ga
ask the program about the truth of each of these formulas individually. Notice that in the drawn world all the premises are true and the conclusion false-- the depicted world constitutes a counter-example to the argument. It proves that the argument is invalid.
We can also use a device that we used in propositional logic. Instead of trying to see if all the premises can be true and the conclusion false at one and the same time, we can make a list of all the premises and the negation of the conclusion. Thus
(Fb∨Ga)⊃(Fa∧Gb), Gb, Fa, ~(Fb∨Ga)
now ask the program if this entire list of formulas is true.
It will tell you that they are. In the jargon, the set of formulas is 'satisfiable'.
In practical cases you will be given the argument and have to produce the counter-example drawing yourself.
The program can do this for you.
Open a new Browser (under the File Menu).
Copy the list of formulas
(Fb∨Ga)⊃(Fa∧Gb), Gb, Fa, ~(Fb∨Ga)
and paste it into the new journal. Select it in the new journal and click Satisfiable?, and look at the drawing.
[Copying to a new journal is not an essential part of Satisfiable? , on this occasion we are just trying to preserve the original file as is.]