1/7/06
and do the 6 exercises of Propositional Exercise 4 (Propex4).
Symbolize the following arguments, using the program's standard conventions. (Answers below-- you can ask the program to do it, step at a time, if you wish.)
a)
If philosophy is hard then philosophy is interesting.
Philosophy is hard.
Therefore
Philosophy is interesting.
b)
Logic is hard and philosophy is interesting.
Therefore
Philosophy is interesting or philosophy is hard.
c)
If philosophy is hard, then logic is hard .
If logic is hard, philosophy is interesting .
Therefore
If philosophy is hard
then philosophy is interesting.
Exercise 1 (of 6) (answers)
a)
H⊃I,
H
∴
I
b)
M∧I
∴
I∨H
c)
H⊃M,
M⊃I
∴
H⊃I
Sometimes English sentences are ambiguous. Try to symbolize the following, identifying ambiguities. Ask the program to do it (Select, Copy, click To Symbols).
a) Philosophy is hard and logic is hard or philosophy is interesting.
a) Philosophy is hard and logic is hard or philosophy is interesting.
b) Either philosophy is hard or logic is hard or philosophy is interesting.
(*The program finds rather more ambiguities in this than there really are.*)
b) Either philosophy is hard or logic is hard or philosophy is interesting.
c) It is not the case that philosophy is hard and logic is hard.
c) It is not the case that philosophy is hard and logic is hard.
When confronted with ambiguities, you just have to make the best you can with them. Often there is an obvious preferred meaning; otherwise courtesy suggests that you try to give an argument meaning that will make it valid. If ambiguities remain unresolved, just list them all.
English uses devices like commas and brackets to suggest intended meaning.
Try Satisfiable (in Deriver 'Games'->Propositional->Satisfiable). Make sure that you can get 10 out of 10 right in 2 minutes.
If it is possible for all of a collection or list of formulas to be true at one and the same time (ie they are simultaneously satisfiable), those formulas are Consistent. Try Consistent (in Deriver 'Games'->Propositional->Consistent). Make sure that you can get 10 out of 10 right in, say, 7 minutes.
If it is possible for all of the premises of an argument to be true, and the conclusion false, at one and the same time, that argument is Invalid (and the assignment that does this amounts to a Semantic Counter Example) . Try Invalid (in Deriver 'Games'->Propositional->Invalid). Make sure that you can get 10 out of 10 right in, say, 7 minutes.
You will need to copy and paste this into the Journal.
Usually when we write arguments out we write the premises in a list separated by commas, then a '∴' and then the conclusion. In the following exercise all the arguments are actually invalid.
See how you get on searching for a counter-example. What you have to do is to assign truth values to the atomic propositions, so that all the premises in an argument come out to be true and the conclusion false. Make your assignments and ask the program about the truth value of the premises and conclusion. If you are struggling with this select the accompanying list of premises with the negation of the conclusion and click Satisfiable? under the Semantics Menu-- the program will tell you which atomic propositions have to be true.
(*The program assumes that to start with all atomic proposition are assigned false. So you really only need tell it about the ones you want to be true. Similarly it will tell you only about the true ones. When the program tells you about satisfiability there will be some extra information that will be explained later in the course.*)
a) Q, P⊃Q ∴ P
assign true <atomic proposition(s)>
assign false <atomic proposition(s)>
a) satisfiability test list Q, P⊃Q,∼P
b) F∨G ∴ F∧G
assign true <atomic proposition(s)>
assign false <atomic proposition(s)>
b) satisfiability test list F∨G,∼(F∧G)
c) ∼A, A ⊃ B, B ⊃C ∴ ∼C
assign true <atomic proposition(s)>
assign false <atomic proposition(s)>
c) satisfiability test list ∼A, A ⊃ B, B ⊃C, ∼(∼C)