PHL310 Valid Reasoning II, past assignments
I'd like to see what system you learned, and where you are at.
Five problems, just for me to see. The first two are just
translations into some kind of formal logic. For problem 1, use a
propositional logic (the smallest things in your language will be
sentences). For problem two, use a quantified logic with
predicates. For 3-5, generate a proof (a syntactic proof, not a
truth table). Use whatever rules or proof methods you remember.
- If both Nemo and Jaws are fish, then neither Patrick
nor Mr. Krab are.
- All men are mortal.
- Premises: (P → Q), (R → Q), (P v R). Conclusion: Q.
- Conclusion: ((P → Q) → (¬Q → ¬P))
- Premises: ∀x(Fx ↔ Gx), ∃xFx.
Here's a homework with two tracks, since some of us are catching
up and some remember the material well. If remember the
propositional logic, do (or at least try) track 2. Else do
1. Read chapters 1 through 5 of A
Concise Introduction to Logic, and answer the following
questions from the book:
2. We can make use of a theorem -- or of a special
sentence called an "axiom" -- in the following way.
If you replace every occurence of a sentence in a theorem or
axiom with a different sentence (but the same replacement in
each case), you also get a theorem or axiom.
So, if ((P → Q) → (¬Q → ¬P)) is a
theorem, you can replace P with R, and Q with (S v T) and get
the following theorem:
((R → (SvT)) → (¬(SvT) → ¬R)).
- Chapter 2 question 5.
- Chapter 3 question 1a and question 2a.
- Chapter 4 question 1b, d, f, h.
- Chapter 5 questions 1 and 2d.
It turns out that we can do away with indirect derivation
if we allow ourselves certain sentences as axioms (an axiom is a
sentence that we assume--it's like a very special premise; but we
treat it the way we treat theorems in that we'll allow versions or
instances of it).
Prove the following:
Premises: (R → S), (¬S v ¬R).
But prove it without using indirect derivation, but rather a
direct derivation and also by using the following (((P → Q) ^
(P → ¬Q)) → ¬P) as your axiom. (If you feel
stuck, first prove it using indirect derivation, and then look at
your proof and think of how you can use that axiom....) The way
to use the axiom is: any time you like, you can write down either
(((P → Q) ^ (P → ¬Q)) → ¬P), or you can
write a sentence that is made from that by replacing each P with
some sentence, and each Q with some sentence. Your justification
on the right is "axiom".
Try this one if you feel ambitious. Using that axiom, can you
prove without indirect derivation:
Premises: (P → R), (Q → R), (P v Q).
You asked for hints. They're not that hard, actually. Consider
first whether you might not just get the conclusion from the
axiom and modus ponens. In that case, the instance of the axiom
you'd want for the first problem is:
(((R → φ) ^ (R → ¬ φ)) → ¬R)
. And for the second would be:
(((¬R → ψ) ^ (¬R → ¬ ψ)) → ¬¬R)
In each case, what will you use for φ, what will you use for