English to Logic

Formalize the following arguments as syntactic sequents of the propositional calculus,
giving an explicit interpretation of your sentence-letters.

The murderer was either Colonel Mustard or Professor Plum.  But it wasn’t
Professor Plum.  So it was Colonel Mustard.

P = The murderer was Colonel Mustard
Q = The murderer was Professor Plum

(P x Q) ^ not Q => P

Picture 22

We can see from the table that (P x Q) ^ not Q => P is a logical implication (tautology) and that we can conclude that P must be true if P x Q ^ not P is true. Therefore it is proved that The murderer was Colonel Mustard.

Either the Master or the Dean was in the library.  But if the Master wasn’t there,
the Dean wasn’t there either.  So they were both in the library.

P = the Master was in the library.
Q = the Dean was in the library.

not (P x Q) => (P ^ Q)

Picture 23

We can see from the table that not (p x q) does imply (p ^ q) and that p and q are true, thus it is proven that the Master was in the library and the Dean was in the library.

You can only buy a Young Persons railcard if you’re under 26 or a student;
otherwise not.  If you can buy a Young Persons railcard, you can get discounted
train tickets.  But you’re not under 26.  So unless you’re a student, you can’t get
discounted train tickets

P = You are under 26
Q = You are a student
R = You can buy a Young person rail card
S = You can get a discount

not (((P v Q) => R) => S) => not S v Q

(I cannot be bothered to do a truth table for this – so  cannot prove that my answer is correct – but I am pretty sure that is – so mybe you would like to do one and let me know?)

If God is willing to prevent suffering, but unable to do so, He is not omnipotent. If
He is able to prevent suffering, but unwilling to do so, He is not loving. If God
exists, He is loving and omnipotent. And if He is both willing and able to prevent
suffering, then there can’t be any suffering – but there is. So God doesn’t exist.

P = God is willing to prevent suffering
Q = God is omnipotent
R = God is able to prevent suffering
S = God is loving
T = God exists
U = There is no suffering

not (((P ^ not R) => not Q) ^ ((P ^ not P) => not S) ^ ((P ^ R) => not U)) => not T

(I cannot be bothered to do a truth table for this – so  cannot prove that my answer is correct – but I am pretty sure that is – so maybe you would like to do one and let me know?)

The protesters will go away if Oxford stops experiments on animals. But this
could only happen with government intervention. So, unless the government
intervenes, they won’t go away

P = The protesters will go away
Q = Oxford stops experiments on animals
R = the government intervenes

not ((Q => P) => R) => not P

Picture 24

Here we can see that not ((Q => P) => R) => not P , not P and not ((Q => P) => R) are all true, thus we can conclude that not ((Q => P) => R) => not P is true.

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3 Comments

  1. Hi, very promising site, am sure to go back! Ive made one of the riskiest decisions in my life, and its to weasel my way into Discrete Structures in my postgrad. I hope I can swallow what Ive chewed! Just a comment re “If God…”, dont worry, no rant here. Strictly speaking, the argument (by the book) is leakproof. But singling “God” out as a being by going as far as name him/her/it/etc, then identifying him as “not existing”, makes the whole chain of argument sound cooky. anyway, jez my two cents.

  2. Faulty premise: If God is able to prevent suffering and unwilling to do so, He is not loving. I am not willing to prevent my children from ever suffering, particularly when they willfully choose to risk it. I love my children and I know that preventing all suffering is not in their best interest. I care more about their long term well being than their immediate gratification. I love them enough to let them learn from their errors.

    The value of (p implies q) is F.

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