Some Logical(?) Arguments

Click on question number to see the analysis

And another selection from Lewis Carroll:

1. Babies are illogical. Nobody is despised who can manage a crocodile. Illogical persons are despised. Therefore, babies can not manage crocodiles.

2. There are no pencils of mine in this box. No candies of mine are cigars. The whole of my property, that is not in this box, consists of cigars. Therefore, no pencils of mine are candies.

3. All puddings are nice. This dish is a pudding. No nice things are wholesome. Therefore, this dish is unwholesome.

4. No interesting poems are unpopular among people of real taste. No modern poetry is free from affectation. All your poems are on the subject of soap bubbles. No affected poetry is popular among people of real taste. No ancient poem is on the subject of soap bubbles. Therefore, all your poems are uninteresting.

5. I trust every animal that belongs to me. Dogs gnaw bones. I admit no animals into my study, unless they will beg when told to do so. All the animals in the yard are mine. I admit every animal, that I trust, into my study. The only animals, that are really willing to beg when told to do so, are dogs. Therefore, all the animals in the yard gnaw bones.


Answer to 1: In the universe of people, let A(x) = "x can manage a crocodile", B(x) = "x is a baby", C(x) = "x is despised" and D(x) = "x is logical". The given statements can be written symbolically (with some equivalent forms) as:
(x)(B(x)~D(x)
(x)(A(x)~C(x)
(x)(~D(x)C(x))
(x)(B(x)~A(x)).
So, for all x, B(x)~D(x)C(x), but by the contrapositive of the second statement we have that for all x, ~W(x)~O(x), so we can conclude that (x)(B(x)~A(x)).

Answer to 2: In the universe of my things, let A(x) = "x is a cigar", B(x) = "x is in this box", C(x) = "x is a pencil" and D(x) = "x is a candy". Symbolically we have:

(x)(C(x)~B(x)),
(x)(D(x)~A(x))
(x)(~B(x)A(x))
(x)(C(x)~D(x))
So, for all x, C(x)~B(x) A(x) (by the contrapositive of the second statement)~D(x). Thus, (x)(C(x)~D(x)).

Answer to 3: In the universe of things, let A(x) = "x is a nice thing", B(x) = "x is a pudding", C(x) = "x is this dish", D(x) = "x is wholesome". Symbolically, we have:

(x)(B(x)A(x)),
(x)(C(x)B(x)),
(x)(D(x)~A(x)),
(x)(C(x)~D(x)).
We have for all x, C(x)B(x)A(x)~D(x) by the contrapositive of the third statement. So(x)(C(x)~D(x)).

Answer to 4: In the universe of poems, let A(x) = "x is affected", B(x) = "x is ancient", C(x) = "x interesting", D(x) = "x on the subject of soap bubbles", E(x) = "x is popular among people with real taste", H(x) = "x was written by you". Symbolically we have:

(x)(C(x)E(x)),
(x)(~B(x)A(x)),
(x)(H(x)D(x)),
(x)(A(x)~E(x)),
(x)(B(x)~D(x)),
(x)(H(x)~C(x)).
Using contrapositives when needed, we have the following chain of implications, for all x, H(x)D(x)~B(x)A(x)~E(x)~C(x).

Answer to 5: In the universe of animals, let A(x) = "x is admitted to my study", B(x) = "x is trusted by me", C(x) = "x is a dog", D(x) = "x gnaws bones", E(x) = "x is in the yard", H(x) = "x is mine", and K(x) = "x is willing to beg when told". Symbolically we have:

(x)(H(x)B(x)),
(x)(C(x)D(x)),
(x)(A(x)K(x)),
(x)(E(x)H(x)),
(x)(B(x)A(x)),
(x)(K(x)C(x)),
(x)(E(x)D(x)).
We have the following chain of implications, for all x, E(x)H(x)B(x)A(x)K(x)C(x)D(x).