Symbolic Logic, Dale Jacquette, Wadsworth, 2001, Chpt. 8, IV(12), p. 435

The argument:

"Everything depends on something. All things that depend on everything are sustained by them. Everything that sustains all things depends on them. Therefore, there is something that sustains and is sustained by something."

1. (x)(∃y)Dxy

2. (x)(y)(Dxy ⊃Syx)

3. (x)(y)(Sxy ⊃Dxy)

4. ∴(∃x)(∃y)(Sxy • Syx)

5. (∃y)Dxy ......... 1 UI x/x

6. Dxa ......... 5 EI y/a

7. (y)(Dxy ⊃Syx) ......... 2 UI x/x

8. Dxa ⊃Sax ......... 7 UI y/a

9. Sax ......... 6,8 MP

10. (y)(Say ⊃Day) ......... 3 UI x/a

11. Sax ⊃Dax ......... 10 UI y/x

12. Dax ......... 9,11 MP

13. (y)(Day ⊃Sya) ......... 2 UI x/a

14. Dax ⊃Sxa ......... 13 UI y/x

15. Sxa ......... 12,14 MP

16. Sxa • Sax ......... 15,9 Conj.

17. (∃y)(Sxy • Syx) ......... 16 EG a/y

18. (∃x)(∃y)(Sxy • Syx) ......... 17 EG x/x

Symbolic Logic, Dale Jacquette, Wadsworth, 2001, Chpt. 8, IV(11), p. 435

The argument:

Not everyone has the good fortune to come from Corinth. There are metaphysicians or there are epistemologists only if all have the good fortune to come from Corinth. Thus, there are no metaphysicians.

and the proof:

1. ¬ (x)Cx
2. (∃x)(Mx ∨Ex) ⊃(x)Cx
3. ∴¬ (∃x)Mx
4. ¬ (∃x)(Mx ∨Ex) ......... 1,2 MT
5. (x) ¬ (Mx ∨Ex) ......... 4 QC
6. (x)(¬Mx • ¬ Ex) ......... 5 DeM
7. ¬Mx • ¬ Ex ......... 6 UI x/x
8. ¬Mx ......... 7 Simp.
9. (x) ¬Mx ......... 8 UG
10. ¬ (∃x)Mx ......... 9 QC

Crossverb puzzles

I haven’t come across many of those (none to be precise) so I am going to redress the balance, though perhaps not here and not just yet. The way I see it is they are long overdue.

We are given to understand that crossword puzzles are a good way to exercise our brain. Likewise, I haven’t heard that wisdom questioned very often. Personally, I find them uninspiring. I wonder if this is because the majority require that the answers be nouns, or at least something akin to nouns.

Where the answers are not nouns, as in full sentences, expressions or verbs indeed, the clues are formulated in terms of a definition or, most often, a paraphrase. Definitions is something that works best with nouns. So nouns cast a long shadow one way or another. Verbs defy definitions – they demand examples of usage.

Conventional crossword puzzles are an amusement best suited to speakers whose first language is the same as the language of the clues. Perhaps you get a kick out of guessing that ‘Polly and Esther’ make ‘polyester’ if you are doing the Daily Telegraph crossword over breakfast in Tunbridge Wells, but from a language learner’s point of view, the educational benefit of this sort of nonsense is zero.

Verbs suffer a bad image, too. The speech of educated classes is full of nouns. The hoi polloi and the lumpen proletariat bark out verbs. Well, I am of the latter stock and proud of it. Layers of education will not smother me’ verbs.

The problem with verbs is that they are polymorphic (where did I pick it up?). As well as tenses, we have to contend with the transitive v intransitive distinction, perfective v imperfective aspect, simple v compound, stative v dynamic, main v auxiliary, and so on.

But I don’t think crossverb puzzles are a lost cause. The ways of setting up the grid are explained in many places on the internet – I am concerned with the content. To start off, I propose a verb spelt downwards in the centre. Let it be: CHUCKED. This will act as an anchor and indicate the tense of all other verbs in the puzzle. The clues have blanks where the verbs would be. The clues confound the most common collocations but do not stray beyond the ordinary. Thus, “The cat _ _ _ _ _ _ the mouse once more,” where perhaps the last cell in the clued verb is crossed with the last letter of the verb given (here: D). Hence, the answer is not ‘caught’ but, for example, ‘tossed’.

A moment’s reflection suggests that there are many ways of setting crossverb puzzles. Why is no one doing it?

Symbolic Logic, Dale Jacquette, Wadsworth, 2001, Chpt. 8, IV(6), p.435

A whimiscal argument to round off a rather uninspiring week:

Something is rotten in the state of Denmark only if all 3-day-old herrings are putrifying. There are at least some 3-day-old herrings if there are at least some 3-day-old mackerels. Thus, we may conclude that something rotten in the state of Denmark is a 3-day-old mackerel only if something is putrifying.

The works:

1. (∃x)Rxd ⊃(x)[(Tx • Hx) ⊃ Px]
2. (∃x)(Tx • Mx) ⊃(∃x)(Tx • Hx)
3. ∴(∃x)(Rxd • Tx • Mx) ⊃(∃x)Px
4. * (∃x)(Rxd • Tx • Mx) ......... ACP
5. * Rad • Ta • Ma ......... 4 EI x/a
6. * Ta • Ma ......... 5 Simp.
7. * (∃x)(Tx • Mx) ......... 6 EG
8. * (∃x)(Tx • Hx) ......... 2,7 MP
9. * Tm • Hm ......... 8 EI x/m
10. * Rad ......... 5 Simp.
11. * (∃x)Rxd ......... 10 EG
12. * (x)[(Tx • Hx) ⊃ Px] ......... 11,1 MP
13. * (Tm • Hm) ⊃ Pm ......... 12 UI x/m
14. * Pm ......... 9,13 MP
15. * (∃x)Px ......... 14 EG
16. (∃x)(Rxd • Tx • Mx) ⊃(∃x)Px ......... 4-15 CP

Reductio ad absurdum

The saying goes that the hardest thing to put up with is a good example. There is actually a harder thing to put up with – a counterexample. In logic, a counterexample works by supposing that our conclusion is the opposite of what it says. Via a series of moves we then show that the opposite cannot hold, so the original conclusion is true. For example:

If elephants are pink, then elephants have dangly trunks.
If elephants have floppy ears, then if they have dangly trunks, then they are either clever or have no feelings.
Elephants are not clever and elephants have feelings.

The conclusion is: Elephants are not pink.

And this conclusion follows from our premises! To show that it does, we can simply negate it: It is not the case that elephants are not pink. The conclusion follows not because ‘Elephants are pink’ (after we cancel double negation) is false, but because the argument a) is not of the kind where the premises are true and the conclusion is false, and b) because the rules of deduction operate is such a way as to yield the conclusion (I leave out the workings here).

In everyday speech we hardly ever use such arguments, but we do use reductio ad absurdum – often unnecessarily and annoyingly, but mostly to sound clever.

It’s a feature of everyday speech that we use rhetoric and hyperbole. With them our speech sounds exaggerated, without them – dull. We have heard this rehearsed many times: - I shall defend to the last my freedom of speech. - So, that means you can go and offend anyone you please because you believe in freedom of speech? The counterexample here is meant to knock down the sweeping claim made by the first speaker.

In such cases the second speaker deserves to be told that freedom of speech is a privilege, not an obligation, and that discretion is the better part of valor, and that failure to draw the distinction is an affliction of small minds.

Another multiply fallacious example is: - I studied philosophy. – No wonder you are poor. Here, a series of hypothetical syllogisms produces the outcome: I studied philosophy. Anyone who studies philosophy is not practically-minded. Anyone who is not practically-minded has no knack for business. Anyone who has no knack for business is poor. So, anyone who studied philosophy is poor. The logic works, but the maxim of cooperation (in communication) doesn’t.

In the end, absurd is as absurd does.

Symbolic Logic, Dale Jacquette, Wadsworth, 2001, Chpt. 8, IV(5), p.435

Prove:

Everything is either an unconscious or immaterial entity. All sentient beings, if they are physically embodied, are material entities. Hence, no sentient physically embodied beings are conscious.

The argument looks daunting, but in fact isn't. It is a good example, perhaps, of how the shape and sound of words stand in the way of our seeing the message. Once we get past the symbolisation, we can practically walk through to the conclusion.

1. (x)[Ex ⊃(¬Cx ∨¬ Mx)]
2. (x){(Sx • Bx) ⊃[Px ⊃(Mx • Ex)]}
3. ∴(x)[(Sx • Px • Bx) ⊃¬ Cx]
4. * Sx • Px • Bx ......... ACP
5. * Sx • Bx ......... 4 Simp.
6. * (Sx • Bx) ⊃[Px ⊃(Mx • Ex)] ......... 2 UI x/x
7. * Px ⊃(Mx • Ex) ......... 5,6 MP
8. * Px ......... 4 Simp.
9. * Mx • Ex ......... 8,7 MP
10. * Ex ......... 9 Simp.
11. * Ex ⊃(¬Cx ∨¬ Mx) ......... 1 UI x/x
12. * ¬Cx ∨¬ Mx ......... 10,11 MP
13. * Mx ......... 9 Simp.
14. * ¬Cx ......... 13,12 DS
15. (Sx • Px • Bx) ⊃¬ Cx ......... 4-14 CP
16. (x)[(Sx • Px • Bx) ⊃¬ Cx] ......... 15 UG

Symbolic Logic, Dale Jacquette, Wadsworth, 2001, Chpt. 8, IV(4), p. 434

The argument is:

Every honcho is both a honcho and a bigwig. Some nonbigwigs are movers and shakers. So, it follows immediately that some movers and shakers are nonhonchos.

The proof:

1. (x)[Hx ⊃(Hx • Bx)]
2. (∃x)(¬ Bx • Mx • Sx)
3. ∴(∃x)(Mx • Sx • ¬ Hx)
4. ¬ Bm • Mm • Sm ......... 2 EI x/m
5. Hm ⊃(Hm • Bm) ......... 1 UI x/m
6. ¬ Hm ∨(Hm • Bm) ......... 5 MI
7. (¬ Hm ∨Hm ) • (¬ Hm ∨Bm) ......... 6 Dist.
8. ¬ Hm ∨Bm ......... 7 Simp.
9. ¬ Bm ......... 4 Simp.
10. ¬ Hm ......... 9,8 DS
11. Mm • Sm ......... 4 Simp.
12. ¬ Hm • Mm • Sm ......... 10,11 Conj.
13. Mm • Sm • ¬ Hm ......... 12 Comm.
14. (∃x)(Mx • Sx • ¬ Hx) ......... 13 EG