Having grown up on a farm where travel and transport were by horse and cart, I have finally worked out half a century later what it is that I’m missing each time I sit in traffic staring at the back of a car in front of me. The cleavage.
The hard edge of the urban cyclist is bent in the shape of the cycle frame; the soft edge is sadistically inclined.
Each time I walk over a pavement that has just been immaculately laid, I sense a water main underneath smirking.
Translate then derive the conclusion.
A horse is an animal. Therefore, whoever owns a horse owns an animal. (Hx: x is a horse; Ax: x is an animal; Oxy: x owns y)
The difficulty here is to translate the conclusion correctly. Before we do that, it helps to paraphrase it:
Therefore, anyone who owns a horse owns an animal.
Therefore, if someone owns a horse, then he owns an animal.
or, less elegantly but, as is almost always the case, more helpfully:
Therefore, if for any person there is a horse that he owns, then there is an animal that he owns.
The last translation in particular gives us a hint as to how to translate it into formal logic. Only ‘he’ is anaphoric – which will make the overall sentence structure a conditional – not the ‘horse’. This is not a donkey type sentence. Hence, two existential quantifiers – one for the antecedent, one for the consequent of the conditional.
1. (x)(Hx ⊃ Ax) ∴ (z)[(∃x)(Hx • Ozx) ⊃ (∃y)(Ay • Ozy)]
24. ~ ~ (z)[(∃x)(Hx • Ozx) ⊃ (∃y)(Ay • Ozy)] 25. (z)[(∃x)(Hx • Ozx) ⊃ (∃y)(Ay • Ozy)] |
AIP 2 Impl 3 CQ 4 CQ 5 DM 6 CQ 7 CQ 8 Impl 9 DM 10 EI 11 Simp 12 EI 1 UI 13 Simp 14,15 MP 11 Com 17 Simp 18 UI 16,19 DS 13 Com 21 Simp 20-23 Conj 2-23 IP 24 DN |
The Big Four – a designation often given to the species at the top of the food chain whose life consists largely of striking poses and cultivating their image in the afternoon sun or popular media channels, punctuated periodically by bouts of feeding frenzy
Collar – part of a garment such as a coat or a shirt that fitted around the neck when there were necks, but recently evolution has surprised us all and started making redundancies in the human body by dispensing first with the neck and joining the base of the head directly with the oesophagus, so collars have been retired.
The press – alternative pronunciation of ‘depress’
Translate then derive the conclusion.
Whoever is a friend of either Michael or Paul will receive a gift. If Michael has any friends, then Eileen is one of them. Therefore, if Anne is a friend of Michael, then Eileen will receive a gift. (Fxy: x is a friend of y, Rx: will receive a gift)
1. (x)[(Fxm v Fxp) ⊃ Rx] 2. (∃x)Fxm ⊃ Fem ∴ Fam ⊃ Re
9. Fam ⊃ Re |
ACP 3 EG 2,4 MP 1 UI 5 Add 6,7 MP 3-8 CP |
Why don’t we trust people who contradict themselves? This is a no-brainer: because one of their statements must be a lie:
Face masks protect against infection. (P)
Face masks do not protect against infection. (Not P)
But another reason, less often articulated though grasped intuitively by any sane person, is that anything follows from a contradiction, including a lie.
If someone is known to have said both of these statements in public, they might want to defend one of them (no matter which) by smuggling in another statement:
Either that or bleach is a cure for viral infection. (Q)
We defend the truth of one statement by disjoining a new statement to it (more on that in the previous post). By the rule of disjunctive syllogism, the new statement will be stripped of company and made to stand out unsupported. Contradictory statements cancel each other out.
Premise 1: P
Premise 2: Not P
Addition 3: P or Q
Inference: Therefore, Q.
A lie abhors a solo appearance. It likes company. For a lie to thrive, it likes to be shielded by truths (and other lies alike).
The language of deductive logic does a rather poor job of tracking the rhythms and patterns of natural language, but it can shed some light on linguistic behaviour. This post is a short introduction to the consequences of contradiction, discussed next.
Two arguing positions require an expenditure of energy: when we challenge the truth of our opponent’s statement and when we defend the truth of ours. How do we do this?
Facts, figures, examples, sometimes definitions work best. Eloquence helps. This is the high art of arguing. The low art – the circus or Punch-and-Judy type of argument – is different, and not entirely unfamiliar to logicians.
To challenge an opponent’s statement, we conjoin a ridiculously false statement of our own (the ridicule amplifies the effect) to our opponent’s statement. ‘And pigs might fly,’ is the English language handy option. The key word is ‘and’. Whether the opponent’s statement is true or false, ‘and’ will make the whole statement false. Conjunction is true only if both conjuncts are true.
How do we defend the truth of our statement? We add a new statement to the argument – by disjunction. This is a more subtle technique. Here is an illustration:
A: There is a deep state in America.
B: Really?
A: Yes. Either that or we are being manipulated by big corporations.
The key word is ‘or’. If we somehow manage to legitimize just one of these statements, the whole comes out true, because a disjunction is false only when both disjuncts are false.
The little turd who built a crude version of your sandcastle in his corner of the sandpit and then came across and kicked down yours? He will re-emerge later in your life as the second undermanager who loops your ‘l’s and closes your ‘o’s.
History is cyclical and now is the age of the cyclist. When civilisation is restored, hard questions will be asked.
I have played with one hand under the card table all my life, so I was not surprised that, when I opened my facebook account, facebook had three tabs for me: enemy requests, enemy suggestions and all my enemies.