But not in the same way that obviousness is. You simply referred to some 'oddity' of 'logical implication'. Which oddity? And does 'logical implication' refer to logical entailment or material implication? It was an ambiguous answer. I couldn't discern whether you understood why the second argument was valid; and this came right after you incorrectly told me the first argument was an identity. At that point, it was far from clear to me that you knew what you were talking about.
Truthtables are useful, especially when checking for consistency. However, they are not the same thing as a good explanation, at least to me.It's pretty easy to explain with a truth tablewe just use the thing that I added to the original response, slap in some rows where p's and q's are 1s and 0s, bold some columns that are equivalent, and we have a cutanddry explanation using a truth table.
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Thread: Weird Logic

12062012, 12:01 AM #11A criticism that can be brought against everything ought not to be brought against anything.

12062012, 12:07 AM #12
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They merely want to redefine a connective/function, creating a new, slightly different language by narrowing the use of the conditional (for various reasons). Since material implication is defined as what it is, and is pure semantics, it is true (a tautology).
It uses English terms with a different meaning, causing confusion, but I've not witnessed any find the actually meaning of the material implication counterintuitive, once they realize it is not English and does not resemble the English "if, then" in any significant way. "It is logic's first surprise" in the wiki article just means in the typical learning process of formal logic, it is the first outright contradiction of English (though "or" is a common surprise, though not a contradiction).
Its conclusion is not the same as its premise, it is not an identity. The conclusion merely entails the premise. All three can lead back to their premise, with additional logic added, becoming an identity.
Maybe I have a strange mind to not find it counterintuitive at all, but formal logic is a language built to be as minimalistic as possible, for the sake of precision, about truth preservation. All conditionals in natural language can be built from it, sometimes in needlessly complex ways, but they remain true. It's much simpler if natural languages like English are never brought into it.
EDIT: made the third paragraph clearer.

12062012, 12:28 AM #13
No, it really is an identity.
~P → P ⊨ ~(~P & ~P) ⊨ ~(~P) ⊨ P
Or alternately we could use bologna's example,
~P → P ⊨ ~(~P) v P ⊨ P v P ⊨ P
If the conclusion of a valid argument implies the premises, then the premises and conclusion are logically equivalent. In this case, while they are different formulae, they share the same consequence class (i.e. set of logical consequences). They are no less an identity than,
P ⊨ ~(~P)
If by 'identity' you mean identical formulae, then obviously that's not what I meant.
In contrast, the first two arguments are not identities; their conclusions cannot 'lead back to the premise' as you claim. That is, the consequence class of each conclusion is a proper subset of the consequence class of its respective premise.
Maybe I have a strange mind to not find it counterintuitive at all, but formal logic is a language built to be as minimalistic as possible, for the sake of precision, about truth preservation. All conditionals in natural language can be built from it, sometimes in needlessly complex ways, but they remain true. It's much simpler if English is never brought into it.A criticism that can be brought against everything ought not to be brought against anything.

12062012, 12:49 AM #14
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Not quite, I meant absent reductio ad absurdum (what is required to conclude P absent ~P, in the case of ~P → P), which is what I remember Identity being defined as. I'm going to guess that has either changed now or you are familiar with different semantics. It's clear now, at least.
Whilst I see the reason for the definition of identity you use, the reason for the definition I was using is the idea that P v ~P, ~(P & ~P) are both always true
Take ~P → P.
In the case of P, then P is true.
In the case of ~P, then P is true.
For all possibilities, P is true (so your definition of Identity fits), however in the second case both ~P and P are true, contradicting ~(P & ~P) and thus making it absurd.
With the original premise, P, and the absurdity of the case of ~P, reductio ad absurdum discharges ~P.
Without the original premise, P, reductio ad absurdum is equally valid in discharging ~P → P as it is ~P, since both statements were used to reach the absurd P & ~P.
The same is true when going from the premise to the conclusion, P and ~P both equally valid for discharge. Despite the premise and conclusion entailing each other prior to P v ~P, ~(P & ~P), it is considered a worthwhile distinction from those similar cases that still lack absurdity after P v ~P, ~(P & ~P) is applied.
Well various areas of statistics and probability are commonly counterintuitive, and that is absent any mental attempt to give English meaning to a nonEnglish term (which I did as well when learning about implication as well). If semantic confusion falls under "counterintuitive" then so be it, but I found when learning about the Monty Hall problem, for example, that my mind attempted to contradict the actual reasoning, and not the semantics, which was a much more jarring experience. That changed the way I thought, whereas formal logic (sentential, at least) did not, it just provided clarity (and as an anecdote I thought I witnessed this same reaction in several others).

12062012, 06:35 AM #15
The Monty Hall problem is fun, isn't it?
What people often fail to realize, and what they also often fail to explain, is that Monty changes the game in a predictable way. Monty has knowledge of where the goats are, and when you choose a door, he always opens a door revealing a goat. Some think of this as changing the mathematical odds, but what he is actually doing is making the first door choice irrelevant to the current situation (You start with three doors but you know for certain that one will be definitely eliminated, so it's like that door never existed)

12062012, 08:55 AM #16garbageGuest

12062012, 01:13 PM #17'One of (Lucas) Cranach's masterpieces, discussed by (Joseph) Koerner, is in it's selfreferentiality the perfect expression of lefthemisphere emptiness and a precursor of postmodernism. There is no longer anything to point to beyond, nothing Other, so it points pointlessly to itself.'  Iain McGilChrist
Suppose a tree fell down, Pooh, when we were underneath it?"
"Suppose it didn't," said Pooh, after careful thought.
Piglet was comforted by this.
 A.A. Milne.

12062012, 01:40 PM #18
A reductio ad absurdum is not required to prove the argument. The proof is extremely simple:
Code:P ⊨ ~P → P (1) P Premise (2) ~P Assumption for discharge (3) ~P → P From (1) and (2), discharge assumption
The law of identity,
P ≡ P
That is a symbol often used to assert identity, and it's customarily extended to cases of logical equivalence:
P ≡ ~(~P)
P ≡ P & P
P ≡ P v P
P ≡ ~(~P & ~P)
...
And so on ad infinitum. These are not, of course, identical formulae, but they are logically identical. That is, they have precisely the same truth conditions and consequence class. If the premise is true, then so must the conclusion be true and vice versathe 'vice versa' is what sets identity apart from deducibility. Deducibility is transitive, while identity is symmetrical. So you can add to the above list:
P ≡ ~P → P
Take ~P → P.
In the case of P, then P is true.
In the case of ~P, then P is true.
For all possibilities, P is true (so your definition of Identity fits), however in the second case both ~P and P are true, contradicting ~(P & ~P) and thus making it absurd.
With the original premise, P, and the absurdity of the case of ~P, reductio ad absurdum discharges ~P.
Without the original premise, P, reductio ad absurdum is equally valid in discharging ~P → P as it is ~P, since both statements were used to reach the absurd P & ~P.
The same is true when going from the premise to the conclusion, P and ~P both equally valid for discharge.
Well various areas of statistics and probability are commonly counterintuitive, and that is absent any mental attempt to give English meaning to a nonEnglish term (which I did as well when learning about implication as well). If semantic confusion falls under "counterintuitive" then so be it, but I found when learning about the Monty Hall problem, for example, that my mind attempted to contradict the actual reasoning, and not the semantics, which was a much more jarring experience. That changed the way I thought, whereas formal logic (sentential, at least) did not, it just provided clarity (and as an anecdote I thought I witnessed this same reaction in several others).A criticism that can be brought against everything ought not to be brought against anything.

12062012, 01:50 PM #19
A bit of both. To entertain, to educate, to ponder, and amuse.
I assumed the latter; which meant that you weren't my target audience. Getting down into the weeds would have been less clear for my target audience.A criticism that can be brought against everything ought not to be brought against anything.

12062012, 02:00 PM #20
Yes. I'm selftaught. I just bought some textbooks and started studying, so that probably made it more difficult. In any case, when I first encountered arguments like those I listed in the opening post (or their purely formal counterparts), I was surprised and confused. It was not immediately apparent to me how the conclusion could be deduced from the premises, and much less apparent how it could be true (even after I was aware of some purely mechanical means of testing for validity). It took practice, especially to familiarise myself with the nature of truthtables of logical connectives. Understanding the basic rules of logic is relatively easy; understanding their consequences is difficult and often surprising.
A criticism that can be brought against everything ought not to be brought against anything.
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