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Thread: Weird Logic

  1. #11
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    Quote Originally Posted by bologna View Post
    Ambiguity is relative.
    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.

    It's pretty easy to explain with a truth table--we 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 cut-and-dry explanation using a truth table.
    Truth-tables are useful, especially when checking for consistency. However, they are not the same thing as a good explanation, at least to me.
    A criticism that can be brought against everything ought not to be brought against anything.

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    Quote Originally Posted by reason View Post
    Right, though even some logicians have objected to material implication on these grounds. In any case, it's especially counterintuitive when the conclusion's antecedent is the negation of the premise.
    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).

    Quote Originally Posted by reason View Post
    No. The third argument is the identity. Bologna figured it out but for some reason said the first argument was the identity.
    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.

  3. #13
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    Quote Originally Posted by erm View Post
    As I said in my first post, it's the closest to an identity, since it commonly leads into one, but it is not. Its conclusion is not the same as its premise, it is not an identity. All three can lead back to the premise, becoming an identity, but none are in their current state.
    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.
    Perhaps you do have a strange mind. Most people struggle to shed their habits of thought and intuitions so easily--it takes practice.
    A criticism that can be brought against everything ought not to be brought against anything.

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    Quote Originally Posted by reason View Post
    If by 'identity' you mean identical formulae, then obviously that's not what I meant.
    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.

    Quote Originally Posted by reason View Post
    Perhaps you do have a strange mind. Most people struggle to shed their habits of thought and intuitions so easily--it takes practice.
    Well various areas of statistics and probability are commonly counterintuitive, and that is absent any mental attempt to give English meaning to a non-English 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).

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    Quote Originally Posted by erm View Post
    Well various areas of statistics and probability are commonly counterintuitive, and that is absent any mental attempt to give English meaning to a non-English 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).
    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)

  6. #16
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    Quote Originally Posted by reason View Post
    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.

    Truth-tables are useful, especially when checking for consistency. However, they are not the same thing as a good explanation, at least to me.
    Was your purpose to give us a problem to solve and then grade our solutions, or to give us an actual point of discussion? 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.

  7. #17
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    Quote Originally Posted by reason View Post
    Perhaps you do have a strange mind. Most people struggle to shed their habits of thought and intuitions so easily--it takes practice.
    Have you?
    'One of (Lucas) Cranach's masterpieces, discussed by (Joseph) Koerner, is in it's self-referentiality the perfect expression of left-hemisphere emptiness and a precursor of post-modernism. 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.

  8. #18
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    Quote Originally Posted by erm View Post
    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.
    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
    Yes, the premise and the assumption are contradictory, but there needn't be a reductio ad absurdum. It's a straightforward assume and discharge.

    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 versa--the '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.
    This is irrelevant to the validity of the argument and the truth of the conclusion. Since '~P → P' is--in an important sense--just another way of writing 'P', of course, if '~P' is true, then 'P' is false and we have contradicted the premise, but '~P' is neither a premise nor entailed by '~P → P'. In other words, '~P → P' does not contradict 'P v ~P', and so all this irrelevant except as a means to construct an unnecessarily complicated proof.

    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.
    If you want to construct a reductio ad absurdum, then yes, both formulae are used to deduce 'P & ~P'. However, while all assumptions one uses must be discharged for a deduction to be valid, premises need not be. In fact, the rule is customarily that premises are never discharged. Since 'P' is the only premise, then, only '~P' can be discharged by the reductio ad absurdum.

    Well various areas of statistics and probability are commonly counterintuitive, and that is absent any mental attempt to give English meaning to a non-English 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).
    Well, my primary goal with this thread was to mildly entertain people who are interested but have little familiarity with formal logic. If I just wanted to discuss logic, there are more interesting subjects and better forums. I was also curious to see if someone who was initially puzzled by these arguments could figure them out.
    A criticism that can be brought against everything ought not to be brought against anything.

  9. #19
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    Quote Originally Posted by bologna View Post
    Was your purpose to give us a problem to solve and then grade our solutions, or to give us an actual point of discussion?
    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.
    Right, and we're back to you explaining why your answer was ambiguous, and it's a fine explanation. But then why does it feel like you're trying to disagree with me about something?
    A criticism that can be brought against everything ought not to be brought against anything.

  10. #20
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    Quote Originally Posted by AffirmitiveAnxiety View Post
    Have you?
    Yes. I'm self-taught. 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 truth-tables 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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