1. ## Weird Logic

All three of these arguments are valid (i.e. if the premise is true, then the conclusion must be true on pain of contradiction).

Socrates is a man.
Therefore,
Socrates is a man or Socrates is immortal.

Socrates is a man.
Therefore,
If Socrates is immortal, then Socrates is a man.

Socrates is mortal.
Therefore,
If Socrates is immortal, then Socrates is mortal.

One of these arguments is actually an identity (i.e. A, therefore, A). Which one?

So if all men are mortal, and Socrates is a man, then each of the conclusions above must be true, but they all seem intuitively wrong.

2. Logic is just a funny beast sometimes. That first one reduces to identity; second is true because of a counterintuitive 'oddity' of logical implication.

Third is even more 'odd.' But, really, it's phrased essentially as (¬q → q) and (p → q) = ¬p ∨ q, therefore we have ¬(¬p) ∨ q, thus q ∨ q, thus q). q is pretty easy to understand!

The thing that people never, ever understand is the word "if"--both in propositional logic and in real life. People also don't understand stats, but that's a subject for another day.

3. They only seem intuitively wrong if you read them in English rather than formal logic. The English readings have ambiguity to them, and some of the meanings in the ambiguity would make the arguments logically invalid. "Or" in English has a variety of uses (such as "only one of these components is true" and "all these components must be possible", but in formal logic it's just "at least one of the components is true"), and conditionals do too (in English "if... then" is taken to imply a connection between the components, whereas in formal logic it never implies such).

The last one is a common start to a reductio ad absurdem, assuming immortal and mortal are contradictory (in English they often are not), as such it could end up as an identity (though it technically isn't, using the system I am familiar with).

4. Originally Posted by bologna
Logic is just a funny beast sometimes.
Indeed.

That first one reduces to identity
No.

second is true because of a counterintuitive 'oddity' of logical implication.

The thing that people never, ever understand is the word "if"--both in propositional logic and in real life.
Or the word 'then'.

5. Originally Posted by erm
They only seem intuitively wrong if you read them in English rather than formal logic. The English readings have ambiguity to them, and some of the meanings in the ambiguity would make the arguments logically invalid. "Or" in English has a variety of uses (such as "only one of these components is true" and "all these components must be possible", but in formal logic it's just "at least one of the components is true"), and conditionals do too (in English "if... then" is taken to imply a connection between the components, whereas in formal logic it never implies such).
Even when translated into formulae, these arguments are not obviously valid.

P: Socrates is a man
Q: Socrates is mortal

P ⊨ P v ~Q

P ⊨ ~Q → P

P ⊨ ~P → P

I hope these logical symbols for material implication and semantic entailment appear correctly for everyone.

The last one is the start of a reductio ad absurdem, assuming immortal and mortal are contradictory (in English they often are not).
No. There is no contradiction. The conclusion is true.

6. Originally Posted by reason
It's a pretty straightforward answer. I doubt that most people know or care about propositional logic and so they just want to hear the point--"logic is odd and sometimes counterintuitive."

I'd go through the symbolic rigamaroo, but I'll just let people cheat and use truth tables.

7. Originally Posted by reason
Even when translated into formulae, these arguments are not obviously valid.

P: Socrates is a man
Q: Socrates is mortal

P ⊨ P v ~Q

P ⊨ ~Q → P

P ⊨ ~P → P

I hope these logical symbols for material implication and semantic entailment appear correctly for everyone.
How are they not obviously valid now?

If established that P is true, then the statement "at least one of the following, P..., is true" is also true, no matter the other components included with P.
If established that P is true, then any conditional concluding P is also true, since in formal logic there is no implied connection between the antecedent and the consequent, only the consequent need be true for the conditional to be true (I exist, therefore if you exist then I exist).

Originally Posted by reason
No. There is no contradiction. The conclusion is true.
As I said, the start of a reductio ad absurdum. The conclusion must be true for the reductio ad absurdum to follow.

8. Originally Posted by bologna
It's a pretty straightforward answer. I doubt that most people know or care about propositional logic and so they just want to hear the point--"logic is odd and sometimes counterintuitive."
Perhaps, but that's an explanation of why you gave an ambiguous answer, and not an explanation why it wasn't actually ambiguous.

I'd go through the symbolic rigamaroo, but I'll just let people cheat and use truth tables.
Even with the truth-table, it's still counterintuitive and difficult to explain why it must be so.

Third is even more 'odd.' But, really, it's phrased essentially as (¬q → q) and (p → q) = ¬p ∨ q, therefore we have ¬(¬p) ∨ q, thus q ∨ q, thus q). q is pretty easy to understand!
Right, but then why did you say the first argument was the identity?

9. Originally Posted by erm
How are they not obviously valid now?
Obviousness is relative.

If established that P is true, then the statement "at least one of the following, P..., is true" is also true, no matter the other components included with P.
Right, because logical disjunction is traditionally inclusive rather than exclusive. The first argument is the most obviously valid, but it may still be rather counterintuitive to those with only a passing familiarity with logic.

If established that P is true, then any conditional concluding P is also true, since in formal logic there is no implied connection between the antecedent and the consequent, only the consequent need be true for the conditional to be true (I exist, therefore if you exist then I exist).
Right, though 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. Most of the confusion stems from peoples' tendency to conflate material implication and causation; these arguments are actually good for dispelling that illusion.

As I said, the start of a reductio ad absurdum. The conclusion must be true for the reductio ad absurdum to follow.
No. The third argument is the identity (A, therefore, A). bologna figured it out but for some reason said the first argument was the identity.

10. Originally Posted by reason
Perhaps, but that's an explanation of why you gave an ambiguous answer, and not an explanation why it wasn't actually ambiguous.
Ambiguity is relative.
Even with the truth-table, it's still counterintuitive and difficult to explain why it must be so.
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.

I'd do it here but I forgot how to make a table in vBulletin
Right, but then why did you say the first argument was the identity?
I fucked up at first glance, realized it, edited my post with the actual answer, and forgot to edit out the original fuckup.

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