When logics die,
The secret of the soil grows through the eye,
And blood jumps in the sun;
Above the waste allotments the dawn halts. -- Dylan Thomas
It was a pointless adventure, convincing Khuno that a logical consequence of the material implication was not "bad logic" (or a figment of my diseased imagination).
Arguments, such as "(b->r) / (-r->-b)", with both logically equivalent premises and logically equivalent conclusions ARE arguments (as has been a settled matter for over 2,300 years). Khuno denied this. I quoted from 3 text-books - which stated flat-out that such arguments ARE arguments. Khuno declared that logic (as taught in universities) was "bad logic". Of course logic, like scientific data, human history, reality, etc. are mere hurdles to be vaulted-over by his pole-vaulting mind. The cultists (notably Norton and Airion) sided with Khuno on this howler, insisting that I did not "understand the context" of Khuno's deep, counter-logical insight.
If "X/X", a valid argument, is not an argument - because it has a premise and a conclusion which are logically equivalent, then "X/-X", an invalid argument, is an argument - because it lacks a premise and a conclusion which are logically equivalent. If "X" cannot be logically derived from "X", material implication (and logic itself) breaks-down. The valid argument modus ponens "(p -> q) & p / q" is a tautology as all valid arguments must be. Further, all arguments are propositions. To prove an argument, it is converted into its corresponding conditional (a proposition).
Logic is not an idealized way of thinking. It's a game, and - in order to play it correctly, one is compelled to follow its rules (conventions) - not dream-up wild excuses as to why it should be subverted. If arguments with both logically equivalent premises and conclusions cannot be arguments, then the material implication goes haywire, rendering every other argument undecideable (or worse).
"(-r -> -b)" is the contrapositive of "(b -> r)"; that is, the law of contraposition (an axiom of replacement) makes both sentences logically equivalent.
The following is a valid argument (though unsound). Due to the truth-functional nature of implication, this argument cannot be other than valid. Since the premises form a self-contradiction, any conclusion flows from "(p & -p)".
-p / q
Whether anyone likes it or not, implication works non-intuitively. The material implication is an inference generator, i.e. one can infer from self-contradictory (false) premises, best captured by the phrase "generative absurdity", but one cannot prove anything with self-contradictory premises.
In the following example - on the other side of the "/" sign is a tautology. It, too, forms an argument:
It's a valid argument, and - depending on what p is, it could be sound. If p="New York City is the capital of New York", then the argument is unsound. If p="Albany is the capital of New York", then it is both valid and sound. Neither arguments with contradictory premises nor tautologous conclusions have been struck-out as "bad logic". Arguments with both logically equivalent premises and conclusions have not been struck-out - either.
Suppose that a professor of logic handed you a paper with "(b->r) / (-r->-b)" on the first line and its truth-table on the second line. If he put his hand over the "(b->r) / (-r->-b)" line and ordered you to determine via inspection of the truth-table, whether or not the right side (the conclusion) follows from the left side (the premise), there's only one answer: it would be self-contradictory to affirm the left-hand side, yet deny the right - because the corresponding truth-table produces a tautology - a logical truth.
As Wittgenstein remarked: "In logic, there are no surprises.", and it should come as no surprise that "(b->r) / (-r->-b)" - a logical truth - is an argument.
In first-order predicate calculus, Phryne's argument can be translated as:
A(x) E(y) [ Bx -> Ryx ] / A(x) E(y) [ -Ryx -> -Bx ]
For every x, there is some y; such that, if x is a belief, then y is a reason for x; therefore, for every x, there is some y; such that, if y is not a reason for x, then x is not a belief.
Finally, here's a trivial proof of the aforementioned first-order argument (which has become a signature taboo) in the khunoinian death to logic cult:
1. A(x)E(y)[Bx -> Ryx] / A(x)E(y)[-Ryx -> -Bx] 2. E(y)[Bz -> Ryz] 1, UI 3. Bz -> Raz 2, EI 4. -Bz v Raz 3, IMPL 5. Raz v -Bz 4, COM 6. -Raz -> -Bz 5, IMPL 7. E(y)[-Ryz -> -Bz] 6, EG 8. A(x)E(y)[-Ryx -> -Bx] 7, UG
1. (a) From Hurley's Logic (Third Edition), page 313:
"Determine whether the following arguments are valid or invalid by constructing a truth table for each.
1. k->-k / -k" (For the logically challanged: (k->-k) = (-k) and ( k->-k / -k) is a valid argument.)
(b) From The Principles of Deductive Logic, by John T Kearns, page 214:
"p / (p&p) 1, 1, Conj" (For the logically challanged: p = (p&p) and (p / (p&p) is a valid argument.)
Immediately above is a conjunction. It's specified in text-books as a rule of inference. To Khuno - since he redefined what an inference is, no inference is occurring in this argument - because the premises and the conclusion are logically equivalent. So - even though we have an inference rule and we employ it, it yields no inference? (self-contradiction)
(c) From Elementary logic, by Benson Mates, page 83:
q / (p&q)" (For the logically challanged: p&q = (p&q) and (p & q / (p&q)) is a valid argument.)
Logicians speak of the notion of containment - in which the conclusion is filled-up by the premises. It stands to reason that the more absolute and more complete the containment, the more logical force an argument has. There is not much of (if any) logical transition from the conjunction of "p" and "q" in the premises to "(p&q)" in the conclusion. This maximal filling of a conclusion by its premises is a logical strength - not a weakness for an argument; hence, there's no good reason to demote the argument "p & q / (p&q)" from the status of an argument to a non-argument.
The following argument contains a logically equivalent premise (via a raft of conjoined premises) and a logically equivalent conclusion. If I am able to understand Khuno, the proof of this argument cannot be a proof of this argument - because there is no inference occurring from the combined premises to the conclusion. This is yet another example of kunoinian reductio ad absurdum.
1. (q->p) 2. (q v p) 3. -(-p & q) -> (p v -q) 4. -(-p&-p) 5. -p -> -q / -(p & q) -> p 6. -p->-q 1, CONTRA 7. -q->p 2, IMPL 8. -p->p 6,7, HS 9. p v p 8, DN, IMPL 10. p 9, TAUT 11. p v (p & q) 10, ADD 12. (p & q) v p 11, COM 13. -(p & q) -> p 12, IMPL
p = Khuno snorts crystal draino
q = Joe gets abducted by aliens
If Joe gets abducted by aliens, then Khuno snorts crystal draino, but either Joe gets abducted by aliens or Khuno snorts crystal draino, and it is not the case that if Khuno does not snort crystal draino and Joe gets abducted by aliens, then either Khuno snorts crystal draino or Joe does not gets abducted by space aliens, yet it is not the case that both Khuno does not snort crystal draino and Khuno does not snort crystal draino, but if Khuno does not snort crystal draino, then Joe does not get abducted by aliens; therefore, if it is not the case that Khuno snorts crystal draino and Joe gets abducted by aliens, then Khuno snorts crystal draino.