Is the following argument a case of the logical fallacy of affirming the consequent?
“A problem may have multiple efficient solutions, GIF and JPEG are two solutions to the same problem, therefore all problems have multiple solutions.”
I’m not exactly red hot when it comes to recognising these fallacies, so any help would be appreciated.
Thanks.
I think it’s attributing to the whole what is true of a part…
Is that the fallacy of composition?
I don’t think it rises above a hasty generalization. There’s not even an attempt at induction.
Fancy name or not, it’s simply stunningly wrong.
Yep. This is just silly. The logic problem with that is going from “may have multiple efficient solutions” to “therefore all problems have multiple solutions.” “Some” does not equal “all”.
Therefore all cats are orange.
No it’s not. I refreshed my memory by looking in a book, of all things.
The fallacy of affirming the consequent looks like this:
If S1 then S2. S2, therefore S1.
In words"
If you take cyanide you will die. John died. Therefore John took cyanide.
Obvious nonsense since John actually drowned.
I think Squink’s hasty generalization is right.
Yep, this is what the debate is over. The guy who I’m debating with is under the impression that there’s multiple eficient solutions to the same problem.
Thanks for the replies, everyone.
Whoa whoa whoa… Hold up there, chief.
Just because the argument was flawed doesn’t mean the conclusion is. From my experience, most problems do have more than one good solution unless they were constructed specifically to force a solution.
No, I explained wrong. Not problems as in maths problems, but as in algorithms to solve a specific task. By good, I mean “most efficient”, i.e. there may be multiple algorithms to solve a specific task, but it’s possible that there’ only a single most efficient algorithm to solve it. Or is that incorrect, too?
There might be a single most efficient algorithm but the difference between that and the next best might be so small as to not make it worthwhile spending the time to find it.
I’m not sure that the question you’re trying to answer is subject to a strictly logical resolution.
It’s possible, but it’s considerably more complicated than you might think. Some problems have a sequence of algorithms for them that asymptotically approach a bound on performance, but there’s no single best. I’m no expert on complexity theory, so I don’t want to comment any further.
It is possible, but there is no reason to belive that it exists or even that “efficiency” as you define it is a valid criterion for evaluation.
Your example is also my example. Gif and jpg may seem like different soultions to the same problem but they are not. One works best in some circumstances and the other works best in other circumstances, so to compare them in terms of “efficiency” may be making a comparison that is meaningless. Good logic built on a false premise yields unknown results. You may get it right, but not because of your logic.
You’re far from defining “efficient” rigorously enough to come close to the question having a definitive answer.
That’s what they want you to think.
If you limit yourself to problems with ‘most efficient solutions’ (however you define efficiency) that can be viewed as the minima of a function, the question becomes “do all those functions have a single global minima?”.
It would be hurtful to think that such a big class has that property, and if it’s not true for a subset of problems, it can’t be true for all problems.
–Of course, that hurtful bit needs a little expansion.
** YES!
**
“Two swallows doth not a summer make”
Simply put, “Two solutions to one problem” are just that, “Two solutions to one problem.”
FYI Tfhere are some problems that have no solutions.
That which proves too much proves nothing.
SSHH. They don’t want us to know that the guy on the grassy knoll shot him.
Well, the trick is in exactly what you mean by “efficiency”. Contrary to Mathochist’s assertion, there is a standard notion of efficiency with respect to a particular resource as a function of the input size, but again there can be complications. For some problems, there are bounds on the product of their running times and memory usage…
That said, if you limit yourself to a specific resource–generally time–and you take a problem where there is an algorithm that obtains the lower asymptotic bound, then yes, there is a single most efficient resource, as we’re dealing with integer-valued functions on the integers.
Even so, it’s possible that there might be multiple algorithms which are all equally efficient, but more efficient than any other algorithm. A tie for first place, in other words.
As far as image formats, GIF and JPEG are somewhat complimentary. Of images which humans are likely to want to render, most will be rendered well and efficiently by JPEG or GIF, but few will do well under both.