How do we prove a statement like “if A then B”?
- Assume A is true
- See if B is true given the assumption
- If it is then “if A then B” is true.
Example:
- Jane speaks (assumption)
- Jane is alive (assumption + real world knowledge)
- Therefore “if Jane speaks then Jane is alive”
BUT logic is awesome. Consider this:
Sentence X: If Sentence X is true then I’m the Pope
Note that here, A = “Sentence X is true” and B = “I’m the Pope”
- Sentence X is true (assumption)
- “If Sentence X is true then I’m the Pope” is true (expanding step 1)
- I’m the Pope (assumption plus step 2)
- This proves “If Sentence X is true then I’m the Pope” is true (using the same method of proving “if A then B”)
- This proves Sentence X is true (rewriting step 4)
- This proves I’m the Pope (from steps 4 and 5)
I’ve just proved I’m the Pope! Hat tip Goatism (which proved everything was a goat — alas the site seems gone). Now where have I gone wrong? Discussion to come. [EDIT: here it is]
7 comments ↓
Self-referential assumption is the mother of all fuckups.
~ Under Siege 2
true true — but if you meant that seriously then self-reference is a major part of a lot of what makes our world work (esp. computing/technology) — eg. godel’s incompleteness theorem, the halting problem, the 50 other results in computer science that have proofs similar to the halting problem etc
[...] ← Logic Is Awesome [...]
Truth is the accordance of a conception with its object/substance. The pope is not always the same person. So, the underlying object, that is, the person, changes. Therefore, it is not true that someone is a pope as the underlying object is not the same all the time. In fact, if you check, a Pope is just a regular guy. So, what is true, is that there is a guy with a title “Pope”. The church has no monopoly on issuing this title and you may well be a Pope if that is the title you want to have- just say to yourself “from here on I’m a pope!”
Logically, a great many things are true but whether the conceptions are in accordance with their physical object is another matter. Getting objective empirical data is, some say, impossible. Logic, therefore, is garbage.
If you use Santa Clause as an example, you will too find that he is just a guy. So, Santa does not exist, but a man who became Santa sure existed.
In the mind, logically, only nothing does not exist.
The pope reference was incidental, Curry’s Paradox can be used to prove anything. If you like you can replace it with a more precise statement like “I Michael Fridman am universally recognised as the leader of the Catholic Church on this 17th Jan 2009″
I don’t follow the 2nd paragraph: how is logic garbage?
Yes, the Curry’s Paradox can be used to prove anything as logically true. But, is logical truth worth anything at all? Are we not using logic dogmatically?
In general semantics the map/territory distinction illustrates how, in experience, we really never come in contact with the underlying objects – something Kant said before.
For example, given the concept of equilateral triangle from geometry one can say “if this sentence is true then a equilateral triangle exists.” Unlike in a claim that one is Pope, here, this statement makes a less profound claim. But, still, sadly, a perfect equilateral triangle probably does not exist in the real world. Yet, such a triangle sure does exist a priori, independently of experience, in the mind and so the statement is true. In the real world, only judgements that are in accordance with the physical object are true.
So, we can at best say that we know little about empirical objects. Consequently, we should not be making claims of, or promise, objective validity of a statement that is based on purely logical judgments.
Logic is a useful tool though. Anything that is logically true, is not necessarily true, but has only a change of being true. So, I say such a sentence should be written this way:
“If this sentence is true, then there is a chance that I may be a Pope.”
Logic is awesome but it cannot be applied dogmatically in a primitive form to every situation. Just ask a quantum physicist…
Tom — I just realised I didn’t link to my follow up discussion — it’s here — http://anadder.com/logic-is-fallible
It addresses some of the points you made. Here’s where I disagree: I don’t think there’s such a thing as dogmatic logic, it’s merely the process of putting together some propositions to come up with some other proposition. Just like a computer if you put garbage in you get garbage out.
There’s no reason to separate logically true and actually true. IF your premises are correct and you follow the laws of logic then your conclusions will be correct in actuality — that’s where its strength lies.
What Curry’s paradox shows is that at least some of our assumptions about the laws of logic aren’t correct — that’s why you can use it to derive anything.
(Your comments about possibilities and modal logic is something I haven’t looked into much, hoping to do so eventually).
And yes, as per my followup post, quantum physics may well cause us to modify our conception of logic — it’s not some static thing and it may change like any other empirical science BECAUSE it’s supposed to be a tool for deriving actual truth (from other true premises)
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