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I am having trouble understanding what defines Entailment operator. On Mathoverflow I posted this question on what I perceive to be paradox of entailment.

Consider: Modus Ponens:

P therefore Q
P


Therefore, Q

My question is could we have arbitrarily defined it such as:

P therefore Q
P


Therefore, P

However, if an advanced civilization receives information in the form of following nested structures/containers or packets, how will they interpret Modus Ponens if in an alien land Modus Ponens is not defined?

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Therefore, it raises the validity of MP. How do we define the implication or entailment operator? Because if we use an argument that relies on implication or entailment operator ipso facto, then it is trivial that logic is circular.

Is there a flaw in my rationale?

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3 Answers

up vote 11 down vote accepted

Let's skip straight to the end.

then it is trivial that logic is circular.

Correct. Logic is circular.

Note that due to Agrippa's Trilemma, there are only three things logic could possibly be founded upon: unsupported axioms we take on faith, circular reasoning, or an infinite regress. Or, of course, a combination of the three.

Lewis Carroll famously demonstrated that Modus Ponens is based upon an infinite regress.

There's no way to prove Modus Ponens, except by one of the three horns of Agrippa's Trilemma.

So it goes.

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+1 for "Note that due to Agrippa's Trilemma, ... infinite regress. Or, of course, a combination of the three." The Albert's original presentation doesn't include the "combination of the three" part, so this casts a nice shadow of doubt on the correct interpretation of the trilemma... –  Thomas Klimpel May 30 '13 at 7:47
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I'm not sure I follow the details of your question: the second argument schema you present is, of course, valid. We might have given that schema the name modus ponens. What would follow exactly? The fact remains that modus ponens (the first schema) is also valid.

In any event, and re your broader worry, logic is rock bottom. That is, there is no non-logical way to define what a logical constant is, or what logical entailment is. The best you can do is provide illustrations and models that might help the recipient of the "definition" know what the intended meaning of entailment is. For example, thus: consider all logically possible worlds. P entails Q iff every such world in which P is a world in which Q. Again, I don't see how this compromises the validity of modus ponens.

You might want to read the SEP entry on logical constants for an elaboration of these ideas.

Incidentally, at least a published philosopher has proposed a counterexample to modus ponens! There is a discussion of the example at math SE. FWIW, nobody has stopped believing in modus ponens because of McGee's paper, but there you go.

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I am conflicted as to delete the question. I realized that second argument is valid, which I mistakenly thought is not. –  Sniper Clown Jul 3 '12 at 1:41
    
Oh, don't keep it for me :) Do delete it if you feel the question is ill formulated. –  Schiphol Jul 3 '12 at 1:45
    
It's too late for delete. Besides the problem of interpretation remain. The reason we are comfortable with MP is because of Earth's linear time and the argument form agrees with it. In my sketch, I was trying to force a paradox. For instance, in an universe where effect precedes cause, can MP be still used to model logical argument of that hypothetical land? –  Sniper Clown Jul 3 '12 at 1:54
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Bear in mind that the conditional in "If p then q" is the material conditional. That is, the following is a perfectly good example of modus ponens: Premise: Italy won the Euros 2012. Premise: If Italy won the Euros 2012, then the flying spaghetti monster exists. Conclusion: The FSM exists. This depends very little on the fact that causes precede effects :) –  Schiphol Jul 3 '12 at 2:19
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The material conditional has nothing whatsoever to do with causation. Really. Read this. –  Schiphol Jul 3 '12 at 12:56
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In order to present said alien civilization with an adequate definition of Modus Ponens, we would need to provide the truth values of both P and Q for every possible case, rather than just one or the other and for only some cases. (If we were to apply more generally your method of revealing only part of the output, we could avoid ever adequately defining any function because we could always keep re-iterating the premises (or just tacking on a tautology, such as (P or ~P)) in our output instead of providing any new information).

What distinguishes MP from the disjunction (P or Q) is that in the latter case, P and Q cannot both be false under the same interpretation, while in the case of logical entailment, it is not possible for Q to be false while P is true. This would be evident to your hypothetical civilization.

The mathoverflow post to which you linked covers efforts to get around the ability to draw unrelated tautological conclusions by attempting to set rules for a relatedness requirement--but this is mostly a matter of psychology, because formal logic does not purport to do anything except preserve the truth, and that it does. The efforts do, however, make logic more useful to humans because they create reasoning-like restrictions.


...P.S. As @Schiphol mentioned, the material conditional has nothing to do with causation. Rather, it dictates what IS true, right now, given that something else IS true, right now. Look up subjunctive arguments to see some truth-functionality issues with trying to bring temporal qualifiers into propositional logic. There are separate, temporal modal logics to deal with that sort of thing, because propositional logic is not really sufficiently expressive.

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My objection rather was for humans generally to engage in a logical discussion, must use a logic that can be sketched with logical operators and connectives as well as entailment. Self-referentially, this very on-going discussion, if were to be represented symbolically, it'd require entailment. At any rate, we are 'stuck in the system'. But, we cannot make that claim either. As Michael Dorfman answered logic is circular. Zen masters on the other hand came close to realizing it and that is why they'd blabber meaningless phrases to stun thinking for Satori. –  Sniper Clown Nov 17 '12 at 20:30
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@Mahmud I'm going to agree with one thing you implied: that we are biologically built to digest only certain types of logic. But this doesn't mean we can't formalize other, "non-intuitive" logics on paper. The circularity of logic does not liken it to "meaningless babbling"--it is a simple consequence of the fact that you cannot have a system, any system, without axioms. You may, however, want to check out D.Hofstadter on logic and Zen. =) –  QuietThud Nov 17 '12 at 21:10
    
...Although seeing as how you managed to fit recursion, Zen philosophy, and "being stuck in the system" into one short comment, I'm going to assume that GEB is the primary source you are working off of. I would suggest, if you haven't done so already, studying some advanced formal logic--it certainly helps to see that there are ways to "jump out of the system" to all kinds of higher levels. In all seriousness, that is what abstract languages are for. –  QuietThud Nov 17 '12 at 22:14
    
By the way, Godelian incompleteness doesn't mean we are "stuck" in any level, it means that we won't ever hit an "ultimate level". Which is quite beyond the considerations of humans engaging in logical discussions! –  QuietThud Nov 17 '12 at 22:19
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Check out Ultimate L: Its wide, airy space allows extra steps to be bolted to the top of the infinite staircase as necessary to fill in gaps below, making good on Gödel's hunch about rooting out the unsolvability that riddles mathematics. Gödel's incompleteness theorem would not be dead, but you could chase it as far as you pleased up the staircase into the infinite attic of mathematics. –  QuietThud Nov 17 '12 at 22:19
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