Talk:Paradoxes of material implication

A curious logical universal expression

--Faustnh (talk) 15:40, 8 March 2009 (UTC)

( ScientificLaw1 AND ScientificLaw2 AND ScientificLaw3 )

--->

(

( ScientificLaw1 ---> ScientificLaw2 )

AND

( ScientificLaw2 ---> ScientificLaw3 )

AND

( ScientificLaw3 ---> ScientificLaw1 )

)

---

Why is this curious? It is an obvious consequence of the axiom of classical logic (A->(B->A)), and is not true with strict implication. — Charles Stewart (talk) 09:05, 9 March 2009 (UTC)

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--Faustnh (talk) 14:12, 9 March 2009 (UTC)

A MUCH MORE interesting re-versioning of the formula above:

( ScLaw1 AND ScLaw2 AND ( ScLaw1 --> ScLaw2 ) )

<--->

( ScLaw1 AND ScLaw2 AND ( ScLaw1 --> ScLaw2 ) AND ( ScLaw1 <-- ScLaw2 ) )

---

I don't find this more interesting; it is just another fairly easy consequence of the K axiom, A->(B -> A), which is equivalent to what the article calls the principle of simplification, and is elsewhere widely known as weakening. These might be worthwhile exercises, appropriate for someone learning how to work with strict and material implication, but from an encyclopedic point of view, they are just not particularly clear ways of expressing some consequences the K axiom. Something that was intuitively arresting would be interesting; these, not so much. — Charles Stewart (talk) 09:14, 10 March 2009 (UTC)

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--Faustnh (talk) 12:55, 10 March 2009 (UTC)

Respectfully, two points:

First, I'm not figuring the formula is a clear way of expressing some consequences of the K axiom.

Second, I'm absolutely aware of that formula is not "having-been-empirically-tested"-true.

By the way, another curious expression: NOT Life --> ( Life --> NOT Life ) ... (and again, I'm aware of this is not "having-been-empirically-tested"-true). Best regards. —Preceding unsigned comment added by Faustnh (talk • contribs) 13:03, 10 March 2009 (UTC)

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--Faustnh (talk) 21:08, 12 March 2009 (UTC)

Another re-versioning (sorry, I'm having a bad week)...

( CauseInput AND ( CauseInput --> EffectOutput ) AND NOT ( ( NOT CauseInput ) --> EffectOutput ) )

<--->

( CauseInput AND ( CauseInput --> EffectOutput ) AND NOT ( ( NOT CauseInput ) --> EffectOutput ) AND EffectOutput AND ( CauseInput <-- EffectOutput ) )

--Faustnh (talk) 21:08, 12 March 2009 (UTC)

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Merge with Paradox of entailment

I don't think there can be any real objection to the merge, although it is not obvious which title the merged article should have. — Charles Stewart (talk) 09:05, 9 March 2009 (UTC) Edit: fixed target article from Paradox of material implication to Paradox of entailment — Charles Stewart (talk) 09:17, 10 March 2009 (UTC)

Merge done, from Paradox of entailment (Talk:Paradox of entailment) to here. — Charles Stewart (talk) 13:17, 20 March 2009 (UTC)

TLC needed

This article needs:

1. Cleaning up: the merge I did isn't ghastly, but it's not really tidy either. The article would probably read better if the PofE was used, both formally and informally, in the lede, and the alternatives were introduced in the supporting sections.
2. The article needs a proper introduction to strict implication, after all these paradoxes were what led to that formulation.
3. There are some (one in the present list) PofMIs that are not valid in non-truth-functional intuitionistic logic. This is worth including, since it illuminates what a logical theory needs to support material implication. — Charles Stewart (talk) 13:27, 20 March 2009 (UTC)

The current article (1 June 2017) is really awful compared to the way it was around 2010-2011. It doesn't even discuss most of the major forms of paradox. Can we just revert to the earlier version? I'm not thinking my own edits, which were pretty good, but someone after me made it even clearer and better and that would be my preferred version. (I'll dig into the history and see if I can find the version I mean.) Howard Landman (talk) 21:23, 1 June 2017 (UTC)

Never mind. I'm not sure what I was looking at. I went through every version from 2010 to 2017 and the latest 05:18, 16 February 2017 version seems fine. But I don't think that it is what I was shown yesterday. Howard Landman (talk) 06:59, 3 June 2017 (UTC)

Switches in series

"(P & Q) -> R" does not correctly represent the logic of two switches wired in series. It is true that if both switches are on, the light will come on. Thus, (P & Q) is indeed a sufficient condition for R: "If (P & Q), then R". However, it is also the case that both must be on, and so it is a necessary condition as well: "If R, then (P & Q)". Thus, the correct formula is the biconditional "(P & Q) if and only if R": "(P & Q) <-> R", not "(P & Q) -> R". — Doug Shaver—Preceding unsigned comment added by Doug Shaver (talk • contribs) 14:59, 27 June 2010 (UTC)

Doug, "(P & Q) -> R" is a correct (i.e. true) statement about two switches in series. It is also true about two switches in parallel. I think you are confusing specificity with correctness. The inference "(P & Q) -> R, therefore either P -> R or Q -> R" gives valid results in the parallel case but invalid results in the series case. Besides, "(P & Q) <-> R" implies "(P & Q) -> R" which still implies "P -> R or Q -> R", so your more specific formulation does NOT escape the paradox. Howard Landman (talk) 02:08, 5 December 2010 (UTC)

Interesting reference

Hermione: "I'm sorry, but that's completely ridiculous! How can I possibly prove that [the Ressurection Stone] doesn't exist? mean, you could claim that anything's real if the only basis for believing in it is that nobody's proved it doesn't exist!" Mr. Lovegood: "Yes, you could. I am glad to see that you are opening your mind a little." -J. K. Rowling —Preceding unsigned comment added by 74.96.158.81 (talk) 02:31, 27 July 2010 (UTC)

Examples of falsity should be clearly marked as false

The article contains unnecessarily confusing text such as [brain pain warning!:] 'Thus the proposition "if John is in Paris, then John is in England" holds because we have prior knowledge that the conclusion is true. If John is not in London, then the proposition "if John is in London, then John is in France" is true because we have prior knowledge that the premise is false.'

If you can follow this and know the subject well (I don't), please rewrite it so that misstatements demonstrating the "paradox" are more clearly demarcated as misstatements. For example, the preceding could be modified (if appropriate) by saying something like:

'The INVALID proposition "If John is in Paris, then John is in England" could nevertheless APPEAR to be valid because the "paradox" places UNDUE emphasis on the part which is valid: that we know John is indeed in England.'

(Boldface or italics are suggested for the words I show in capitals.)

The prose of the article should never be appearing to actually suggest that false statements are true. That's not clarity.

Parsiferon (talk) 11:17, 19 November 2010 (UTC)

There is nothing "invalid" or "false" about those statements, as far as classical logic is concerned. If, indeed, John is in England, then "If John is in Paris, then John is in England," is simply a true statement, no less "valid" than "If John is in Liverpool, then John is in England."

These statements bother our intuitions because the "if-then" statements of natural language are not truth-functional and hence are not well-captured by the "if-then" statements of symbolic logic. There have been various attempts to formalize the natural language use of conditionals, but none have been wholly successful, if I understand correctly. Thus, the semantics of the natural language usage are vague and unclear (we know what's true when we see it). Phiwum (talk) 13:31, 19 November 2010 (UTC)

I agree with Parsiferon. I liked this section much better after my last major edit of it. It seems to be slowly degrading and has gotten more murky and confused (and even incorrect!) since then.

Phiwum: NO truth-function can adequately represent the natural-language meaning of "if". Material implication is the best candidate, and it fails miserably. It is NOT necessarily the case that natural language is too "vague", it is simply that truth-functionality does not capture all of it; in particular, classical logic has no notion whatsoever of "relevance", whereas natural languages (and many non-classical logics) do. The reasoning method "Translate IF to M.I., use classical logic, translate M.I. to IF" is unsound and gives manifestly wrong answers, which is the whole point of this article. Howard Landman (talk) 01:58, 5 December 2010 (UTC)

The discussion of the "paradox of entailment" in this article is incorrect.

The discussion of the "paradox of entailment" in this article is incorrect. The "paradox of entailment" consists of a conditional in which the antecedent is a contradiction. The conditional is true regardless of the truth value of the consequent. In case the consequent is false, this can appear paradoxical. No deduction is involved. Modus ponens cannot apply to such a conditional because the antecedent is false. In this discussion, a seemingly paradoxical conditional is presented and modus ponens is implicitly, but incorrectly, appealed to in order to justify a manifestly false deduction. I hope that whoever is responsible for this section will correct it.

Dagme (talk) 19:44, 2 January 2011 (UTC) (84.100.243.205 (talk) 21:20, 1 March 2015 (UTC)) http://mindnewcontinent.wordpress.com/

Another nice one

Here is another nice one: (A ---> B) OR (B ---> A) is always true. In words: Because it is raining, my pencil tip broke OR because my penciltip broke, it is raining.

It not possible that neither "because it is raining, my pencil tip broke" nor "because my pencil tip broke, it is raining" are true.

That is no. 5 $(p\to q)\lor (q\to r)$.

My personal hunch: temporally and spatially unconected events don't belong in real world examples of logical statements.

Your personal hunch is wrong.

And yet another one

This one allegedly due to Aristotle:

either a or !a (at any point in time)

so it is today true (or false) that the sun will shine on the 8th of July, 2049.

This famously implies Determinism. ;)

Paradox of entailment

This is not related to the principle of explosion. It is the principle of explosion.

Who calls this property "the paradox of entailment"? If it's a common name, I've never heard it. Can we at least have a citation for this? Phiwum (talk) 02:32, 23 October 2016 (UTC)

MY personal hunch

Is different levels - syntactic (implication), semantic (inference), pragmatic. The paradoxes exist when we give from semantic into syntactic "machine" wrong information. And we get wrong information back. I was taught that way. Constantine^hehe 12:22, 11 December 2019 (UTC)