Talk:Modus ponens

Miscellaneous

One of the "conversation" chapters in Gödel, Escher, Bach is by Lewis Caroll, and is about MP and how it can be extended to absurdity. -- Tarquin 05:58 Aug 27, 2002 (PDT)

It's not correct to put "sic" after "premiss." That's not a spelling error, it's an archaic spelling. — Preceding unsigned comment added by 64.199.25.9 (talk) 20:00, 1 May 2012 (UTC)

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I'm putting in an article about that. To help, can someone clarify if "modus ponens" is the correct term to use with this argument:

1. ∀x∀y:equalsame(x,y) ⇒ x=y
2. equalsame(a,b)
3. ∴ a=b

In other words, does the existence of the quantifiers prevent me from calling this "modus ponens"?

--Ryguasu 23:34 Dec 3, 2002 (UTC)

I would insert the step

• equalsame(a,b) ⇒ a=b

which follows from 1 by specialization. Then the remainder of the argument would be modus ponens. AxelBoldt 03:24 Dec 4, 2002 (UTC)

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Is this the same as a sylogism?

It's a type of syllogism. — Gwalla | Talk 21:07, 16 Sep 2004 (UTC)

self-referential premises

   If the argument is modus ponens and its premises are true, then it is sound.
   The premises are true.
   Therefore, it is a sound argument.

For the purposes of my following statements:

this argument

the argument whose text you see here

the referenced argument

the argument referred to in this argument, whose soundness is argued

I assume "the premises are true" in the second line refers to the the premises mentioned in the first line, the premises of the referenced argument, as opposed to the premises of this argument, the argument presented here directly. The premises of this argument are not well premised as true by the second line due to expectations of the reader that the premises of the referenced argument are to be addressed explicitly at this point. However, the use of the definite article over the possessive pronoun suggests that the author is not referring to the referenced argument for this premise...

In short, I just realized you're messing with people.

And to lead into this modus ponens with "instances of its use may be either sound or unsound" is pure genius since this instance may indeed (or ininterpretation) be either.

You got me all worked up.

Psychology

I've heard that the "modus ponens" is considered something every man is born with (in order to be able to make transactions, like: - "I give you A, if you give me B" - you give me B -> I give you A), while the "modus tollens" is something that needs reflection first. I don't exactly know if this is true/unargued, but this should be mentioned perhaps. Also I've heard that the full name of the modus is "modus ponendo ponens" (and his 'counterpart' "modus tollendo tollens"), if this is true, it might be added too.

I don't want to change this by myself, because I'm not really sure whether it's true or not, as mentioned.

Truth Tables

Should the truth tables of modus ponens be added to this article? --Vince.Buffalo 05:41, 19 August 2006 (UTC)

It should probably have a disclaimer that the given truth table applies ONLY to classical two valued logic, while Modus Ponens applies to a good deal more. 155.101.224.65 (talk) 16:39, 27 October 2008 (UTC)

Can you provide reference? yayay (talk) 17:00, 27 October 2008 (UTC)

Just pick any paper on a three valued logic with truth tables and Modus Ponens for examples of some of the various truth tables other than the one in the article. Lukasiewicz's Multivalued logic and his infinite valued logic are, for example, are not two valued but have the rule of modus ponens A good reference for the infinite valued calculus is: A. Tarski and J. Lukasiewicz, "Investigations into the Sentential Calculus" appearing as Chapter IV in Tarski's "Logic, Semantics, and Metamathmatics" And for the multiple valued logic is probably: Lukasiewicz J. (1913) Die logishen Grundlagen der Wahrscheinichkeitsrechnung. What would be a reference for the truth table given applying to anything other than classical two valued logic? Nahaj (talk) 18:55, 27 October 2008 (UTC)

Trivial Truth

Could someone include some discussion of the following problem ...

Here are the truth functions of modus ponens:

((P > Q ) & P) > Q

 1  1  1    1  1   [1]  1
 1  0  0    0  1   [1]  0
 0  1  1    0  0   [1]  1
 0  1  0    0  0   [1]  0
                    *

Underneath the main (conclusion) operator, all lines of the truth table are true, hence the argument is valid.

But there is a problem once you start filling in the variables. The usual example is If one is a man, then one is mortal. Socrates is a man, hence Socrates is mortal. That works. P is true, Q is true, and the conclusion, by the magic of modus ponens comes out true. But what about this: If the moon orbits the earth, then I am wearing white carpenter's pants. Again, the first premise, P, is true. And take my word for it that the second premise, Q, is also true. Given the foregoing, the conclusion is valid. But why? It doesn't seem like the moon orbiting the earth should have any bearing on what I am wearing today, does it?

There are only two ways I have to deal with this, and I hope someone can help. First is simply to say that propositional logic doesn't account for modalities--whether the moon necessarily or possibly orbiting the earth has any impact on my choice of pants. Granted, modal logic, temporal logic, fuzzy logic, and some applications of predicate logic capture all of that. But as to basic bone-headed propositional logic, the conclusion seems odd, because it leaves open the possibility of a modus ponens sentence returning an invalid result--which it shouldn't be able to do.

So I think I have a second answer that works better. Because basic propositional logic doesn't account for time, modality, probability, etc. Given that, propositional logic describes a world in which all true propositions are necessarily related to each other (or necessarily not related to each other.) For instance, in the world that propositional logic can describe--every time a butterfly flaps its wings, there either must or must not be a hurricane.

That's about all I have to describe it, but I'd love to hear what anyone else has to say.

Thx.

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I think you misunderstand the notion of a logically valid argument. A valid argument is one in which the conclusion is *guaranteed* simply by virtue of the form of its premises. In your example, you seem to just be assigning truth-values to propositions. It makes no sense to say "given that the earlier two propositions are true, the conclusion is valid," since validity is a property of arguments, not of individual propositions (a common category mistake people make when first learning about logic). "Validity" just means "truth-preserving."

In standard form the modus ponens argument similar to yours would go:

1. The moon orbits the earth.

2. If the moon orbits the earth, then I am wearing white carpenter pants.

3. Therefore, I am wearing white carpenter pants. (1,2 modus ponens)

In this case, 3 is guaranteed by 1 and 2. Whether or not the argument is *sound* has to do with the truth of 1 and 2, and what I believe you are saying is that 2 is absurd (false). This does prevent the argument from being sound, but the argument itself is still valid.

eliminated link to "falsity implies anything"

I stated this incorrectly in the history page - modus ponens is just a form, and as such truth-value assignments are irrelevant. My real justification for deleting the link is that it really is too unrelated to an article on modus ponens (it would go well in an article on conditional statements). [Prior unsigned comment from 2006-11-03T00:01:47 130.64.31.145]

Note also that there are systems with modus ponens that do not have the property that falsity implies everything. Nahaj (talk) 19:00, 27 October 2008 (UTC)

(P->Q ^ Q->R) -> P->R

isn't this called hypothetical syllogism as well?

   If P, then Q.
   If Q, then R.
   P.
   Therefore, R.  —Preceding unsigned comment added by Michael miceli (talk • contribs) 14:17, 1 October 2007 (UTC) 

Another rule

$P\to Q,\neg P\vdash true$

Does this argument form have a name? Thanks, --Abdull (talk) 16:01, 8 January 2010 (UTC)

This isn't really a rule. I mean, if you have "true" on the right, typically you could have anything at all on the left and it would be a valid inference. Knorlin (talk) 17:13, 5 November 2022 (UTC)

The converse

Should we say something about the converse not necessarily being true? I.e., in the example,

If today is Tuesday, then I will go to work.

Today is Tuesday.

Therefore, I will go to work.

Just because you go to work doesn't mean it is Tuesday; it could be Wednesday. Tisane (talk) 04:56, 9 March 2010 (UTC)

I agree, but I'm not an expert. There's also the fellacious argument "today is not Tuesday, therefore I will not go to work" which does not follow from the previous argument. This error, I believe, has been catalgogued elsewhere according to some vague recollection I have--143.210.122.139 (talk) 14:18, 8 April 2011 (UTC)

So, "If today is Tuesday, then I will go to work. I will go to work. Therefore it is Tuesday." That's called the fallacy of affirming the consequent. Knorlin (talk) 17:15, 5 November 2022 (UTC)

Modus ponens is argument of symbolic logic.

'If today is Tuesday, then I will go to work and today is Wednesday' means symbolically 'A imp B, C'. By this way can't be already contrived modus ponens.

The truth table of implication:

• 1 1 true
• 1 0 false
• 0 1 true
• 0 0 true

'If today is Tuesday, then I will go to work and today is Tuesday' means symbolically 'A imp B, A'. The whole judgement is based on the the implication. If we do the correct conversion by denying of the conclusion, i. e. not B (I won't go to work), then we know securely, that musn't be Tuesday and premises are denied, too (denied is at least the second). The judgement is therefore correct. Chomsky (talk) 15:57, 15 November 2011 (UTC)

((P → Q) ∧ P) → Q

Section titles may be changed and removed in whatever way, but I do not think that this propositional theorem is off-topical. Incnis Mrsi (talk) 05:54, 19 April 2013 (UTC)

Comment

The sentence near the beginning saying that modus ponens "must not be confused with a logical law" is potentially confusing. It depends upon exactly what you mean by "logical law". I would think that most logicians would be happy to call it a logical law. I think that the distinction being made in the texts referred to is between an axiom or theorem (a necessarily true formula) and an inference rule. Modus ponens is an inference rule rather than an axiom. But it still could be called a logical law, since, as far as I know, "logical law" does not have a precise technical meaning. Sifonios (talk) 11:03, 6 November 2014 (UTC)

You are entirely correct about this, and I've wondered how to approach your point. Since the early 1800's "modus ponens", under a different name Principle of sufficient reason and not formalized as an "inference rule" as it is today, was considered a "law" by both Hamilton (1830's) and Russell (1900 to 1910). But see in particular the quote at footnote #10 in this article; also more at Laws of thought where Hamilton identifies the two. Somewhere along the line between his Principles of Mathematics and Principia Mathematical Russell singled out his expression of the "principle of sufficient reason" (he never called it that in anything he wrote, to my knowledge; he was abysmal at footnoting and sourcing so we can't trace how his ideas evolved) and adopted it/singled it out in Principia Mathematica as "Anything implied by a true elementary proposition is true", the very first of his "Primitive propositions (Pp)" *1.1 to which he added *1.2 through *1.7 to form his axiom-set, or if you prefer "laws of thought" (by this time the moniker "laws of thought" seemed to be passé by the mathematicians but still used by philosophers -- cf Russell 1912 for example. By the time of Hilbert and Gödel (1920's-1930's), modus ponens had been reclassified as a "rule" as opposed to an axiom. It would seem that the contemporary literature starting in particular with Gödel separates his axioms from his "sentence formation rules", the "rule of inference (modus ponens)", "rule of substitution", and a tacit "rule of specification". Unfortunately I don't have enough sources to make sense of exactly what's happened here. BillWvbailey (talk) 15:43, 6 November 2014 (UTC)

Responsibility in MP of establishing the antecedent - not so clear in the article.

I am not a specialist in this domain, but the article doesn't seem to be explicit concerning the issue that, when attempting to use MP in an argument, the responsibility of establishing soundness will be primarily dependent on establishing the antecedent: This be complex in areas of eg implied causality:"if it rains, flowers will bloom" may at first glance appear reasonable, even coherent, But what is missing are all the intermediary logical steps to determine that raining implies flowering - and yet to be established at all is that "A implies B" can ever be coherent beyond some hermeneutic stance. (20040302 (talk) 11:30, 30 July 2016 (UTC))

Supposed "counterexample" makes no sense

The supposed "contradiction" is as follows:

Either Shakespeare or Hobbes wrote Hamlet.
If either Shakespeare or Hobbes wrote Hamlet, then if Shakespeare didn't do it, Hobbes did.
Therefore, if Shakespeare didn't write Hamlet, Hobbes did it."

This is in fact a perfectly valid argument. The article's analysis of it, however, is fraught.

The first premise seems reasonable enough, because Shakespeare is generally credited with writing Hamlet. The second premise seems reasonable as well, because with the set of Hamlet's possible authors limited to just Shakespeare and Hobbes, eliminating one leaves only the other.

But the conclusion is dubious, because if Shakespeare is ruled out as Hamlet's author, there are many more plausible alternatives than Hobbes.

Emphasis mine. This is such a blatant contradiction I'm surprised that it made it into a Wikipedia article, let alone was apparently published by an actual philosopher. If indeed it is the case that the set of authors is only Shakespeare or Hobbes, then it obviously follows that if it was not one it was the other. Where this analysis fails is by assuming that the other premises would still hold if it is suddenly discovered that Shakespeare was not the author. We can only say premise 1 is true because Shakespeare DID write Hamlet. If that were not the case, the argument would still be valid - the conclusion would still follow from the premises - but the first premise would be false and therefore the conclusion could not be determined.

So the section finishes:

That these kinds of cases constitute failures of modus ponens remains a minority view among logicians, but there is no consensus on how the cases should be disposed of.

The apparent implication being that McGee has somehow torn a massive hole in the foundations of propositional logic which has stumped the eminent professors of the field, when in fact it is a very basic conflation of natural language with formal reasoning.

I recommend this entire passage is rewritten in a way that explicitly states these apparent "counterexamples" are fallacious and explains why. I would do it myself but I'm new to Wikipedia and am not sure if it would constitute original research. I'm going to comment it out for now, however, because the information it presents is totally misleading to readers.

2A01:4C8:1408:FF0B:BA30:3428:7426:C9BF (talk) 11:41, 29 January 2020 (UTC)

Let me address this, as the author of the offending material. The intent was not to make any actual claims about the validity of modus ponens, just to document some debate about it among philosophical logicians. McGee's 1985 article, appearing in one of academic philosophy's leading journals, caused quite a stir. There continues to be discussion about it. See, for instance, a piece by Mandelkern in Philosophy and Phenomenological Research, so new it is not yet in print but is viewable online if you have access (https://onlinelibrary.wiley.com/doi/abs/10.1111/phpr.12513). In the first paragraph he writes, "McGee (1985) showed that there is good reason to take seriously the possibility that Modus Ponens (MP) is false. . . ." I did not mean to exaggerate the significance of McGee's position, and I don't think I did exaggerate it. He's a person of note in the discipline (who still teaches at MIT), he took this stance on MP, most of the discipline thinks he's wrong, but there's no consensus on where he went wrong. That's why his 1985 paper and a 1989 follow-up piece in The Philosophical Review continue to get referenced and discussed.

McGee's article features an example about the 1980 presidential election involving Carter, Reagan and Anderson. There are also examples about someone sighting a fish (possibly a lungfish) and someone digging for gold but possibly finding silver. All of those examples require a bit of setup, whereas the Shakespeare/Hobbes example doesn't, and so that keeps the section from being any longer than it already is. That was my thinking, although I would have preferred not to introduce a new example (the whole "original research" rule).

I'm happy to entertain suggestions about how the section might be improved, although I think anyone who would set out to explain where McGee goes wrong should first spend some time thinking about the problem and reviewing the literature. In the meantime, please un-comment the section. There's nothing so majorly wrong with the section to warrant removing it.Knorlin (talk) 16:18, 29 January 2020 (UTC)

It seems like the passage should at least mention that although "If Shakespeare didn't write Hamlet, Hobbes did it" may appear to be false, it is actually a true statement according to how the Material conditional functions in classical logic. NicolinoChess31415926 (talk) 22:25, 23 May 2023 (UTC)

Done! Knorlin (talk) 19:04, 24 May 2023 (UTC)

Addendum: Actually, it might not be a bad idea to add a few sentences outlining the different reactions to McGee by other philosophers. This would be good added info. Knorlin (talk) 18:33, 29 January 2020 (UTC)

Reference to Principia Mathematica

In the following sentence, the author of Principia Mathematica has been abridged to be "Russell." Whitehead is a co-author of the work, so the absence seems conspicuous. Is this intentional? Is there any objection to adding Whitehead as well? modify 18:45, 17 July 2020 (UTC)

Enderton, for example, observes that "modus ponens can produce shorter formulas from longer ones",[8] and Russell observes that "the process of the inference cannot be reduced to symbols. Its sole record is the occurrence of ⊦q [the consequent] ... an inference is the dropping of a true premise; it is the dissolution of an implication".[9]

Boolean algebra section added

See Talk:Boolean_algebra#CS_guy_tackles_modus_ponens_with_mixed_results — MaxEnt 02:23, 14 October 2022 (UTC)

The section on relation to other mathematical frameworks is something of a bag of odds and ends. The “Subjective logic” framework developed by user Josang and presented here (and also more fully in a separate entry) doesn’t have widespread currency, so far as I know. However, it may be deserving of attention, and I guess there’s little harm in letting that material stand. The recently added material on “Boolean algebra,” however, has some issues. As user MaxEnt acknowledges, it's awkward to have ${\textstyle \rightarrow }$ appear in both the object language and the metalanguage. He may also be aware that his use of “${\textstyle =}$” doesn’t clearly distinguish between ${\textstyle x=1}$ meaning “${\textstyle x}$ is true” and ${\textstyle x=1}$ meaning “${\textstyle x}$ is a necessary truth.” I have tried to address these problems and in doing so have switched to a more narrative style of exposition. Someone else might see room for further improvement. Or perhaps MaxEnt will take exception to my changes—let's see. Knorlin (talk) 08:18, 5 November 2022 (UTC)

Probability calculus material trimmed back

Treating $\Pr(Q|P)$ as the probability of a conditional sentence is a dubious move; see Lewis's triviality result. The cited observation by Hailperin is about $\Pr(P\rightarrow Q)$, not $\Pr(Q|P)$. So I cut one paragraph and corrected the other. Knorlin (talk) 07:47, 20 November 2023 (UTC)