Talk:Modus ponens

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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 (talk) 20:00, 1 May 2012 (UTC)

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)

Is this the same as a sylogism?

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

self-referential premises[edit]

   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.


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[edit]

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. (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[edit]

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.


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"[edit]

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]

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[edit]

isn't this called hypothetical syllogism as well?

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

Another rule[edit]

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

The converse[edit]

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-- (talk) 14:18, 8 April 2011 (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[edit]

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)


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.[edit]

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))