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Since long ago I've been feeling uneasy about the modus ponens. It's sometimes presented as if it were a sort of discovery or a law of how things are.

I think it simply describes what the sentence structure "If ... then ..." means. It's basically the definition of implication.

How is this treated formally in mathematics? Is "implication" a primitive black box at first and then we assert the modus ponens as an axiom?

Once you know the semantics of implication (if-then sentences), you don't need the modus ponens any more. And learning the modus ponens teaches you all there is to know about the meaning of if-then sentences.

I mean, "modus ponens" and "semantics of implication" cover the exact same thing.

You can specify the semantics of implication with a truth-table. Why do you ever need to deal with modus ponens then?

EDIT: Additional clarification. I think the modus ponens is a superfluous step in most discussions. I think the moment that you prove to yourself, the moment you realize, the moment that it clicks in your head that "If p then q"; and then later on you notice that p is true, you automatically realize that q is true.

Otherwise what does it mean that you truly realized "If p then q"? If this belief cannot work with the p belief to yield q automatically, in what sense do these beliefs really encode their meanings?

By automatical I mean that you don't have to go through another step like

Okay, so whenever I have an if-then sentence and I also have its antecedent among my beliefs, I can apply modus ponens and append the consequent to my list of beliefs. Since I'm in such a situation now, I can do this.

But this would just introduce another meta modus ponens. Because you have the modus ponens rule, which itself has an if then structure on a meta level: If you have such two sentences, you can also have a third. In order to apply it, you need to see that you indeed have two such sentences and realize by meta modus ponens that this situation in conjunction with modus ponens mean that you can indeed apply the rule. This leads to an infinite regress.

I think it's best to stop on the first level already and accept q automatically. Just as you don't appeal to meta modus ponens to actually make modus ponens work (you assume modus ponens automatically kicks in), you could already assume that the "If p then q" sentence works automatically too.

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    $\begingroup$ A small technicality: modus ponens is not typically treated as an axiom, but rather as a rule of inference. In answer to your question, the modus ponens rule is precisely how the "semantics of implication" is treated in formal reasoning; in some sense, the rule of inference in this case defines the meaning of implication. $\endgroup$ – Dustan Levenstein May 14 '15 at 14:48
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    $\begingroup$ Your confusion about the infinite regress is actually exactly why we need to distinguish between axioms and rules of inference: axioms are merely statements about a system, wheras rules of inference tell you what kinds of steps are valid in a proof deriving new statements from previous statements. We simply chop off the regress at the point of rules of inference. $\endgroup$ – Dustan Levenstein May 14 '15 at 15:06
  • $\begingroup$ There is an amusing little story by Lewis Carroll "What the Tortoise Said to Achilles". [If we think of "modus ponens" as an axiom, then we would need a meta modus ponens, then a meta meta modus ponens and so on...] ditext.com/carroll/tortoise.html $\endgroup$ – GEdgar May 15 '15 at 12:53
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Modus ponens is the rule of inference that allows us to conclude $$p\to q,p\vdash q $$ and indeed is the one rule that gives the syntactic construct expressed by the $\to $ symbol its semantic (or actually just part of it; do you know what's missing?). Similarly, the rules of inference $$\begin{align}a\land b&\vdash a\\a\land b&\vdash b\\a,b&\vdash a\land b\end{align} $$ do the same for the $\land$ symbol and "and" semantics. At any rate, it is comforting that there is only a handful of rules neede to express the exact meaning of "if" and "and", ans it is certainly simpler to agree that a certain explicit handful of syntactic rules of inference shall hold and much less error-prone than saying "hey, the $\to$ stands for if-then, just as you probably learned in kindergarden, but onl yin the material sense, not necessarily in the modal sense or that there need not be this causality thing between left and right and stuff"

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Taken literally, this is not quite true. A connective $c$ might be said to satisfy modus ponens provided that

if $c(p,q)$ and $p$ are true, then $q$ is true too

at every row of the truth-table. However there are many connectives satisfying this condition, for example conjunction, biconditional, joint denial, and absurdity.

There are some ways you might get the claim you want. First, you might propose that the material conditional is the "weakest" connective satisfying modus ponens. That is to say, you might propose

if $c$ satisfies modus ponens, then at every row of a truth-table, $c(p,q)$ is true only if $p\to q$ is true.

Alternatively, you might notice that a natural deduction system really contains two rules for conditional, the other being conditional proof. You might try to show that conditional is the only connective satisfying both of these rules. If you get stuck, you might then try to prove the weaker claim, that conditional is the only truth-functional connective which satisfies modus ponens and conditional proof in the context of the rest of the rules of the system.

Note that the idea of "satisfying conditional proof" is somewhat less straightforward. Here, you want to use the idea that for a connective to satisfy an inference rule is for the inference rule, interpreted by that connective, to be such that it always transforms valid arguments into valid arguments.

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Modus Ponens is what gives semantic implication utility! For example, the sentence if two equals three then I'm the king of the world is true by definition (since the antecedent is false), however, this sentence is never useful!

Consider now the following:

If $x$ is a natural number then $x+1$ is a natural number

This proposition is true for every natural number $x$, and moreover, Modus Ponens allows you to conclude, for example, that $6$ is a natural number since we know $5$ is a natural number.

It seems Modus Ponens is useless, and somehow it is semantically (it can be deduced from the definitions semantically), but not sintactically, in the context of formal proofs, you need to add the condition whenever you have $p\implies q$ and $p$ then you have $q$.

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I think the modus ponens is a superfluous step in most discussions. I think the moment that you prove to yourself, the moment you realize, the moment that it clicks in your head that "If p then q"; and then later on you notice that p is true, you automatically realize that q is true.

But "modus ponens" is neither more nor less than a fancy semi-formal name for exactly the reasoning step you describe here! (And by extension, for the formal rule that embodies that reasoning, when you're speaking about a syntactic proof system).

In other words you're saying "modus ponens is superfluous because it is just modus ponens".

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  • $\begingroup$ I'd say modus ponens describes what a reasoning agent does from an external point of view. But it is not represented in the reasoning agent's mind. The agent itself doesn't think "Hey, we have this modus ponens thing, let's apply it". It only thinks "Hey, we have p->q and p, so let's add q". Only an external observer can point out that "Look the agent derived a new sentence. I noticed a pattern in when and how it derives new sentences and one of these patterns I named modus ponens". $\endgroup$ – isarandi May 15 '15 at 15:48
  • $\begingroup$ @isarandi: Yes, and so? If you want to design a machine that replicates what your agent is doing, then you're going to need names for the patterns you construct it to follow, such that you can speak about them to your fellow engineers. $\endgroup$ – Henning Makholm May 15 '15 at 15:52
  • $\begingroup$ I guess my original issue was that the modus ponens is often invoked in semi-formal philosophical arguments, where you yourself do the reasoning. If you can't trust yourself to automatically apply it, how can you trust yourself to apply "meta modus ponens" automatically, i.e. automatically decide whether modus ponens is applicable. I think these kinds of formal-looking listings of claims in semi-formal philosophy (like theological arguments etc) is just a way to make the argument seem objective, when in fact all the difficult parts have been hidden into the premises. But I digress. $\endgroup$ – isarandi May 15 '15 at 16:01
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"Why do you ever need to deal with modus ponens then?"

Because we're humans and not computers or say integrated circuits. Think e.g. a 10 year old child and the two ways of presenting "if semantics" to them. Which one would be grasped more easily, more naturally? This question is rhetorical, I think.

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Let's think of a game like Clue. I don't remember how the game actually works, but let's suppose it's something like this:
In the game you have to discover the details of a murder by taking note of hints that appear during the game.
For instance, you could conclude, at some point, that "if the culprit is right handed, it's not a woman". So, your notes of the things you know at this point would include this line:
"right handed->woman"
But, you don't know if the culprit is actually right handed, or if it's a woman.
But at some later point you happen to acquire the information that the culprit is right handed, so you add this line:
"right handed".
Now, because of modus ponens, you can add this line, without receiving any additional information from the game:
"woman".

So, implication and modus ponens are actually very different: the first only looks at the truth values of its components, while the second looks at their relationship:
let's take
A="I'm playing a game"
B="I'm not the culprit"
C="I have brown hair".
Let's say in our case they are all happen to be true.
$A\land B\to C $ is true, and I can write it down.
$A\land B \vdash C $ is different: it's not something we would write on the list, it's a meta statement that means "if A and B are on the list, I can infer C".
It's also wrong, since there's no clear connection between A, B and C, nor the truth of C is an universal constant (it just happens to be true in my specific situation. it's probably not true for every innocent player, at least if we interpret the I as anyone who's saying the sentence and not me specifically).

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