## Modus Ponens as Substitute for Syllogism

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This is a pretty basic question, but I haven't been able to find a clear answer in the various sources I've looked at, including the Kneales' chapter on conditionals in 'The Development of Logic'.

In propositional logic, when modus ponens stands in for a syllogistic/categorical argument, with the antecedent being the conjunction of the original syllogism's premises, and the consequent the conclusion, is it the standard interpretation to regard the affirmation of the antecedent as thereby entailing the conclusion only on the assumption that the original argument was valid, i.e the validity is external to the conditional itself, as opposed to that validity somehow being transferred to the conditional so long as the antecedent-as-premises is affirmed?

And more broadly, is propositional logic just a logic of "this is what follows, assuming not only that the statements p, q, etc, are true but also that the rules of inference do actually represent valid arguments" (insofar as validity is the aim, rather than, say, causal arguments)?

In answer to Graham, I'm adding this to the original question because of the text limit for comments.

In propositional logic, syllogistic, i.e. categorical, arguments are regularly expressed using modus ponens, with the conjunction of the two premises (e.g. "all men are mortal & Socrates is a man") serving as the antecedent of the conditional, if p then q, and the consequent ("Socrates is mortal") as the conclusion. Stating this in the form of a conditional statement is recognized to not be sufficient for the conclusion to be entailed by the premises per se, but when the conjunction of the premises, 'p', is then affirmed to be true, which turns the conditional into a modus ponens argument, then the logic textbooks describe the conclusion, 'q', as then being 'inferred'. But surely this can't be the case, and my question was "is it generally recognised that this is not the case".

As an example of why this can't be so, take the enthymeme "Socrates is a man, therefore Socrates is mortal", which is an invalid argument as it stands, but if you add the missing premise "all men are mortal" then it becomes a valid syllogism, yet modus ponens doesn't differentiate between the two - so long as the premises are affirmed to be true, then the conclusion that "Socrates is mortal" can in both cases be 'inferred'. Surely this means that whatever validity there was in the original argument has not been carried over to the modus ponens formulation.

Yes, a valid syllogism doesn't ever become invalid when expressed using modus ponens, but only because the validity provided by the original logical form has disappeared (if this were not the case, then we should not be able to 'infer' a consequent that stands for a syllogism's conclusion, from the affirmation of an antecedent that stands for an enthymeme's premise, yet we can). It seems to me that the conclusion is only 'inferred' if the syllogism has already been proved outside the system and then assumed within it. And even then, what kind of inference is "this is a valid argument and its premises are true, so it's a sound argument too, which means its conclusion is also true"? This merely declares that a valid argument, made elsewhere, is sound. You don't infer something from an affirmation that something is the case, but rather, if an inference is valid to the effect that something must follow from something else, then an affirmation of that first thing entitles you to affirm, not infer, the second thing.

In other words, in a syllogism, a conclusion is true on the condition that the premises are true, only because such an argument is already valid, whereas with modus ponens, when used to express such an argument, the conclusion is only true on the condition that the premises are true AND that it's a valid argument to begin with. And affirming the antecedent merely affirms the former. If that is right, then propositional logic, which is heavily reliant on material implication, is not 'truth-preserving' in the sense that actually valid arguments are truth-preserving, i.e. where you can only go from true premises to true conclusions because there's a valid form that is prior to any claim of soundness, but instead, if we grant from the outset that certain statements are true and that certain material implications are also true, i.e. truly stand for valid arguments, then from this we can say (not 'infer' or 'conclude', except indirectly and implicitly) that other things have to be true. Again, for all I know all of this all might be commonplace, but the purpose of my original question was just to check if that was so.

I made an edit mainly to break the text into smaller pieces. You may roll this back or continue editing. Welcome! – Frank Hubeny – 2018-08-14T19:53:55.023

There are two different notions of consequence, semantic and syntactic. For semantic inference it makes no difference what the rules of inference are, it is defined in terms of truth in models, for syntactic inference whatever rules of inference are adopted are automatically "valid" since following them is how validity is defined, see Implies vs. Entails vs. Provable on Math SE. But in either case it makes no difference whether premises are true for a conditional to be valid, so I am not sure what the "original argument" is doing here.

– Conifold – 2018-08-14T21:26:20.217

@ScottB. Please divide this question into even shorter statements. – Mark Andrews – 2018-08-14T23:16:37.030

It's not clear what you are asking. Perhaps provide an example of an "original sylogism" and how "modus ponens stands in for" it. – Graham Kemp – 2018-08-15T02:20:07.553

Hi Graham, I've replied to you by editing the question. – Scott B. – 2018-08-15T21:40:46.687

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In propositional logic, syllogistic, i.e. categorical, arguments are regularly expressed using modus ponens, with the conjunction of the two premises (e.g. "all men are mortal & Socrates is a man") serving as the antecedent of the conditional, if p then q, and the consequent ("Socrates is mortal") as the conclusion. Stating this in the form of a conditional statement is recognized to not be sufficient for the conclusion to be entailed by the premises per se,

No, "All men are mortal" is the conditional statement (a universal one to be precise). "Socrates is a man" is another predicate. They conjointly entail the consequent "Socrates is mortal."

Ɐx (Man(x)→Mortal(x)), Man(Socrates) Ⱶ Mortal(Socrates)

but when the conjunction of the premises, 'p', is then affirmed to be true, which turns the conditional into a modus ponens argument, then the logic textbooks describe the conclusion, 'q', as then being 'inferred'. But surely this can't be the case, and my question was "is it generally recognised that this is not the case".

It is not. It is the case that q will be infered from p and p→q using the rule of 'modus ponens'.

p→q, p Ⱶ q

PS:

You seem to be confusing the rule of modus ponens with the the tautology: ((p → q) ˄ p) → q , which can be proven by using that rule of inference.

0. |___
1. |  |_ (p → q) ˄ p      Assumption
2. |  |  p → q            ˄ Elimination (1)
3. |  |  p                ˄ Elimination (1)
4. |  |  q                → Elimination (2,3)  aka Modus Ponens
5. |  ((p → q) ˄ p) → q   → Introduction (1-4)


Hi Graham, thanks for the response, as to the first point, I don't dispute the validity of the way predicate logic represents syllogistic argument, only the way propositional logic does. Your line of predicate logic isn't a proof and still requires UI and then MP. And my point was precisely that MP adds nothing unless there's a prior categorical demonstration such as you gave: Ɐx (Man(x)→Mortal(x)), Man(Socrates) Ⱶ Mortal(Socrates). I.e. to prove that argument, Man(Socrates) Ⱶ Mortal(Socrates) must be instantiated then Man(Socrates) affirmed, from which we then conclude Mortal(Socrates) by MP. – Scott B. – 2018-08-16T21:11:14.837

As to the second point, I know that MP says you can infer q from (p>q and p), but in my question I was arguing that i) unless there's a prior demonstration of the kind that is explicit in predicate logic, or (I believe) assumed in propositional logic, then you cannot meaningfully 'infer' anything, and ii) even if you do have either an explicit or assumed categorical argument prior to your 'proof' by MP, such a proof merely amounts to the assertion that the argument presented using modus ponens is sound. – Scott B. – 2018-08-16T21:51:25.307

Also, I don't see that there's any difference in modus ponens as a purported argument form and modus ponens expressed as a truth-functionally tautologous conditional statement. The very fact that it can be represented as a tautology is an indication to me of its ultimate vacuity. – Scott B. – 2018-08-16T22:03:13.317

@ScottB Another way to say that is that it is "self-evidently true", or axiomatic, that Q is inferable from P and P implies Q. Modus ponens is a fundamental rule of inference essentially justified by "that's what implication means". The Socratic sylogism can be seen to be composed of two such rules ... modus ponens and universal instantiation ... and so is not fundamental. – Graham Kemp – 2018-08-16T22:43:45.763

IMO, logical deduction shouldn't itself be expressed axiomatically, but rather, self-evident axioms such as the laws of identity and non-contradiction provide the fundamental basis for logically-valid form that says: if certain things involving identity are the case, these other things must follow. Although such axioms express how things metaphysically must be (and so are tautologous), the point of a logical argument is to show specific consequences of necessity, not to be a vacuous imitation of it, as if a valid argument itself rather than its conclusion were what is necessarily true. – Scott B. – 2018-08-20T20:30:35.203

In other words, an axiomatic tautology says how things must be, while a logically-valid argument says, given these axioms which say how things must be, this thing must follow from that. Why should a valid argument that is based on axiomatic tautologies itself be expressed as a modus ponens argument which is in effect such a tautology? An axiom is a basis for a theorem (axiom + inference = conclusion), so why should this be expressed as if the whole argument were a just an axiom? The only way to turn a valid argument into an axiomatic truth is to render it as a tautology in the vacuous sense. – Scott B. – 2018-08-20T21:01:12.700

It's odd that in logic 101 textbooks they always say "an argument isn't true or false, it's either valid or not and if it's valid, then it's sound or not - rather it's the conclusion that is true or false", but then they immediately go on to present modus ponens as being a necessarily true argument (a tautology, not a valid argument with a necessarily true consequence). Either they were right the first time or else modus ponens can't be a necessarily true argument, because arguments aren't true or false, only premises and conclusions are. Which is it? – Scott B. – 2018-08-20T21:12:38.350

@ScottB. The tautology (p ˄ (p → q)) → q is not modus ponens. It is a statement that is true for all asignments of its literals. Modus ponens is a rule of inference to derive a statement from other statements of the specified schema; the valid argument that from P and P→Q you can derive Q. They are different structures. – Graham Kemp – 2018-08-20T23:33:48.160

Your claim is that affirming the antecedent turns a conditional statement, which all are agreed doesn't entail its consequent, into a MP argument whereby that consequent can be deduced from the truth of that antedent (meaning, if the antecedent stands for premises of a valid argument, then the conclusion can be deduced as a true consequent)? If this is so, then MP is an assertoric argument, despite the conditional now being one of its premises. which asserts that a conclusion is entailed by premises, no? – Scott B. – 2018-08-23T19:58:35.373

And given that any assertoric valid argument can also be expressed as a hypothetical conditional, indicating that if the premises are true, then the conclusion follows, e.g. "IF (all men are mortal &. etc), then S is mortal", it seems to me that "if p & (if p, then q), then q" is just such a hypothetical rendering of MP: if the premises of MP are true, then its conclusion is. – Scott B. – 2018-08-23T19:58:50.957

Except, unlike the syllogism case, the result is a tautology. This is because affirming the antecedent effectively strikes out the "if" from the conditional premise, because for a conclusion to be entailed by premises there must be a valid argument form, which in this case can only be "p therefore q". (This applies if MP is being used to represent a valid argument whose conclusion is entailed - I'm aware there are other interpretations of the conditional's function). – Scott B. – 2018-08-23T19:59:06.457

At the same time, by affirming "p", we are saying in effect that "p therefore q" is not only valid but is also sound. And yet there is no formal validity in the bare-boned argument "this therefore that", which is why i.) MP can provide no more than a confirmation of the soundness of a valid argument made elsewhere, with its own validity being derivative of that, and ii.) when stated as a hypothetical, it's a tautology: "if the premises are true, and the conclusion follows from the premises, then the conclusion is also true." – Scott B. – 2018-08-23T19:59:23.930

This tautology is necessarily true because it represents not a valid argument but the fact that the truth of a valid argument's premises would, by defintion, entail the truth of its conclusion. i.e., although it is supposed to be basic that arguments themselves are not true or false, but rather, if an argument is valid then a true conclusion is entailed by true premises (the 'necessity' here being provided by logical form) yet MP, stated hypothetically is necessarily true only vacuously: "an argument with premises that entail a conclusion entails that conclusion if the premises are true". – Scott B. – 2018-08-23T20:13:00.973