# Validity of a simple logical argument

## Is the argument valid?

• ### The argument doesn't make sense

• Total voters
11
• Poll closed .
OK, I think I can surmise that nobody knows here any justification given by mathematicians, logicians, philosophers etc. that the definition of validity you use here is correct.
Oh, well, such is life.
EB

which of your many threads would you like someone to post into?

Again, speakpigeon, you're coming across like a petulant schoolboy looking for help with your school homework.
I suggest you look at the arguments for and against classical logic being considered "the one right logic". But if your question is why the definition of validity is what it is for classical logic, then go look up the history of truth preservation in logic, and then start with the axioms upon which classical logic is built. Eventually you'll come to the definition as given.
Of course, if you omit, or change, one or more of the axioms of classical logic in the formation of a different logic, then you can end up with arguments that might be valid under classical logic and invalid under this other logic, or arguments where the question of validity is not applicable.
But that's homework for you to do.

You've insulted me for "bad logic" and yet you can't produce any justification given by professional experts, like mathematicians, logicians, philosophers etc. that the definition of validity you use here is correct.
Whoa.
EB

How do you know? Can you give me a workable definition of validity in which the argument I gave is not valid?
Recall that what you're comparing to is this: "An argument is valid if the truth of the premises guarantees the truth of the conclusion."
or, equivalently: "An argument is valid if it is impossible for the premises to all be true and for the conclusion to be false, at the same time."
Can't you produce any justification given by professional experts, like mathematicians, logicians, philosophers etc. that the definition of validity you use here is correct?
EB

You've insulted me for "bad logic" ...
And examples of where I have insulted you for such would be... where, exactly?
...and yet you can't produce any justification given by professional experts, like mathematicians, logicians, philosophers etc. that the definition of validity you use here is correct.
No sequitur on your part, which ironically is bad logic. Producing justification or not regarding the specific definition used is irrelevant to whether or not you have been insulted for "bad logic": the former is a matter of regurgitation regarding the history of logic, the latter a matter of application of that logic.

OK, I think I can surmise that nobody knows here any justification given by mathematicians, logicians, philosophers etc. that the definition of validity you use here is correct.
I'm not using one, as you have been reminded.
James has posted a common and reasonable one - it would work for me, if I needed such a thing.
If you have a better one, let's see it.
You've insulted me for "bad logic" and yet you can't produce any justification given by professional experts, like mathematicians, logicians, philosophers etc. that the definition of validity you use here is correct.
Argument from authority doesn't work here, especially mistaken authority, as you have demonstrated.

The rules of inference in classical logic seem to imply the principle of explosion (ex contradictione quodlibet ECQ). There are some very simple proofs. So the question becomes, how can the rules of classical logic best be modified so as to eliminate volatility. But introducing changes creates the danger of implications rippling through the logical system and creating inconsistencies elsewhere.

This is the province of so-called 'para-consistent logic' of which there are several different varieties. See the little history in section 1.2 in the article below (by Graham Priest, a very big name):

https://plato.stanford.edu/entries/logic-paraconsistent/

"In antiquity, however, no one seems to have endorsed the validity of ECQ. Aristotle presented what is sometimes called the connexive principle: "it is impossible that the same thing should be necessitated by the being and by the not-being of the same thing" (Prior Analytic II 4 57b3)...

...The principle was taken up by Boethius (480-524) and Abelard (1079-1142), who considered two accounts of consequences. The first one is a familiar one: it is impossible for the premises to be true but conclusion false. The first account is thus similar to the contemporary notion of truth-preservation. The second one is less accepted recently: the sense of the premises contains that of the conclusion. This account, as in relevant logics, does not permit an inference whose conclusion is arbitrary. Abelard held that the first account fails to meet the connexive principle and that the second account (the account of containment) captured Aristotle's principle.

Abelard's position was shown to face a difficulty by Alberic of Paris in the 1130's. Most medieval logicians didn't, however, abandon the account of validity based on containment or something similar. But one way to handle the difficulty is to reject the connexive principle. This approach, which has become most influential, was accepted by the followers of Adam Balsham or Parvipontanus... The Parvipontanians embraced the truth-preservation account of consequences and the "paradoxes" that are associated with it. In fact, it was a member of the Parvipontanians, William of Soissons, who discovered in the twelfth century what we now call the C.I. Lewis argument for ECQ...

The containment account, however, did not disappear. John Duns Scotus (1266-1308) and his followers accepted the containment account. The Cologne School of the late fifteenth century argued against ECQ by rejecting disjunctive syllogism..."

This little proof of ECQ is from the 'Principle of Explosion' article in Wikipedia

1. P & ~P ...our premise, some arbitrary contradiction
2. P ...from 1 by conjunction elimination
3. ~P ...from 1 by conjunction elimination
4. P or Q ...from 2 by disjunction introduction, where Q is any arbitrary proposition
5. Q ...our conclusion, from 3 and 4 by disjunctive syllogism

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Here's a not-forbiddingly-technical 12 page philosophical logic paper, written by a professor at the U. of Leeds, that discusses various arguments for and against ECQ and outlines the motivations for various kinds of paraconsistency (there are several). The paper is aimed at logic teachers faced with inevitable resistance by beginning logic students to accepting ECQ, weighing the various arguments commonly used to justify it against a position once widely held by earlier medieval logicians that the author calls Ex Contradictione Nihil (ECN) basically Aristotle's position in the Prior Analytics. He argues that most of the arguments for ECQ aren't slam dunks and how some of the earlier medieval ideas are making belated comebacks in some of the newer logical developments of the 20th century.

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This may be of interest;
Quantum logic has been proposed as the correct logic for propositional inference generally, most notably by the philosopher Hilary Putnam, at least at one point in his career. This thesis was an important ingredient in Putnam's 1968 paper "Is Logic Empirical?" in which he analysed the epistemological status of the rules of propositional logic.
Putnam attributes the idea that anomalies associated to quantum measurements originate with anomalies in the logic of physics itself to the physicist David Finkelstein. However, this idea had been around for some time and had been revived several years earlier by George Mackey's work on group representations and symmetry.
https://en.wikipedia.org/wiki/Quantum_logic

The rules of inference in classical logic seem to imply the principle of explosion (ex contradictione quodlibet ECQ). There are some very simple proofs. So the question becomes, how can the rules of classical logic best be modified so as to eliminate volatility. But introducing changes creates the danger of implications rippling through the logical system and creating inconsistencies elsewhere.
Yes, there's no absolute proof either way. It all comes down to the arbitrary logical principles (axioms) you choose to accept. In standard mathematical logic, the EFQ is a consequence of the definition of the material implication.
So, it's funny but true that the definition of validity in standard mathematical logic follows from the definition of the material implication. No other justification. Drop material implication, and the definition of validity becomes unnecessary. And the material implication itself is not justified.
This muddle just shows that mathematicians, like Socrates, are mortal beings and hence, imperfect creatures.
EB

Here's a not-forbiddingly-technical 12 page philosophical logic paper, written by a professor at the U. of Leeds, that discusses various arguments for and against ECQ and outlines the motivations for various kinds of paraconsistency (there are several). The paper is aimed at logic teachers faced with inevitable resistance by beginning logic students to accepting ECQ, weighing the various arguments commonly used to justify it against a position once widely held by earlier medieval logicians that the author calls Ex Contradictione Nihil (ECN) basically Aristotle's position in the Prior Analytics. He argues that most of the arguments for ECQ aren't slam dunks and how some of the earlier medieval ideas are making belated comebacks in some of the newer logical developments of the 20th century.

Good find. A good summary of my complaint...
Nobody who has taught elementary logic can have escaped student objections when teaching the metatheorem of classical logic stating that any proposition whatsoever follows from a contradiction. ‘That doesn’t make sense’, comes the complaint, ‘it’s absurd’.
And however familiar one might be working with ECQ, however adept at responding to students (in one or more of the ways surveyed below), the suspicion remains that there is substance to the protest.
Surely, it doesn’t follow as a matter of logic from an arbitrary contradiction that the moon is made of cheese.
That provides some badly needed perspective for our Dogmatic Outfit here.
EB

Good find. A good summary of my complaint...
You had a complaint? Where was it raised?
That provides some badly needed perspective for our Dogmatic Outfit here.
It adds nothing new. If one wants to change what "validity" means, or change the underlying axioms of classical logic, then you'll arrive at a different answer. But hey, I get that you have a need to stroke your ego. So go for it. Stroke away.

In standard mathematical logic, the EFQ is a consequence of the definition of the material implication... No other justification. Drop material implication, and the definition of validity becomes unnecessary. And the material implication itself is not justified.

But what about the "little proof" in post #148?

1. P & ~P ...our premise, some arbitrary contradiction
2. P ...from 1 by conjunction elimination
3. ~P ...from 1 by conjunction elimination
4. P v Q ...from 2 by disjunction introduction, where Q is any arbitrary proposition
5. Q ...our conclusion, from 3 and 4 by disjunctive syllogism

This is from Wikipedia, who got it from Lewis and Langford's Symbolic Logic. Historians of logic note that (according to a plausible interpretation) it is found in John of Salisbury's 12th century Metalogicon, attributed to William of Soissons earlier in the same century. Which is plausible since William was a Parvipontanian and they were champions of ECQ.

Hewett writes: "The proof is classically valid and suffices to licence any instance of ECQ. Any attempt to resist this conclusion will therefore require rejection of at least one classically acceptable rule, or else the introduction of new constraints (on which more below). Some relevance logicians have seen v-introduction as the guilty party (after all, this is the point at which apparently irrelevant material is imported into the argument), whereas others have focused on disjunctive syllogism."

My point being that ECQ doesn't seem to merely be the result of how we define material implication.

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But what about the "little proof" in post #148?

1. P & ~P ...our premise, some arbitrary contradiction
2. P ...from 1 by conjunction elimination
3. ~P ...from 1 by conjunction elimination
4. P v Q ...from 2 by disjunction introduction, where Q is any arbitrary proposition
5. Q ...our conclusion, from 3 and 4 by disjunctive syllogism

This is from Wikipedia, who got it from Lewis and Langford's Symbolic Logic. Historians of logic note that (according to a plausible interpretation) it is found in John of Salisbury's 12th century Metalogicon, attributed to William of Soissons earlier in the same century. Which is plausible since William was a Parvipontanian and they were champions of ECQ.

Hewett writes: "The proof is classically valid and suffices to licence any instance of ECQ. Any attempt to resist this conclusion will therefore require rejection of at least one classically acceptable rule, or else the introduction of new constraints (on which more below). Some relevance logicians have seen v-introduction as the guilty party (after all, this is the point at which apparently irrelevant material is imported into the argument), whereas others have focused on disjunctive syllogism."

My point being that ECQ doesn't seem to merely be the result of how we define material implication.

That's why it is accepted today. Modern mathematicians found it convenient to adopt the material implication but realised it led to explosion. The EFQ provided a convenient historical support for the choice. The EFQ principle is merely a principle you can choose to accept and different methods of logic accept different sets of axioms. It's unfortunately very easy to get sidetracked by the formalism itself, but it's even easier when you dogmatically decide that human logical intuition isn't reliable, even though the only non-arbitrary foundation for any method of logic is human logical intuition.

The proof here is only valid in the context of an arbitrary method of proof specifying which rules of inference are accepted. Change the rules and the proof here becomes invalid. The problem with this particular proof is that it give a result with is contrary to our logical intuition, which means that some or all rules are just wrong.

My overall assessment is that the various notions of consequence (together with implication and validity) as used throughout the whole of mathematical logic are essentially wrong and indeed seriously wrong. All formal methods of logic produced so far are essentially seriously wrong except, as far as I can tell, Aristotle's syllogism and perhaps the Stoics' method, both being essentially founded on the Modus ponens, and both based on the human logical intuition. Modern mathematical logic cut itself from logic as a mental capability. Wrong move.
EB

Modern mathematical logic cut itself from logic as a mental capability.
Mathematics requires self-consistency and rigor.
All formal methods of logic produced so far are essentially seriously wrong except, as far as I can tell, Aristotle's syllogism and perhaps the Stoics' method, both being essentially founded on the Modus ponens, and both based on the human logical intuition.
They protect you from such common errors of intuition as the "modal fallacy", though. So they are handy to keep around.

How does Modal logic deal with the issue of :
unknown variable + unknown variable equaling a definitive result?
A maybe plus a maybe doesn't normally equal a definitive.... ( I would have intuitively thought...)

Mathematics requires self-consistency and rigor.
So why is it mathematicians still don't use formal logic to prove their theorems?
And what's not self-consistent about the logical reasoning of a human being?
Could you explain what is logically inconsistent or not rigorous for example about Aristotle's syllogistic?
EB

How does Modal logic deal with the issue of :
unknown variable + unknown variable equaling a definitive result?
A maybe plus a maybe doesn't normally equal a definitive.... ( I would have intuitively thought...)
It doesn't.
EB

So why is it mathematicians still don't use formal logic to prove their theorems?
They prefer not to waste their time.
And what's not self-consistent about the logical reasoning of a human being?
Intuition, you were talking about. Not reasoning.