## Is it possible to prove inference rules?

• 69
Are inference rules, such as modus ponens, disjunctive syllogism and others, possible of being proved? I mean, are they axioms, which validity is transcendental and exist by itself, or can they be deducted from the very three logical principles?
I ask this because I've seen the exposition of this inference rules in a lot of logic books, but I haven't seen, until then, why are those propositions true, even though they are intuitive.
• 967
If you mean traditional logic, aka "laws of thought," there are different ways to axiomatize it. But modus ponens, for example, can simply be proven from the truth table.
• 69
Yeah, i'm talking about traditional logic. But by proving it by truth table wouldn't you already be relying on the inference as a valid method, since the truth table is the conjunction of the premises implying in the conclusion? I mean, why wouldn't it be a petitio principii?
• 967
But by proving it by truth table wouldn't you already be relying on the inference as a valid method, since the truth table is the conjunction of the premises implying in the conclusion?

In classical logic implication p->q is a given, so yes, it is a valid method by definition. It is defined as a truth function f(p, q) with a known truth table. Other logics may treat implication differently though.
• 763
Are inference rules, such as modus ponens, disjunctive syllogism and others, possible of being proved? I mean, are they axioms, which validity is transcendental and exist by itself, or can they be deducted from the very three logical principles?

No. "Proof" is defined by the axioms and inference rules one adopts. Ergo, there's no way to independently prove the validity of such things because proof and validities are what you get from the above things. There's no transcendental rules that cannot be violated.

A proof of that (in the colloquial sense of "proof") is that even in formal logic, you kinda have to develop the logic twice. Your construction of the formal system is done within a metatheory which itself can have its mathematical properties investigated, it's axioms brought to light, etc. And sure, you could keep going, checking out the meta-meta theory, etc., but you're just doing the same thing as you were in the object theory.

So the notion of a purely independent proof, of "laws of thought" or absolute, inescapable presuppositions that need no proof is just an incoherent idea. You just pick the ideas that seem reasonable or right to you - whether for pragmatism, interest or what have you - and let things play out. Like, the formal logic "LP", the Logic of Paradox, is a propositional logic which does not validate Modus Ponens. Of course, to be of any use beyond vague mathematical interest it can be augmented with a non-truth-functional conditional, but this is just to give evidence to my point about why an absolute background proof of such things doesn't make sense. Nearly any rule can, by some means, be done without in a formalism. The only real no-no everyone* agrees on is avoiding triviality.

*Outside very sparse philosophical interest; I've seen one wishy-washy "defense" of trivialism that was more for interest than believability.
• 69
• 171
Inference are by definition NOT deductions. Ergo they are NOT proofs. They are simply refined conjectures. And that is what Science is all about. That is why Science is NOT Philosophy.
• 69
What is the definition of inference? Why can't it be treated as a deduction? I mean, from "A→B" and "B→C" you deduce that "A→C", right?
• 171
Surely you know the answer to your own question and therefore you are simply being verbose (as in the fallacy of verbosity).
• 69
I'm not being verbose. You said inferences are by definition not deduction. I'm just asking you what definition are you using, because I pointed an exemple of a deductive argument made by an inference.
But then I realized that an inductive argument contains inferences as well, so an inference cannot be a deduction. You could just answer my question instead of being defensive.
• 93
The claim that
the truth table is the conjunction of the premises implying in the conclusion

does not seem correct. There is nothing inherent in the definition or concept of a truth table that identifies it as being anything other than a tabular representation of the possible binary values assigned to some variable. Consider the simplest truth table:

A | A
_____
T | T
F | F

Which just says the variable "A" has the values assigned to it. There is nothing here about "conjunction" or "implication". Truth tables can further be used to define what certain operators like conjunction and implication mean by showing how the values assigned to variables change when the operator is applied to them. At this point truth tables are used to introduce the notions of conjunction or implication or whatever, but they don't purport to prove anything about these operators. Once you have defined what the operators are, you can construct tautologies that are the functional equivalent of the rules of inference that show that the rules preserve the truth values of the variables they're applied to because they are, well, obviously true as tautologies. The rules of inference then are just short hand ways of constructing these tautologies that are more convenient to work with.
• 69
You are right. When I said about truth tables I was thinking on truth tables of arguments, by which an argument can be proved or invalided by checking if the conjunction of the premises implies in the conclusion. I totally forgot about the simpler cases, like what you pointed.
• 34

Actually the method of tables rely on the interpretation of the logical connective, in regards to enunciative logic.

Modus ponens is indeed a completely different thing: it is a valid argumentative form. Thus, it presupposes an enunciative theory to be connected to.

No. "Proof" is defined by the axioms and inference rules one adopts. Ergo, there's no way to independently prove the validity of such things because proof and validities are what you get from the above things.

There are two errors: 1 proof is not 'defined' either by axioms or inference rules. Proof is the results of applying inference rules by means of axioms. This is not at all a definition, just as the egg is not a definition of the chicken.

2 It is false that you can not prove validity of 'such things'(axioms? rules of inference?) independently(even if your statement is incomplete, because you do not specify independent from what). The proof of a logical theory is obtained by verifying the coherence of axioms, i.e. through the non contradiction principle in a certain form: iff from the set of axioms you can not derive, through rule of inference, a contradiction, then the set of axiom is coherent. You can even proof the independency of some axioms from others by verifying that the same theorem deducible by n axioms+ x axioms is deducible even just through n(or x) axioms.

Of course do not exist an absolutely independent way to proof something, because a demonstration itself is defined as the derivation of something from something which is different, and this IN ACCORD TO A RULE(or how else is to be recognized as a relation of derivation and not a succession?). There is always a reference to establish the recognition of a result from the relative process. And this you say:

So the notion of a purely independent proof, of "laws of thought" or absolute, inescapable presuppositions that need no proof is just an incoherent idea.

But not because it is incoherent, but because is IMPOSSIBLE.

I am not saying that rules of inferences are eo ipso processes: yet to us at least there is a correspondent process, through which we infer from something to something different.

The rules do not claim for proof because they are a process. Just as you do not search the proof of the friction causing your car to adhere to the ground, but an explanation of it, which is a completely different thing, and not at all a logical proof.

Being logic the form of thought in general, it is a necessary condition to be SATISFIED in order to have some way at all to distinguish a process and its results. Once satisfied, we have the condition under which proofing has a form at all, recognizable.

Inference are by definition NOT deductions. Ergo they are NOT proofs. They are simply refined conjectures. And that is what Science is all about. That is why Science is NOT Philosophy.hks

Here is a classic confusion( which MindForged did not make) between a method to obtain a proof and a proof, i.e. between say something THROUGH a language and to say something ABOUT a language.

Inference are what renders possible to distinguish at all a logical difference, and since this is the basis of thought in general, without it would be missing the GROUND to proof something. This is the reason why inferences can not be proved, because by RULES OF INFERENCE a ground to proof is furnished, and THEN inferences are made to obtain proofs, after the logical form has been interpreted, and thus there is an object at all as the material to realize the structure of inferences, and Then to infer.

Inferences are not at all relying on definitions, because they are of a different status(logically): a definition is an EQUIVALENCE(hence it presupposes a logical structure too, and is a PROPOSITION) while INFERENCE is the PROCESS through which from NOT EQUIVALENT propositions, one is obtained from others. And a process is not defined, but IDENTIFIED, such as gravity is not defined ny the gravitational law, but its EFFECTS are recognizable insofar as they are measurable through that law, which is so far to be a definition of gravity that it is gravity that made possible to discover the laws. The laws of course are conformant to a structure of general thought, which is the only postulate we need.

Your deeply confused view is evident in the next sentence

They are simply refined conjectureshks

in which you clearly confuse an operation on propositions(inference) with a proposition( conjectures are propositions).

Science is so far from being a pretty little thing of conjectures, that conjectures are propositions not rigorously proved, while proofs are propositions rigorously proved and those, in a systematic unity, constitutes what every rational being call 'science'. A bunch of conjectures, without a system in which they may be proved are just grammatical fantasies.

That is why Science is NOT Philosophy.hks

Being clear that you got wrong giving your account on what science is(or should be in accord to your thoughts of course) I would be curious to hear the difference between science and philosophy, being each science a coherent and consistent system of conclusions obtained through rule of inferences from objective premises. If philosophy is not science is just a fable, as everything with no math and no logic

.
'm not being verbose. You said inferences are by definition not deduction. I'm just asking you what definition are you using, because I pointed an exemple of a deductive argument made by an inference.
But then I realized that an inductive argument contains inferences as well, so an inference cannot be a deduction. You could just answer my question instead of being defensive.

This is a very educate behavior and to me an index of a truly inquisitive, instead of rhetoric, nature.

You are totally correct Nicholas: you recognized that our Great Master Hks would like to teach us(or to put blame on our supposed inferiority in regard to his great mind) Logic, but he don't even know the difference between an operation(inference) and the validity of an argument, for both induction and deduction are inferences, and both can be valid, hence they differ logically(i.e. in respect to validity) just in respect to the truth of the premises. He confuses deductive/inductive(which is psychological) and deductively/inductively VALID(which is logical).

41
The claim that
the truth table is the conjunction of the premises implying in the conclusion
— Nicholas Ferreira

does not seem correct. There is nothing inherent in the definition or concept of a truth table that identifies it as being anything other than a tabular representation of the possible binary values assigned to some variable. Consider the simplest truth table:

A | A
_____
T | T
F | F

Which just says the variable "A" has the values assigned to it. There is nothing here about "conjunction" or "implication". Truth tables can further be used to define what certain operators like conjunction and implication mean by showing how the values assigned to variables change when the operator is applied to them. At this point truth tables are used to introduce the notions of conjunction or implication or whatever, but they don't purport to prove anything about these operators. Once you have defined what the operators are, you can construct tautologies that are the functional equivalent of the rules of inference that show that the rules preserve the truth values of the variables they're applied to because they are, well, obviously true as tautologies. The rules of inference then are just short hand ways of constructing these tautologies that are more convenient to work with.

Here is a person who reflects before writing. Not surprisingly he said something clear, in a very educate manner, and also correct; also with some interesting elements of originality: «truth tables are used to introduce the NOTIONS of...» in a somewhat 'diagrammatic' view of recognition(not merely linked to processes).
• 763
There are two errors: 1 proof is not 'defined' either by axioms or inference rules. Proof is the results of applying inference rules by means of axioms. This is not at all a definition, just as the egg is not a definition of the chicken.

This is either nonsense or splitting hairs in a way that changes nothing. What counts as a "proof" is determined by the axioms and the inference rules. Me referring to that as a definition seems perfectly comprehensible. What counts as a "proof" in Intuitionistic logic is very clearly and certainly distinct from what counts as a "proof" in Classical logic. Because the rules for a valid proof is not the same in different formalisms.

2 It is false that you can not prove validity of 'such things'(axioms? rules of inference?) independently(even if your statement is incomplete, because you do not specify independent from what).The proof of a logical theory is obtained by verifying the coherence of axioms, i.e. through the non contradiction principle in a certain form: iff from the set of axioms you can not derive, through rule of inference, a contradiction, then the set of axiom is coherent. You can even proof the independency of some axioms from others by verifying that the same theorem deducible by n axioms+ x axioms is deducible even just through n(or x) axioms.

You are rather proving my point. "Coherency" here is exactly another way of saying "non-contradictory". In other words, it's reliance on another axiom (Non-contradiction). I explained what I meant by the incoherency of the idea of a purely independent proof of all axioms. Let's quote myself, then:

Reveal
"Proof" is defined by the axioms and inference rules one adopts. Ergo, there's no way to independently prove the validity of such things because proof and validities are what you get from the above things. There's no transcendental rules that cannot be violated.

A proof of that (in the colloquial sense of "proof") is that even in formal logic, you kinda have to develop the logic twice. Your construction of the formal system is done within a metatheory which itself can have its mathematical properties investigated, it's axioms brought to light, etc. And sure, you could keep going, checking out the meta-meta theory, etc., but you're just doing the same thing as you were in the object theory.

So let's make this even clearer, since apparently it wasn't. What counts as a proof requires one to adopt some set of rules by which to establish what will count as a proof. But the reasoning employed in the metatheory (what we're using to reason about the construction of the logic in question) doesn't have some inherent correctness to them, one just ends up presuming some set of inference rules and axioms in the background and those show up in the object theory because of that. So for example, classical logic can be constructed from a boolean algebra, as the two are basically equivalent, so we see that a boolean algebra of sets naturally gives us a certain kind of logic (and the reverse can be done as well). But we know numerous metatheories exist independently of the others using other set theories and such, but you never get to some independently proven axioms or something. You have to assume something is just off the table to get going. I'm not saying this is a problem, it's just how things have to be.

But not because it is incoherent, but because is IMPOSSIBLE.

Sure, I took those words to be mean same thing. I used the word incoherent the idea itself is without meaning because I'm skeptical one could even conceive of how it could even be done. The idea of an independent proof of all axioms makes the mistake of forgetting that what constitutes a proof is determined by some set of axioms and inference rules. The inference rules in proof systems are, after all, taken to be primitive. If they could be proven, we would not take them to be primitive.
• 34
This is either nonsense or splitting hairs in a way that changes nothing. What counts as a "proof" is determined by the axioms and the inference rules

Actually there is a HUGE difference between what you are saying and the actual way in which us in logic prove things.

Your ambiguous use of the term 'determining' it seems to refer to CONSISTENCY.

Let me explain you the very big difference between my(correct) statement and yours(wrong).

If you say that the axioms DETERMINE what a proof is, you are saying that they are CONSISTEN, i.e. that for every propositions you can by means of those axioms, DETERMINE IF that proposition is or is not a tautology.

Now, such was the view of the old proof theory purposed by Hilbert, PROVED INCONSISTENT by a man, Kurt Godel.

This lead to another proof theory: _Natural deduction by Gentzen(and Sequence Calculus, also by him).

In this new theory(which disposes of both Natural deduction and Sequence Calculus) the proofs are not at all 'determined' by the axioms, but the axioms(if there are any:in ND there are no axioms) RELY ON THE 'NATURALITY' of the RULES OF INFERENCE. It is so different from the former theory that Gentzen could prove CONSISTENCY of arithmetic(Peano arithmetic) in 1936. This is important also because of other aspects, but I reserve myself to expose those aspects only later or requested.

It seems to me that you have not clear that AXIOMS and RULES OF INFERENCE are not at all the same thing; and also(here):

"Coherency" here is exactly another way of saying "non-contradictory".

that you have not clear that logical consequence is a SEMANTIC characterization of a set of valid forms, while logical derivation is a SYNTACTIC characterization of s set of valid forms. These are so far from being the same that a theorem(by Godel: compactness theorem) has to be proved, in order to demonstrate their EQUIVALENCE(which is not identity).

What counts as a proof requires one to adopt some set of rules by which to establish what will count as a proof.

But not what is a proof, yet what it will be considered as 'closed' in regards to the set chosen in respect to the operations and relations defined on it.

But the reasoning employed in the metatheory (what we're using to reason about the construction of the logic in question) doesn't have some inherent correctness to them,

This is somehow unclear, for Godel's second incompleteness theorem is taken in some context as stating that the consistency of a theory can only be proved in a stronger theory.

one just ends up presuming some set of inference rules and axioms

And this is false, for axioms are NOT presupposed at all in actual proof theory. So the conjunction is false.

So for example, classical logic can be constructed from a boolean algebra, as the two are basically equivalent, so we see that a boolean algebra of sets naturally gives us a certain kind of logic (and the reverse can be done as well). But we know numerous metatheories exist independently of the others using other set theories and such, but you never get to some independently proven axioms or something.

This is correct. But does not imply:

You have to assume something is just off the table to get going. I'm not saying this is a problem, it'

Just as the chicken assume nothing at all to generate an egg. If a chicken could reason, it would try to explain the egg thing trying to distinguish products and processes and then trying to derive from those processes, as independent from the univocity of resulting in a egg, other things, verisimilarly of the same kind somehow.

Sure, I took those words to be mean same thing. I used the word incoherent the idea itself is without meaning because I'm skeptical one could even conceive of how it could even be done. The idea of an independent proof of all axioms makes the mistake of forgetting that what constitutes a proof is determined by some set of axioms and inference rules. The inference rules in proof systems are, after all, taken to be primitive. If they could be proven, we would not take them to be primitive.

Just a minor remark: primitive notions establish what is an object, in order to distinguishes operations in their properties, but no operations is a primitive notion at all. The fact that our thought is relational is rather factual, and is somehow obtained as an awareness by reflection and maybe something else as a condition, but it is not primitive in the same sense that the notion of 'set' is primitive.

By the way, I think this may be just a minor query, and we agree on the main points.

Thank you for clarifying your thoughts.
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