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? — Nicholas Ferreira
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? — Nicholas Ferreira
the truth table is the conjunction of the premises implying in the conclusion — Nicholas Ferreira
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. — MindForged
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. — MindForged
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
They are simply refined conjectures — hks
That is why Science is NOT Philosophy. — hks
'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. — Nicholas Ferreira
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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. — Mentalusion
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. — Ikolos
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. — Ikolos
"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. — MindForged
But not because it is incoherent, but because is IMPOSSIBLE. — Ikolos
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 — MindForged
"Coherency" here is exactly another way of saying "non-contradictory". — MindForged
What counts as a proof requires one to adopt some set of rules by which to establish what will count as a proof. — MindForged
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, — MindForged
one just ends up presuming some set of inference rules and axioms — MindForged
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. — MindForged
You have to assume something is just off the table to get going. I'm not saying this is a problem, it' — MindForged
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. — MindForged
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