I didn’t want to hijack the other thread, but I see a lot of very flowery and poetic statements about how Logic = Truth and then comparing Logic to various flora and fauna. These statements have me very confused.
My understanding was that Logic was the study of reasoning processes. To say something is “logical” means that the process of reasoning used to arrive at the conclusion was rational.
A person can use logical reasoning methods to arrive at a false or incorrect conclusion. Logic does not guarantee a true or correct conclusion, it just means that the methodology was sound.
I’ve seen many people on the internet claim, “My conclusion is logical / I used logic, therefore my conclusion is the only correct conclusion.” I believe such statements are false.
If they start in the wrong place (i.e. with incorrect or incomplete starting assumptions, then yes - the best possible logic can lead to the wrong conclusion.
But correct logic based on complete and correct starting premises should yield the correct conclusion.
Certainly the claim “My conclusion is logical / I used logic, therefore my conclusion is the only correct conclusion.” would almost always be false, since there are many conclusions that you can logically deduce from a set of assumptions.
“My conclusion is logical / I used logic, therefore no conclusion that contradicts mine is correct” is at least a step or two closer.
You are absolutely correct. A system of logic is a set of rules by which to deduce conclusions from premises. It says nothing about whether the premises are correct.
There are many different systems of logic: boolean, propositional, intuitionist, etc… With the same premises, a particular conclusion may be true in some systems of logic but not in others. Amateur internet logicians generally act as if they’re not aware of this.
Godel’s famous theorem has shown that all logical systems (excepting trivial ones) are incomplete. That is to say, there are some statements that are true in any system, but can’t be logically proven to be true in that system.
If someone claims that a certain conclusion is logical, they ought to be able to state the premises and then show the reasoning that lead to that conclusion. Most folks who advance conclusions as logical in internet debates have never studied formal logic and wouldn’t even know where to begin.
Logic is a way to convince people who share your assumptions that your conclusions are true. And it works very well for that. But most of the time in debates, I find it’s best used to work backwards, probing people for differences.
“Okay, we disagree on X. Well, according to logic based on premise Y, X is true. Oh, you don’t agree with Y? Well, logically, Y follows from Z. You agree with Z? Well then something is wrong with your logic from Z to Y. Let’s try and find out where.”
Often times you’ll find weaknesses in your own assumptions and conclusions, but at the very least you’ve learned something new about how people think and why people’s conclusions may differ from yours. It’s more enlightening than just assuming everyone who thinks differently than you is an idiot.
I think that’s only true if you use inconsistent semantics. Roughly, if you take true assertions about the world, translate them into your logic appropriately, and then draw a conclusion, this conclusion is again a true assertion about the world, no matter what logic you used. This is essentially Carnap’s principle of tolerance.
For a concrete example, computers constructed using different logics are still only comput the same kinds of things – there was a Russian computer, Setun I think, that used a three-valued logic; nevertheless, you can always simulate this computer on a more traditional one using bivalent logic. Thus, the bivalent logic arrives at all the same, and only the same, conclusions the trivalent one does – though they’ll be phrased differently, and the translation may be nontrivial.
Intuitionist logic can certainly lead to different conclusions than the other major systems of logic. Of course, intuitionist logic hasn’t been taken seriously by the great majority of mathematicians for several generations, so some folks may regard it as irrelevant by now.
Most deductive languages seek to unite two concepts of entailment biconditionally (soundness and completeness): strict syntactic entailment (no metatheoretical concept of truth, a semantic term), and semantic entailment, where true expressions entail only true expressions. Compactness is the key move that allows the two to be united, where possible.
Because they’re not.
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Goedel’s theorem only applies to languages that can express number. Completeness is possible in other languages.
**They could begin by taking one of my classes. **
Well, if we have a sentence S and two logics L and L’, then, if the sentence S is true in L, but not in L’, then obviously it can’t have the same content in both cases; so for at least one of your logics, if both are consistent, you’re doing something wrong re semantics. Or, the other example, if I can build a universal computer using intuitionist logic, I can emulate it on a classical computer, which I can describe completely using propositional logic; similarly, I can emulate the classical computer on the intuisitionistic one. So there’s no real difference there (other than maybe one of convenience).
A logical conclusion drawn from true premises will necessarily be true.
The problem is, in the real world we seldom have premises that are known to be absolutely true, and which are simple enough to be used in formal logic.
Not before it was combined with experimentation. If you read the Greek and Roman natural philosophers, you will see that they used logic extensively and came to wildly wrong conclusions. This was because they were using incorrect premises.
You are right that there are some logics that are complete, but they are not necessarily what we would call “trivial.” Sentential logic and first-order predicate logic (FOPL) come to mind. Indeed, Gödel also proved the completeness of FOPL.
Logic that is powerful enough to prove up arithmetical truths (say by incorporating the Peano postulates and relations for addition, multiplication, and well-ordering), however, must be incomplete (or unsound).
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