what I think is “of course there are strong and weak beliefs!” but true and false is only defined relative to who is asking and why (in some cases), so you need to consider the context in which you’re applying LoEM.
Like in my comment to Richard_Kennaway about probability, I am not just talking about beliefs, but about what is. Do we take it as an axiom or a theorem that A or ~A? Likewise for ~(A and ~A)? I admit to being confused about this. Also, does “A” mean the same thing as “A = True”? Does “~A” mean the same thing as “A = False”? If so, in what sense do we say that A literally equals True / False, respectively? Which things are axioms and which things are theorems, here? All of that confuses me.
Since we are often permitted to change our axioms and arrive at systems we either like or don’t like, or like better than others, I think it’s relevant to ask about our choice of axioms and whether or not logic is or should be considered a set of “pre-axioms.”
It seemed like tailcalled was implying that the law of non-contradiction was a theorem, and I’m confused about that as well. Under which axioms?
If I decide that ~(A and ~A) is not an axiom, then I can potentially have A and ~A either be true or not false. Then we would need some other arguments to support that choice. Without absolute truth and absolute falsehood, we’d have to move back to the concept of “we like [it] better or worse” which would make the latter more fundamental. Does allowing A and ~A to mean something get us any utility?
In order for it to get us any utility, there would have to be things that we’d agree were validly described by A and ~A.
Admittedly, it does seem like these or’s and and’s and =’s keep appearing regardless of my choices, here (because I need them for the concept of choice).
In a quasi-philosophical and quasi-logical post I have not posted to LessWrong yet, I argue that negation seems likely to be the most fundamental thing to me (besides the concept of “exists / is”, which is what “true” means). “False” is thus not quite the same thing as negation, and instead means something more like “nonsense gibberish” which is actually far stronger than negation.
It’s really hard to answer these sorts of questions universally because there’s a bunch of ways of setting up things that are strictly speaking different but which yields the same results overall. For instance, some take ¬P to be a primitive notion, whereas I am more used to defining ¬P to mean P⟹⊥. However, pretty much always the inference rules or axioms for taking ¬P to be a primitive are set up in such a way that it is equivalent to P⟹⊥.
If you define it that way, the law of noncontradiction becomes (P∧(P⟹⊥))⟹⊥ is pretty trivial, because it is just a special case of (P∧(P⟹Q))⟹Q, and if you don’t have the rule (P∧(P⟹Q))⟹Q then it seems like your logic must be extremely limited (since it’s like an internalized version of modus ponens, a fundamental rule of reasoning).
I have a bunch of experience dealing with logic that rejects the law of excluded middle, but while there are a bunch of people who also experiment with rejecting the law of noncontradiction, I haven’t seen anything useful come of it, I think because it is quite fundamental to reasoning.
Also, does “A” mean the same thing as “A = True”? Does “~A” mean the same thing as “A = False”?
Statements like A=⊥ are kind of mixing up the semantic (or “meta”) level with the syntactic (or “object”) level.
Like in my comment to Richard_Kennaway about probability, I am not just talking about beliefs, but about what is. Do we take it as an axiom or a theorem that A or ~A?
Are you aware that there are different logics with different axioms? PMC amd LEM are both axioms in Aristotelean logic: LEM does not apply in fuzzy logic; and PNC does not apply in paraconsistent logic.
False” is thus not quite the same thing as negation,
No: negation is a function, false is a value. That’s clear in most programming languages.
False” is thus not quite the same thing as negation, and instead means something more like “nonsense gibberish”
Like in my comment to Richard_Kennaway about probability, I am not just talking about beliefs, but about what is. Do we take it as an axiom or a theorem that A or ~A? Likewise for ~(A and ~A)? I admit to being confused about this. Also, does “A” mean the same thing as “A = True”? Does “~A” mean the same thing as “A = False”? If so, in what sense do we say that A literally equals True / False, respectively? Which things are axioms and which things are theorems, here? All of that confuses me.
Since we are often permitted to change our axioms and arrive at systems we either like or don’t like, or like better than others, I think it’s relevant to ask about our choice of axioms and whether or not logic is or should be considered a set of “pre-axioms.”
It seemed like tailcalled was implying that the law of non-contradiction was a theorem, and I’m confused about that as well. Under which axioms?
If I decide that ~(A and ~A) is not an axiom, then I can potentially have A and ~A either be true or not false. Then we would need some other arguments to support that choice. Without absolute truth and absolute falsehood, we’d have to move back to the concept of “we like [it] better or worse” which would make the latter more fundamental. Does allowing A and ~A to mean something get us any utility?
In order for it to get us any utility, there would have to be things that we’d agree were validly described by A and ~A.
Admittedly, it does seem like these or’s and and’s and =’s keep appearing regardless of my choices, here (because I need them for the concept of choice).
In a quasi-philosophical and quasi-logical post I have not posted to LessWrong yet, I argue that negation seems likely to be the most fundamental thing to me (besides the concept of “exists / is”, which is what “true” means). “False” is thus not quite the same thing as negation, and instead means something more like “nonsense gibberish” which is actually far stronger than negation.
It’s really hard to answer these sorts of questions universally because there’s a bunch of ways of setting up things that are strictly speaking different but which yields the same results overall. For instance, some take ¬P to be a primitive notion, whereas I am more used to defining ¬P to mean P⟹⊥. However, pretty much always the inference rules or axioms for taking ¬P to be a primitive are set up in such a way that it is equivalent to P⟹⊥.
If you define it that way, the law of noncontradiction becomes (P∧(P⟹⊥))⟹⊥ is pretty trivial, because it is just a special case of (P∧(P⟹Q))⟹Q, and if you don’t have the rule (P∧(P⟹Q))⟹Q then it seems like your logic must be extremely limited (since it’s like an internalized version of modus ponens, a fundamental rule of reasoning).
I have a bunch of experience dealing with logic that rejects the law of excluded middle, but while there are a bunch of people who also experiment with rejecting the law of noncontradiction, I haven’t seen anything useful come of it, I think because it is quite fundamental to reasoning.
Statements like A=⊥ are kind of mixing up the semantic (or “meta”) level with the syntactic (or “object”) level.
G2g
Are you aware that there are different logics with different axioms? PMC amd LEM are both axioms in Aristotelean logic: LEM does not apply in fuzzy logic; and PNC does not apply in paraconsistent logic.
No: negation is a function, false is a value. That’s clear in most programming languages.
That doesn’t follow at all.