


Modal Logic
◊P ↔ ¬□¬P // Possibly(P) ↔ ¬Necessarily(¬P) □P ↔ ¬◊¬P // Necessarily(P) ↔ ¬Possibly(¬P) ∀P (P ∈ Analytical_Knowledge(P) ↔ □P) A universal truth predicate that refutes the Tarski Undefinability Theorem can be easily derived simply as membership in the set of Analytical_Knowledge. ∀P (□P ≡ Sound_Deductive_Proof(P) ≡ Theoerem(P)) Assumes Haskell Curry definition of axiom (see linked paper). Copyright 2020 Pete Olcott Remove sci.lang after this one post Remove sci.lang after this one post Remove sci.lang after this one post 


On 20200307 10:54, olcott wrote:
> Modal Logic > ◊P ↔ ¬□¬P // Possibly(P) ↔ ¬Necessarily(¬P) > □P ↔ ¬◊¬P // Necessarily(P) ↔ ¬Possibly(¬P) > ∀P (P ∈ Analytical_Knowledge(P) ↔ □P) > A universal truth predicate that refutes the Tarski Undefinability > Theorem can be easily derived simply as membership in the set of > Analytical_Knowledge. > ∀P (□P ≡ Sound_Deductive_Proof(P) ≡ Theoerem(P)) > Assumes Haskell Curry definition of axiom (see linked paper). Do you actually know any modal logic? If not, you probably shouldn't be using it. You understand that □P does not necessarily imply □□P, right? So if □P is true but □□P is false, what consequences does that have on the theoremhood of P? André 


On 3/7/2020 12:12 PM, André G. Isaak wrote:
> On 20200307 10:54, olcott wrote: > Do you actually know any modal logic? If not, you probably shouldn't be > using it. > You understand that □P does not necessarily imply □□P, right? So if □P > is true but □□P is false, what consequences does that have on the > theoremhood of P? > André Can you provide a concrete example of that? P ≡ "Cats are not dogs" (□P ≡ Sound_Deductive_Proof(P) Theoerem(P)) So are you saying that the necessary statement: "Cats are not dogs" is not necessarily necessary? Please give me an example where "Cats are not dogs" is not necessarily necessary and do not commit any fallacy of equivocation error of changing the set of semantic properties assigned to the terms cat and dog. Whensoever two elements of the universal set have different sets of properties we can know with complete certainty that they are not the same element. 


On 20200307 11:53, olcott wrote:
> On 3/7/2020 12:12 PM, André G. Isaak wrote: > Can you provide a concrete example of that? > P ≡ "Cats are not dogs" > (□P ≡ Sound_Deductive_Proof(P) Theoerem(P)) > So are you saying that the necessary statement: "Cats are not dogs" "necessary" has no modelindependent meaning. What makes this a necessary statement as opposed to a contingent one? > is not necessarily necessary? > Please give me an example where "Cats are not dogs" is not necessarily > necessary and do not commit any fallacy of equivocation error of > changing the set of semantic properties assigned to the terms cat and dog. I asked if you had ever actually studied modal logic. The answer, apparently, is no. □ and ◊ are simply quantifiers. They have no interpretation independent of a model, and you don't provide a model. Whether "Cats are not dogs" is true, necessarily true, necessarily necessarily true, etc. depends entirely on the model you are using. André 


On 3/7/2020 11:26 PM, André G. Isaak wrote:
> On 20200307 11:53, olcott wrote: > "necessary" has no modelindependent meaning. So you don't necessarily have to keep breathing to remain alive? > What makes this a > necessary statement as opposed to a contingent one? Semantic logical entailment. > I asked if you had ever actually studied modal logic. The answer, > apparently, is no. The only thing that I know ab out modal logic is this >>>> ◊P ↔ ¬□¬P // Possibly(P) ↔ ¬Necessarily(¬P) >>>> □P ↔ ¬◊¬P // Necessarily(P) ↔ ¬Possibly(¬P) The above precisely define the English words in terms of each other. This utterly eliminates the error of equating impossible with "I really don't brelieve it" The only expressions of language that are impossibly true are direct contradictions. > □ and ◊ are simply quantifiers. They have no interpretation independent > of a model, and you don't provide a model. Whether "Cats are not dogs" > is true, necessarily true, necessarily necessarily true, etc. depends > entirely on the model you are using. > André So in other words you believe that cats may actually be dogs some of the time? 


On 20200307 23:23, olcott wrote:
> On 3/7/2020 11:26 PM, André G. Isaak wrote: > So you don't necessarily have to keep breathing to remain alive? > Semantic logical entailment. > The only thing that I know ab out modal logic is this Then you really know virtually nothing whatsoever about modal logic. You should therefore stop using it until you actually know what it is. Though I doubt this will benefit you, here is an incredibly brief explanation: A model in modal logic consists of: (A) A set of possible worlds, each of which can be thought of as a model in the nonmodal sense. Exactly what is meant by 'possible' is entirely dependent on what the model is being used for. (B) An accessibility relation which defines which worlds are 'visible' to one another. □P means that P is true in the world under consideration and in all worlds visible to that world. ◊P means that P is true in at least one world visible to the world under consideration. □□P means that P is true in the world under consideration, in all worlds visible to that world, and in all the worlds visible to those worlds including ones which might not be visible to the world under consideration. And so forth. Unless you explain which possible worlds are under consideration and the accessibility between those worlds, □ (universal modal quantifier) and ◊ (existential modal quantifier) don't mean anything. > The above precisely define the English words in terms of each other. > This utterly eliminates the error of > equating impossible with "I really don't brelieve it" > The only expressions of language that are impossibly true are direct > contradictions. Wait a minute  so 'impossibly false' is a type of false, but 'impossibly true' is also a type of false? You really need better terminology. > So in other words you believe that cats may actually be dogs some of the > time? Again, that depends on the model under consideration. What worlds are being considered? alternate evolutionary histories? things which can be imagined? worlds corresponding to our desires? André 