NOTE: Section 2.5 is the important bit, dealing with the relevant Epistemic Principles and their frame conditions.
Now then...
Ka[X] and Ka[V] are such that
Ka[X] -> X Meaning IF Agent a Knows that X, THEN X is the case, X is True
Ka[V] -> V Meaning IF Agent a Knows that V, THEN V is the case, V is True
This is because to Know [], we mean that:
1. [] is Justified.
2. [] is True* (Relevant Bit)
3. [] is Believed to be so.
4. [] satisfies some condition X, X being the defeater to Gettier-style epistemic-defeaters.
Given all this... I don't see how you can make much headway unless the relation of S5 holds for both your Ka[X] and Ka[V]. Namely:
[5] ~Ka[X] -> Ka[~Ka[X]] AND ~Ka[V] -> Ka[~Ka[V]]
Which reads "If Agent 'a' doesn't know that X, then Agent 'a' knows that he doesn't know that X" and likewise for V.
This doesn't work out when we sub for the values of X and V.
What about S4? Let's see:
[4] Ka[X] -> Ka[Ka[X]] AND Ka[V] -> Ka[Ka[V]]
Which reads "If Agent 'a' knows that X, then Agent 'a' knows that he knows that X" and likewise for V.
This does work. Sub in values for X and V, and it does so.
So your system is at least at S4 (or maybe better) but less than S5.
Would you consider that sufficient to "defeating" the Trilemma?
If so, what say you to the objection that the Trilemma must be broken in all possible worlds (as per possible worlds semantics a la Kripke, Lewis et al) for it to be truly "beaten", ergo you need S5 strength?
Aristotle deals with this Trilemma (thousands of years before it was called that) in Posterior Analytics.
Here's the relevant excerpt:
Some hold that, owing to the necessity of knowing the primary premisses, there is no scientific knowledge. Others think there is, but that all truths are demonstrable. Neither doctrine is either true or a necessary deduction from the premisses. The first school, assuming that there is no way of knowing other than by demonstration, maintain that an infinite regress is involved, on the ground that if behind the prior stands no primary, we could not know the posterior through the prior (wherein they are right, for one cannot traverse an infinite series): if on the other hand-they say-the series terminates and there are primary premisses, yet these are unknowable because incapable of demonstration, which according to them is the only form of knowledge. And since thus one cannot know the primary premisses, knowledge of the conclusions which follow from them is not pure scientific knowledge nor properly knowing at all, but rests on the mere supposition that the premisses are true. The other party agree with them as regards knowing, holding that it is only possible by demonstration, but they see no difficulty in holding that all truths are demonstrated, on the ground that demonstration may be circular and reciprocal.
Our own doctrine is that not all knowledge is demonstrative: on the contrary, knowledge of the immediate premisses is independent of demonstration. (The necessity of this is obvious; for since we must know the prior premisses from which the demonstration is drawn, and since the regress must end in immediate truths, those truths must be indemonstrable.) Such, then, is our doctrine, and in addition we maintain that besides scientific knowledge there is its originative source which enables us to recognize the definitions.
I think the key insight here is contained in the last sentence, the "originative source which enables us recognise the definitions." This reminds us of Euclid's book of geometry, which is a series of proofs and deductions, but which rests on axiomatic definitions of things like a "line" or a "point" which Euclid assumes the human mind has enough intuitive power to grasp by itself. The recognition of this intuitive mind, which the Greeks called the νους, the nous, is what is missing from all of modern philosophy imo. Descartes set philosophy on a course of rationalism, of ignoring the noetic mind in favour of the ratiocinative discursive mind, and Kant completed that in his Critique of Pure Reason. Without the intutition of the nous, the noetic mind, and its immediate apprehension of being and truth and other transcendentals, the human mind becomes a logic box trapped in its own circular definitions, cut off from the world and from wisdom and first principles.
Hmmmmm, the recent explosion in artificial intelligence started with engineers who dropped the Law of the Excluded Middle. Fuzzy Logic is arguably the biggest breakthrough in philosophy since Aristotle. And I believe that Fuzzy Logic is the best defense against Post Modernism, an opinion based on spending many hours debating Post Modernists four decades ago as an undergraduate.
Real world things only approximate their Platonic ideals. And thus if one stress tests any logical rule hard enough one can find edge case exceptions. Marxists have been using this as a rhetorical device since Marx: focus all effort deconstructing capitalism and then declare victory when your opponent runs out of steam. Waffle or refer to a Tome of Extraordinary Unreadability should the defender of capitalism try to turn the tables and deconstruct Marxist arguments.
The Left is currently playing this game with abortion. They are focusing on edge cases such as ectopic pregnancies and 10 year old rape victims. Never mind that Democratic governors were pushing for borderline infanticide quite recently. By ruling out the Excluded Middle the pro abortion forces manage to lump edge cases with their hideous agenda.
By treating Humanness as a fuzzy quality, we can quash 90+% of abortions in short order. And maybe do better longer term. And I submit that Christians who claim that abortion a week after conception is equivalent to murder are practicing self deception. Revealed Preference says so.
Disbelieve? Try this thought experiment. Suppose that the Supreme Court ruled that the Servants of Moloch had a Constitutional right to sacrifice two year olds to their god. Would today's Christians limit their protests to political campaigning? Or would they form enraged mobs and burn the priests of Moloch and lynch the judges who granted them the right to human sacrifice even if it leads to civil war?
Good Article!
Let's Denote the proposition X as "Existence Exists".
Let's Denote the proposition V as "Evidence from sense perceptions are not wholly unreliable".
Let Ka[] be denoted as "Agent 'a' knows that []".
Let ~Ka[] be denoted as "Agent 'a' doesn't know that []"
Let ⟨Ka[]⟩ be equal to ~Ka~[], namely denoting "Agent 'a' doesn't know that not-[]"
Relevant: https://plato.stanford.edu/entries/logic-epistemic/
NOTE: Section 2.5 is the important bit, dealing with the relevant Epistemic Principles and their frame conditions.
Now then...
Ka[X] and Ka[V] are such that
Ka[X] -> X Meaning IF Agent a Knows that X, THEN X is the case, X is True
Ka[V] -> V Meaning IF Agent a Knows that V, THEN V is the case, V is True
This is because to Know [], we mean that:
1. [] is Justified.
2. [] is True* (Relevant Bit)
3. [] is Believed to be so.
4. [] satisfies some condition X, X being the defeater to Gettier-style epistemic-defeaters.
Given all this... I don't see how you can make much headway unless the relation of S5 holds for both your Ka[X] and Ka[V]. Namely:
[5] ~Ka[X] -> Ka[~Ka[X]] AND ~Ka[V] -> Ka[~Ka[V]]
Which reads "If Agent 'a' doesn't know that X, then Agent 'a' knows that he doesn't know that X" and likewise for V.
This doesn't work out when we sub for the values of X and V.
What about S4? Let's see:
[4] Ka[X] -> Ka[Ka[X]] AND Ka[V] -> Ka[Ka[V]]
Which reads "If Agent 'a' knows that X, then Agent 'a' knows that he knows that X" and likewise for V.
This does work. Sub in values for X and V, and it does so.
So your system is at least at S4 (or maybe better) but less than S5.
Would you consider that sufficient to "defeating" the Trilemma?
If so, what say you to the objection that the Trilemma must be broken in all possible worlds (as per possible worlds semantics a la Kripke, Lewis et al) for it to be truly "beaten", ergo you need S5 strength?
Aristotle deals with this Trilemma (thousands of years before it was called that) in Posterior Analytics.
Here's the relevant excerpt:
Some hold that, owing to the necessity of knowing the primary premisses, there is no scientific knowledge. Others think there is, but that all truths are demonstrable. Neither doctrine is either true or a necessary deduction from the premisses. The first school, assuming that there is no way of knowing other than by demonstration, maintain that an infinite regress is involved, on the ground that if behind the prior stands no primary, we could not know the posterior through the prior (wherein they are right, for one cannot traverse an infinite series): if on the other hand-they say-the series terminates and there are primary premisses, yet these are unknowable because incapable of demonstration, which according to them is the only form of knowledge. And since thus one cannot know the primary premisses, knowledge of the conclusions which follow from them is not pure scientific knowledge nor properly knowing at all, but rests on the mere supposition that the premisses are true. The other party agree with them as regards knowing, holding that it is only possible by demonstration, but they see no difficulty in holding that all truths are demonstrated, on the ground that demonstration may be circular and reciprocal.
Our own doctrine is that not all knowledge is demonstrative: on the contrary, knowledge of the immediate premisses is independent of demonstration. (The necessity of this is obvious; for since we must know the prior premisses from which the demonstration is drawn, and since the regress must end in immediate truths, those truths must be indemonstrable.) Such, then, is our doctrine, and in addition we maintain that besides scientific knowledge there is its originative source which enables us to recognize the definitions.
http://classics.mit.edu/Aristotle/posterior.1.i.html
I think the key insight here is contained in the last sentence, the "originative source which enables us recognise the definitions." This reminds us of Euclid's book of geometry, which is a series of proofs and deductions, but which rests on axiomatic definitions of things like a "line" or a "point" which Euclid assumes the human mind has enough intuitive power to grasp by itself. The recognition of this intuitive mind, which the Greeks called the νους, the nous, is what is missing from all of modern philosophy imo. Descartes set philosophy on a course of rationalism, of ignoring the noetic mind in favour of the ratiocinative discursive mind, and Kant completed that in his Critique of Pure Reason. Without the intutition of the nous, the noetic mind, and its immediate apprehension of being and truth and other transcendentals, the human mind becomes a logic box trapped in its own circular definitions, cut off from the world and from wisdom and first principles.
Hmmmmm, the recent explosion in artificial intelligence started with engineers who dropped the Law of the Excluded Middle. Fuzzy Logic is arguably the biggest breakthrough in philosophy since Aristotle. And I believe that Fuzzy Logic is the best defense against Post Modernism, an opinion based on spending many hours debating Post Modernists four decades ago as an undergraduate.
Real world things only approximate their Platonic ideals. And thus if one stress tests any logical rule hard enough one can find edge case exceptions. Marxists have been using this as a rhetorical device since Marx: focus all effort deconstructing capitalism and then declare victory when your opponent runs out of steam. Waffle or refer to a Tome of Extraordinary Unreadability should the defender of capitalism try to turn the tables and deconstruct Marxist arguments.
The Left is currently playing this game with abortion. They are focusing on edge cases such as ectopic pregnancies and 10 year old rape victims. Never mind that Democratic governors were pushing for borderline infanticide quite recently. By ruling out the Excluded Middle the pro abortion forces manage to lump edge cases with their hideous agenda.
By treating Humanness as a fuzzy quality, we can quash 90+% of abortions in short order. And maybe do better longer term. And I submit that Christians who claim that abortion a week after conception is equivalent to murder are practicing self deception. Revealed Preference says so.
Disbelieve? Try this thought experiment. Suppose that the Supreme Court ruled that the Servants of Moloch had a Constitutional right to sacrifice two year olds to their god. Would today's Christians limit their protests to political campaigning? Or would they form enraged mobs and burn the priests of Moloch and lynch the judges who granted them the right to human sacrifice even if it leads to civil war?
I would hope the latter.
And as for irrefutability, consider Godel's Incompleteness Theorem. Binary logic disproves itself.
Fuzzy logic deals with strange loops by having a truthfulness of 0.5 for statements which lead to strange loops.