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?
These are good thoughts but I haven't studied the modal theory of knowledge enough to intelligently comment on your question.
I can say that I generally reject "justified true belief" and its various Gettier-responsive variants as a theory of knowledge. I think knowledge is either merely true belief (Sartwell) or something closer to the causal or reliabilist definition. But I haven't written it out in any great detail and as far as I can tell the wider philosophical community hasn't really sorted it out either...
Externalism in Epistemology needs a certain "Architecture" of the World "outside us".
This would be the "Architecture" of cause-effect chains being 'really out there', that they are reliable; that they are tracked by the sum total (or even by individual) human perceptions and senses, etc. It often inevitably moves one to "Naturalism" of some shade.
What I find interesting is that you (as a thinker) try to incorporate Theodicies into your worldview, whilst trying to simultaneously be an Externalist.
How exactly does that work out? Naturalism is ultimately at odds with worldviews that try to speak about Good vs Evil as 'out there' in the world.
I came to religion by way of science. I was an atheist and materialist for many years. However, my encounter with the Simulation Hypothesis persuaded me that atheism was irrational; if we could be living in a simulation, the Simulator might be God. I therefore became an agnostic.
Becoming an agnostic opened me up to exploring whether there was evidence for the supernatural, the paranormal, and for the existence of God, which I had heretofore rejected prima facie. When I did explore that, I found abundant evidence of the supernatural, the paranormal, and most of all, God. I found the alternative theories to God to require more elaborate epicycles than any theism. So I became a "natural theist," but not an adherent of any particular religion. I haven't ever had a religious experience or spiritual moment. I just got here by stubbornly thinking too much.
Having become a theist, I was then faced with the problem of evil, a problem which drove me away from faith in my youth. E.g. if our universe was created (which I now believe it was), why isn't it better/perfect? Not having any pre-commitment to any religious answer, I have been exploring the various answers offered by the great faiths. Right now, I believe the answer is that there is some sort of principle of chaos, entropy, or evil that challenges order, syntropy, and good. This seems to work both scientifically and philosophically.
On the other hand, my concern with Munchausen's Trilemma dates back 20+ years, back to law school, well into my atheist materialist phase. My exploration of that topic has been parallel to my investigation of theism and not closely related.
So, whereas Aristotle or Aquinas or Rand could assert that they had a unified theory that coordinated metaphysics, epistemology, and ethics, I can't currently claim to have that; it's an ongoing project. As such I don't have any assurances I've not stumbled into some contradictions; it'll be an ongoing task for me to sort those out.
My current answer is that the "Architecture" of cause and effect is more-or-less the Logos of ancient philosophy, and that human perceptions reliably find the universe to be intelligible because they partake of the Logos. In contrast, a materialist scientist just has to talk about "the irrational effectiveness of mathematics" and "the wonderful happenstance" and so on.
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?
Fabius, these are really interesting thoughts. I agree that fuzzy logic is really useful, but I think it has to be used probabilistically and not ontologically. E.g.
"The truth value of this statement is 0.5". Is that because:
a) We don't know its truth due to uncertainty and limited information? or
b) It's truth value is 0.5.
There's an interesting and complex argument to be had about how the probabilistic nature of quantum mechanics and the measurement problem applies to this. Arguably the quantum world is fuzzy and measurement leads to the excluded middle...
Our legal system is such a mess in part because it's based way too much on multiplying categories where continuous functions are more important. John Campbell wrote an editorial on this back in the Golden Age of science fiction; Jerry Pournelle reprinted it in one of his anthologies. Campbell's example was using a continuous function for determining speeding ticket fines. I vaguely recall that Campbell had a linear function of speed over speed limit. I tend to think something more nonlinear would be more appropriate. Damage is a function of energy, which is proportional to velocity squared. Throw in another power of velocity to factor in greater probability of an accident due to less time to react. A third power function would make a few over the speed limit not worth pulling people over as it should be. But replacing the step function with that third power function reduces the charm factor: sweet talking a cop from 16 over to 14 over doesn't have the same impact...
Good essay on Holistic Politics (is that your blog?)
That said, I don't think the world is continuous. I think it's ultimately discrete (quantized). That seems to be what quantum mechanics reveals. A discrete world solves Zeno's Paradox, it solves the "infinite coastline" paradox, and more.
Jul 21, 2022·edited Jul 21, 2022Liked by Tree of Woe
Shhhh, that's my old Good Cop mode, back from the days when there was such a thing as an open minded progressive. When Silicon Valley muted my channels back in 2020 and the Left started plagiarizing the Nazi playbook, I've gone on the offensive in more ways than one, here at Substack, and eventually a book -- even if I have to hawk it at gun shows and whatnot.
As for quantum mechanics...maybe. Can't say I fully understand quantum mechanics, despite a bunch of courses at the graduate level. Learning to treat electrons as infinite masses in order to make Feynmann's scattering methods work was quite a WTF moment. Then again, even in QM 1, they treat electrons as point sources; a point source with a finite charge would have infinite mass using the classical formula for energy for an electric field. (Or so I thought back in the day. I need to redo the calculation to see if vanishing volume offsets field strength...)
What I can say, despite layers of rust on my degree, is that those discrete energy levels presume infinite time. Those discrete energy states blur up when electrons only spend a finite time in said states. Also, one generally uses the Schrodinger equation in continuous space to find those discrete states.
The nice thing about quantum mechanics is that it creates wiggle room for free will. Or, at least I hope it does...
I don't think fuzzy logic flouts the law of the excluded middle. The law of the excluded middle is that a thing cannot be both true or false with respect to the same thing, at the same time, in the same manner, in the same mode, etc. The introduction of "maybe" or "if", of modal logic and probability, does not refute this in any way, as that deals with cases in which the manners and modes, quantities and qualities, are variable or unknown.
Godel's Incompleteness Theorem doesn't show that binary logic disproves itself. It shows there are some true statements that it cannot prove. But my irrefutable axioms don't depend on that at all. They depend on the fact that you can't disprove them without using them.
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?
These are good thoughts but I haven't studied the modal theory of knowledge enough to intelligently comment on your question.
I can say that I generally reject "justified true belief" and its various Gettier-responsive variants as a theory of knowledge. I think knowledge is either merely true belief (Sartwell) or something closer to the causal or reliabilist definition. But I haven't written it out in any great detail and as far as I can tell the wider philosophical community hasn't really sorted it out either...
Fair enough!
Externalism in Epistemology needs a certain "Architecture" of the World "outside us".
This would be the "Architecture" of cause-effect chains being 'really out there', that they are reliable; that they are tracked by the sum total (or even by individual) human perceptions and senses, etc. It often inevitably moves one to "Naturalism" of some shade.
What I find interesting is that you (as a thinker) try to incorporate Theodicies into your worldview, whilst trying to simultaneously be an Externalist.
How exactly does that work out? Naturalism is ultimately at odds with worldviews that try to speak about Good vs Evil as 'out there' in the world.
Great question!
I came to religion by way of science. I was an atheist and materialist for many years. However, my encounter with the Simulation Hypothesis persuaded me that atheism was irrational; if we could be living in a simulation, the Simulator might be God. I therefore became an agnostic.
Becoming an agnostic opened me up to exploring whether there was evidence for the supernatural, the paranormal, and for the existence of God, which I had heretofore rejected prima facie. When I did explore that, I found abundant evidence of the supernatural, the paranormal, and most of all, God. I found the alternative theories to God to require more elaborate epicycles than any theism. So I became a "natural theist," but not an adherent of any particular religion. I haven't ever had a religious experience or spiritual moment. I just got here by stubbornly thinking too much.
Having become a theist, I was then faced with the problem of evil, a problem which drove me away from faith in my youth. E.g. if our universe was created (which I now believe it was), why isn't it better/perfect? Not having any pre-commitment to any religious answer, I have been exploring the various answers offered by the great faiths. Right now, I believe the answer is that there is some sort of principle of chaos, entropy, or evil that challenges order, syntropy, and good. This seems to work both scientifically and philosophically.
On the other hand, my concern with Munchausen's Trilemma dates back 20+ years, back to law school, well into my atheist materialist phase. My exploration of that topic has been parallel to my investigation of theism and not closely related.
So, whereas Aristotle or Aquinas or Rand could assert that they had a unified theory that coordinated metaphysics, epistemology, and ethics, I can't currently claim to have that; it's an ongoing project. As such I don't have any assurances I've not stumbled into some contradictions; it'll be an ongoing task for me to sort those out.
My current answer is that the "Architecture" of cause and effect is more-or-less the Logos of ancient philosophy, and that human perceptions reliably find the universe to be intelligible because they partake of the Logos. In contrast, a materialist scientist just has to talk about "the irrational effectiveness of mathematics" and "the wonderful happenstance" and so on.
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.
Fabius, these are really interesting thoughts. I agree that fuzzy logic is really useful, but I think it has to be used probabilistically and not ontologically. E.g.
"The truth value of this statement is 0.5". Is that because:
a) We don't know its truth due to uncertainty and limited information? or
b) It's truth value is 0.5.
There's an interesting and complex argument to be had about how the probabilistic nature of quantum mechanics and the measurement problem applies to this. Arguably the quantum world is fuzzy and measurement leads to the excluded middle...
Gnope. The ontological aspect is double plus important. Try this quick demo (It's mostly pictures):
https://www.holisticpolitics.org/Abortion/FuzzyLogic.php
Our legal system is such a mess in part because it's based way too much on multiplying categories where continuous functions are more important. John Campbell wrote an editorial on this back in the Golden Age of science fiction; Jerry Pournelle reprinted it in one of his anthologies. Campbell's example was using a continuous function for determining speeding ticket fines. I vaguely recall that Campbell had a linear function of speed over speed limit. I tend to think something more nonlinear would be more appropriate. Damage is a function of energy, which is proportional to velocity squared. Throw in another power of velocity to factor in greater probability of an accident due to less time to react. A third power function would make a few over the speed limit not worth pulling people over as it should be. But replacing the step function with that third power function reduces the charm factor: sweet talking a cop from 16 over to 14 over doesn't have the same impact...
Good essay on Holistic Politics (is that your blog?)
That said, I don't think the world is continuous. I think it's ultimately discrete (quantized). That seems to be what quantum mechanics reveals. A discrete world solves Zeno's Paradox, it solves the "infinite coastline" paradox, and more.
Shhhh, that's my old Good Cop mode, back from the days when there was such a thing as an open minded progressive. When Silicon Valley muted my channels back in 2020 and the Left started plagiarizing the Nazi playbook, I've gone on the offensive in more ways than one, here at Substack, and eventually a book -- even if I have to hawk it at gun shows and whatnot.
As for quantum mechanics...maybe. Can't say I fully understand quantum mechanics, despite a bunch of courses at the graduate level. Learning to treat electrons as infinite masses in order to make Feynmann's scattering methods work was quite a WTF moment. Then again, even in QM 1, they treat electrons as point sources; a point source with a finite charge would have infinite mass using the classical formula for energy for an electric field. (Or so I thought back in the day. I need to redo the calculation to see if vanishing volume offsets field strength...)
What I can say, despite layers of rust on my degree, is that those discrete energy levels presume infinite time. Those discrete energy states blur up when electrons only spend a finite time in said states. Also, one generally uses the Schrodinger equation in continuous space to find those discrete states.
The nice thing about quantum mechanics is that it creates wiggle room for free will. Or, at least I hope it does...
No one fully understands quantum mechanics. That's why at West Point they called the course "Magic"!
I don't think fuzzy logic flouts the law of the excluded middle. The law of the excluded middle is that a thing cannot be both true or false with respect to the same thing, at the same time, in the same manner, in the same mode, etc. The introduction of "maybe" or "if", of modal logic and probability, does not refute this in any way, as that deals with cases in which the manners and modes, quantities and qualities, are variable or unknown.
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.
Godel's Incompleteness Theorem doesn't show that binary logic disproves itself. It shows there are some true statements that it cannot prove. But my irrefutable axioms don't depend on that at all. They depend on the fact that you can't disprove them without using them.