9:20 nitpick on the proof of infinite primes: The contradiction is not that the product of primes plus 1 is both prime and composite. The contradiction is that it has a prime factor different than all the primes.
I assume there are multiple proofs of this, because from what I've read there is definitely a proof that derives the contradiction that the number N, such that N is the product of all primes smaller than or equal to p, plus 1, is both prime and composite. (Apparently this kind of proof is erroneously attributed to Euclid. But I assume the proof works, even if it came from somebody other than Euclid.)
@@KaneB This sounds very dubious to me, could you find a source? I don't see how one could possibly conclude N+1 is *either prime or composite*, let alone *both prime and composite*.
@@blackeyefly I just looked this up again and you're right; I misread it initially. I conflated two different ways of presenting the proof. The way that it's usually presented, N = the product of all primes less than or equal to p, plus 1. The way Williamson presents it, N = the product of all positive integers less than or equal to p, plus 1, i.e. p factorial plus 1.
Weird thought, but it's interesting to consider that there could be isolate possible worlds such that there's not any close possible worlds and so counterfactuals stated in that isolated world would also be vacuous.
I think saying counterpossibles are trivially true is putting the cart before the horse. To decide whether something is possible in the first place it seems like you have to start by imagining it were true and then showing that you get a contradiction or something.
IN DEFENSE OF REASON - COUNTERPOSSIBLES This video really gets the head spinning - well done! Indeed this kind of problem is one that can leave philosophy students believing that reason itself is impossible - a trend I have been seeing. The clencher is the problem at the end - where the rules of inference one can make about impossible worlds can be countered by creating an impossible world where that rule of inference does not exist. The example given in the video was "if conjunction elimination did not exist, then Aristotle and Descartes would have been sad". It's hard to know what to name this problem, but I will call it "thwarting counterpossible inference and logic" - TCIAL. As a Lakatosian, I would like to come up with a method to incorporate these anomalies into the research programme of counterpossible logic somehow. Of course, an opponent would just create a world where anomaly incorporation did not exist. Which at the moment leaves me at a loss, but I will try to get back to this later, maybe. The other option for a Lakatosian would be to exclude the anomalies by definition. This would be something like "there can be no worlds that defy the laws of inference" and declare that statement to be itself a law of inference. While this would clearly degenerate the counterpossible research programme, it would at least save it from direct refutation. However, this would be a bad idea. We might want to evaluate the consequences of a world where conjunction elimination does not exist So okay. Let's go ahead and evaluate those consequences. In a world where conjunction elimination did not exist, you could not derive "Aristotle was sad" from "Aristotle and Descartes were sad". While this seems strange to us, to the Aristotle and Descartes of that world, it would be commonplace. As such, we can simply say that the laws of inference are not universal across worlds. They only apply to the worlds in which they hold. So, we can say this also holds for counter-possible rules of inference as well. As such, there is a world where no rules of inference can apply, but that does not stop us from using them elsewhere. Whether this method is incorporation or exclusion, I do not know. Perhaps a little of each. Perhaps the research programme of counterpossibles is both progressive and degenerate.
I don't think that a statement "conjunction elimination holds" is even something truth-apt, in this world or in any other. So this worry of yours looks like a total non-issue to me. Look into Browerian intuitionism, and especially Heyting's version of it.
@@СергейМакеев-ж2н Browerian intuitionism uses conjunction elimination, so I'm not sure what your point is. I mean if we have P&Q, is it not truth apt to conclude A in Browerian intuitionism?
@@InventiveHarvest I don't mean the intuitionistic logic system (though I am a fan of that too), but Brower's and Heyting's philosophical view on the nature of logic itself. Which too is called intuitionism. (It's confusing, I know.) They believe that there is no "one true logic" out there in the world, that logic is just a tool to solve a problem, and you are free to choose whichever tool you like. There is no property of the world itself that determines which tool is the "right" one. And Heyting in particular said that you can even choose different logics for different problems, or invent a new logic for every new problem. There is no need for a "one size fits all" logic for all areas of life.
@@СергейМакеев-ж2н@user-qm4ev6jb7d @user-qm4ev6jb7d okay, so when I talk about worlds, possible and impossible, I am not advocating for moral realism. Worlds are basically just places for hypotheticals. All of the worlds that evaluate the hypotheticals according to classic logic are possible worlds, and all worlds that do not are impossible worlds. A world where conjunction elimination does not exist is just a world that corresponds to a system of logic without the conjunction elimination rule. The problem is that we are trying to make rules of inference to deal with such impossible worlds. If you say that you want to use conjunction elimination in order to say when a counterpossible is vacuous or not, then you will have a problem with the world where that rule does not exist. My solution to this is to say that inference rules to evaluate counterpossibles are not universal. This is basically an intuitionist approach to counterpossible inference.
From the 7+5 example, it seems that counter factual are often not about considering actual counter factual worlds, but ways of pointing to and describing very specific causal relationships (in a sense, how two pieces of knowledge seem to “move with each other). When stating the counter factual, the way these facts are related to each the is structurally preserved, almost formally, even though the world is not in fact possible. This also applies to all the scientific counterpossibles. When you assume, for example, the sun is made up of an ideal gas, it entails very specific simplifications of your calculations, and that’s the sort of “important part.” It’s like causality and relational it’s is a web, and when we make cojnterpossible statements, we are pointing to an independent (or can practically be treated as independent part of the web, even if it’s in fact contradictory with other parts of the web.
Commenting more for algorithm engagement than serious criticism. At 8:43, there’s a minor error: p! is the product of all positive natural numbers less than or equal to p rather than the the product of all primes less than or equal to p
It seems as though we need to distinguish between modal and epistemological statements. Because I haven't replicated enough of the relevant experiments myself, I don't personally know that water is H2O. My knowledge of its molecular structure is entirely dependent on some level of trust in social processes to do part of the work of developing knowledge. For all I know personally, water could be have an different structure, or it could be something like an element in the sense that people imagined when they first talked about air, fire, earth, and water as elements. But with the resources of my cultural environment, I know that it's H2O. I think the previous paragraph is correct -- even though I don't think there are any possible worlds with alt-water that replicates all observable characteristics of water (including the ones that led us to infer its molecular structure) while not having that molecular structure. What I do by eating, drinking, sweating, and metabolizing could not be exactly replicated with alt-water. There could be a world with a lot of superficial similarities, but not enough for alt-water to be water. I think the previous two paragraphs, even if I've made mistakes in one or both, are compatible. One is epistemological. The other is modal.
It's also worth pointing out that P -> (Q | ~Q) is valid when using a paraconsistent implication! (As well as the binary case.) It's a fallacy of relevance, and only the full, relevant implication can see that P => (Q | ~Q) is invalid.
I don't buy the idea that all conditionals are material and apparent counterexamples are to be explained as heuristics for evaluating them. Material conditionals are easy to evaluate. We need heuristics for hard problems, not easy ones. Whether Q holds on the supposition of P just is what is meant by if P then Q, at least in typical cases. So to call suppositional inferences a heuristic is making the tail wag the dog. The suppositionalist is also better able to explain the fact that we can conditionalise almost any kind of speech act, not just propositions. A conditional command is a command within the context of a supposition; it is not an implication. Likewise for other speech acts.
The concept of close and distant possible worlds can be thought of like using different camera lenses. Imagine a world where you own a full-frame camera. If you dream about using a 75mm lens to photograph your coffee cup, this dream might be considered a "closer" possible world compared to a dream where you're using a 14mm lens. The 75mm lens makes the cup seem larger and more detailed, just like a close possible world shares more similarities with your actual reality. If you dream about owning a full-frame camera, this might be considered a "closer" possible world to reality than a dream about owning a time machine because the distance between you and the coffee cup doesn't change just because you're using a different lens or camera but not so in the advent of a radically advanced technology. If technology becomes a gold standard for conceiving possible worlds then it seems a counter possible world is interdependent on any decentralised observer’s imagination. However imagination is indexical to the technology sector. This entails corporate power within a political system based on minimal trust of imagination. I can trust my lens to inform me of a possible world but know its false due to being situated neither in my walking or dream state but within a transnational corporation as if the mind is colonised with counter possible worlds based on the kind of diet derived from the agricultural sector that is foundational for my purported walking state.
@39:45 or so, you mention the notion of “maximal but inconsistent” sets of propositions as models for impossible worlds semantics. I’m not sure what you mean. Of course I understand what you mean by inconsistent, but in the case of consistent sentences, the notion of maximality used in classical model theory for propositional logic along these lines entails that the set is deductively closed (and complete), and when the assumption of deductive closedness or completeness is applied to any set of inconsistent formulas, the resulting complete set of formulas is the set of all formulas. You can change the language if you like from propositional logic to some other logic, with a different semantics or a different notion of deductions, or you can introduce notions of paraconsistency perhaps, but it’s because I thought that the account that you were presenting was intended to use the context of classical propositional logic; hence my confusion.
The word "maximal" is being used in a broader sense here. It's just intended to mean something like "complete", as in a complete description, so that nothing is left out. Nothing is indeterminate in virtue of a failure to specify one way or the other (though things might be specified to be indeterminate).
@@KaneB Thanks for your response. I still find that unclear, if the context is to remain propositional logic, because if S is a deductively closed inconsistent set of formulas (i.e. a model for an inconsistent propositional theory) that is maximal in the sense that you describe and p is a propositional formula such that p is not in the set S then shouldn’t its negation, ~p, be in S? If both p and its negation are not in S, then the question “According to theory S, does p hold?” remains unanswered by the theory at hand, making the deductively closed theory S non-maximal in your sense, I’d say. If you grant me that, then it seems to me that you’re notion of maximality entails completeness, and then inconsistency together with deductive closeness entail that for every propositional formula p both p and its negation are in S. Summing up, it seems that this leads to having only one inconsistent model, i.e., any two inconsistent theories have the same model, making them semantically equivalent. This leads me to propose that Timothy Williamson’s account should be replaced by accounts that used non-maximal sets of formulas for inconsistent worlds. Another approach that seems to make more sense to me is to use a hierarchy of propositional languages (or more generally, languages), in which some languages provide a way to have meta-theoretical discussions about the notions of worlds that are models of inconsistent theories for other languages lower in the hierarchy. I expect there are philosophers and mathematical logicians who have assessed this previously. Edit: As I posted, I noticed that I’d missed your bit about specifying some things as indeterminate. It seems to me that that fits under my original comments about changing the logic system. Rather than using classical propositional logic, we can use a three-valued propositional logic system, in which the truth value “indeterminate” is “between” false and true: false < indeterminate < true; in a sense meaning that our logic system accommodates three distinct levels of ontological commitment for asserted propositional variables, and hence (by a recursive definition of satisfaction a la Tarski) for any given propositional formula. In this case I’d think that in order to use the Williamson notion of an impossible world, we must also do away with the law of contradiction in our deduction system. …
@@writerightmathnation9481 I'm not sure why we would insist on using classical propositional logic here. When I say "maximal" and "complete", I'm not using these terms in any particular formal sense. This is because different philosophers will have different views about the appropriate formal machinery. If there is a way of modelling this in classical propositional logic, that's nice; if not, so much the worse for classical propositional logic in this context -- or so, I assume, the nonvacuist would say.
@@KaneB I do not mean to suggest that I’d “insist” on using propositional logic. But I think it would be helpful to use it as a starting place to formalize these ideas of impossible worlds for use in other logic systems as well. Regarding non-classical logic and possible worlds semantics, I’m sure much has been done, but again, most of the non-classical systems of which I’m aware can be analyzed quite robustly using variants, or extensions, of the approaches used in the formal (and informal) study of classical logic. Indeed, one reason that I enjoy your videos so much is that I can usually see (at least, intuitively) how to formalize your approach as variants of classical logic. The reason that I mention classical propositional logic is that you mentioned in the video that a traditional(classical?) way to model possible worlds semantics is to represent a possible world with a maximal consistent set of propositions (aka propositional formulas), and in fact that is a very classical way to transition from classical propositional semantics to a possible worlds semantics for various modal logic systems in formal logic. This in fact is a nice way to think of Kripke semantics for propositional logic, and it’s a good start for the predicate calculus as well. If you aren’t using the term “maximal” in any formal sense, then how does one assess in a specific instance whether a given set of propositions actually is maximal inconsistent? The maximality condition isn’t related to the logic system one uses in reasoning in this impossible worlds semantics?
Does the statement “if P then Q” have any meaningful difference from the statement “A possible cause of Q is P”(the reason I say “possible” is because the first statement being true does not mean there aren’t other possible causes for Q). If there is no meaningful difference between those two statements, then in the scenario of P being impossible, the first statement would just be false.
Forgive my ignorance, but if there are no P-worlds, then wouldn’t “nothing” be true in them rather than “anything?” Or I guess the idea is that since they don’t exist, we can claim whatever we like about them without fear of being proven wrong. That does make some sense now that I think about it. Perhaps it’s also related to whether one believes in platonism, i.e., whether an impossible P-world actually exists in some intangible realm of “existence.”
Thats a quirk of the mathematics. Since proving something requires you to exhaust all options, there being no options will make something trivial to prove. You just exhaust the 0 options
Yeah, if you think about it, the statement “if P had been, Q would have been” is simply asking wether P causes Q. The whole “think of a hypothetical world” thing is just some tool we use to think about causality. If P is impossible, the first statement is false, because P does not cause Q(because P does not exist)
Hi Mister B, i have a question abt your work. Who long did the ppt take you for this? How many vids do you typically have planned at any given time? Do you have a target topic you want to reach talking about with your work ? Greetings, Kate
for Removing Nonvacuist Intuitions at 17:33 the more accurate train of logic is not that 0=1, but that the answer is in base 9 ... and the number of correct answers is still true 0 for a world in which 5 + 7 = 13 in decimal, then it would still bee more accurate to assume that the math is using different axioms for addition and equalities (thus one would not be able subtract numbers from both sides without a full list of axioms)
I dont understand the vacuoist intuition. I get that if a situtation isnt occuring in any possible world (it is impossible) then there is no way to refute the conditional. But isnt it equally true to say that there is no way to confirm it? If this triangle was four sided it would have sides. According to the vacuoist since there is no world in which a triangle has four sides there is no way to refute it. But there is also no way to confirm it. So shouldnt the truth value be indeterminate as opposed to vacuously true?
Self referential pararoxes (liar paradox, quine's paradox, russel paradox, etc) imply the existence of statements that are neither true nor false. Why wouldn't counterpossibles fall into this same category?
@about 17:39, you explain the Timothy Wiliamson argument but there’s a flaw in that argument that is easy for a mathematician to spot. The phrase “if 5+7 had been equal to 13” connotes to a mathematician that the mathematical structure providing context to this counterpossible statement is just “some other semantics for the plus symbol and the use of numerals”. For a somewhat different example, consider the theory of abelian groups, which is usually formulated in a language that uses a plus symbol, essentially because the behavior (according to the semantics in a particular model that comprises a specific abelian group) of the binary operation is very well aligned with the behavior of the plus sign as used in ordinary addition of integers, which is the “common man” usual context for using the plus sign. A specific abelian group one might consider is the group of modulo six integer arithmetic, in which there are only six elements but each has infinitely many names. In this context, 0=6=12=…= 6k, 1=7=13=…=6k+1, …, 5=11=17=…=6k+5, and similarly for negative numbers, e.g. -2=4=…=6k-2, etc. The equal signs here do not mean equality of integers but congruence modulo 6, or, in the terminology I used above, a statement like 0=12 or 0=6 or 3=9 means that the symbol (numeral) on the left of the equal sign and the numeral on the right of the equal sign are just different names for the same mathematical object in the group under consideration. From this we DO NOT conclude that 0=1; in fact in arithmetic modulo 6, the numeral 0 is definitely not a name for the same modulo six number as is the numeral 1. In this case, the statement 1+7=2 is evaluated to true by this semantic interpretation, because the rule for computing in the group of integers modulo 6 is “add the given numerals as integers and then compute the remainder upon division by six”. Since 8 has a remainder of 2 upon division by 6, the numerals 8 and 2 name the same element of modulo six arithmetic. In this sense, the mathematician considering the hypothesis that the question in the quiz includes “1+7=2” would perhaps say “maybe”, because even abelian group theory isn’t the only alternative meaning for the use of a plus sign in mathematics, and event the use of numerals with a plus sign varies quite a bit depending upon the subdiscipline and the topic of a given publication, or, in the case of a quiz, the course topic. Thus, in the scenario at hand in Timothy Williams’ argument, a mathematician just responds “the plus sign must not mean the same as usual addition for that quiz, if 5+7 was intended to be 13”. Merely the antecedent “if 5+7 had been equal to 13” is not enough information to tell a mathematician what the plus sign is being used for on the quiz in question. Thus the truth value of the counterpossible at hand cannot be determined from the available context. …
Hey just quick heads up, "square the circle" doesnt nean what u think it means, its like... - given an idealised compass - given a straight-edge - u can put compass origin on intersections Make a square of equal area to circle, circle is pi*r**2 which is transendental, lets make things tractable and assume r is integer, u only get rationals out of intersections, therefore no transendental, therefore no equal area I had to google what that phrase meant, ngl
"All counterpossibles are vacuously true"(X) seems, to me, to be a simple restatement of the principle of explosion(Y) - In case I'm using the wrong name here, I mean the "rule" that anything and/or everything follows from a contradiction. Is there any particular reason to treat X and Y as different propositions? (maybe my belief that impossibilities are contradictions is a false belief?)
Y should follow from X. If A then B is vacuously true if A is false in classical logic because you basically need that whenever A is true, so is B there are “no cases to check,” as it were. If A is true and false, the. If A then B is true for all B, by statement X. This is your principle of explosion, statement Y. So in a sense they’re not the exact same statement but Y is a trivial corralary of X.
You shouldn't take anything I say in the lecture videos as an expression of my own views, unless I explicitly state that I'm expressing my own views. As for paraconsistent logic, no actually, I wouldn't say that I believe in it. I don't believe in any logic.
Intuituvely I would argue for vacuism. If P then (Q and not Q) is obviously not rational and false. But this is assuming P is a rational statement. If P in itself is non rational it would make sense non rational conclusions would follow. Even trying to reason about the by definition unreasonable seems naive, especially using reasonable language :D
Keep in mind that vacuism is the position that *all* counterpossibles are vacuous, not just some. And most counterpossibles are not of the form "if P then Q and not Q". You can construct a counterpossible where the impossible part contains a logical contradiction embedded *into* a conjunction where the non-contradictory part is sufficient to entail the conclusion. Arguably this is the form of "If Hobbes squared the circle, African children wouldn't have cared.". There's also the issue of, it is not intuitively obvious that logical possibility and metaphysical possibility are equal in size. For something to be a counterpossible, it must simply be the case that it takes the form "If P were true then Q would be true." and there are no possible worlds where P. But "there are no possible worlds where P" and "P is logically contradictory" are not trivially equivalent claims. You'd have to make a positive claim that they are.
#RM3 does not suffer this nonsense. In binary logic it's all true, but binary logic sucks. You need 3 truth values to get the full picture. The conclusion is not relevant. True => Both (or inconsistent, or unknown) is False in #RM3. 3 truth values captures the irrelevance, and makes so called "informal fallacies" into "formal fallacies". Like P => (Q | ~Q) --- so even if the moon is made of cheese, you still cannot claim it is raining in Ecuador. It could be cheesing.
The material conditional: ua-cam.com/video/r0d0RBsyi0c/v-deo.html
Putnam's twin earth argument: ua-cam.com/video/tnpGTKCh2Sg/v-deo.html
Modal realism: ua-cam.com/video/gcC5g4A9IM4/v-deo.html
If my grandma had wheels, she would have been a bike.
Do no not cite the deep magic to me Witch, I was there when it was written.
lady gaga - born this way album cover
If my grandma had wheels, it would have been a waste of money because she never learnt to drive.
ua-cam.com/users/shortsMJQJ6Jsxjl4?si=ucPNcsYTvNDr128a
@@luszczi from von ribbentrop at the nazi war crimes tribunal, wasn't it?
"Impossible worlds, and possible worlds more generally" - Kane Baker
Ah yes, the famous superset of impossible worlds: possible worlds
This channel really shows off just how much interesting philosophy is out there and how developed it has become
9:20 nitpick on the proof of infinite primes: The contradiction is not that the product of primes plus 1 is both prime and composite. The contradiction is that it has a prime factor different than all the primes.
I assume there are multiple proofs of this, because from what I've read there is definitely a proof that derives the contradiction that the number N, such that N is the product of all primes smaller than or equal to p, plus 1, is both prime and composite. (Apparently this kind of proof is erroneously attributed to Euclid. But I assume the proof works, even if it came from somebody other than Euclid.)
@@KaneB This sounds very dubious to me, could you find a source? I don't see how one could possibly conclude N+1 is *either prime or composite*, let alone *both prime and composite*.
@@blackeyefly I just looked this up again and you're right; I misread it initially. I conflated two different ways of presenting the proof. The way that it's usually presented, N = the product of all primes less than or equal to p, plus 1. The way Williamson presents it, N = the product of all positive integers less than or equal to p, plus 1, i.e. p factorial plus 1.
Weird thought, but it's interesting to consider that there could be isolate possible worlds such that there's not any close possible worlds and so counterfactuals stated in that isolated world would also be vacuous.
Engagement comment for Engagement
Engaging with engagement comment
I think saying counterpossibles are trivially true is putting the cart before the horse. To decide whether something is possible in the first place it seems like you have to start by imagining it were true and then showing that you get a contradiction or something.
IN DEFENSE OF REASON - COUNTERPOSSIBLES
This video really gets the head spinning - well done! Indeed this kind of problem is one that can leave philosophy students believing that reason itself is impossible - a trend I have been seeing.
The clencher is the problem at the end - where the rules of inference one can make about impossible worlds can be countered by creating an impossible world where that rule of inference does not exist. The example given in the video was "if conjunction elimination did not exist, then Aristotle and Descartes would have been sad". It's hard to know what to name this problem, but I will call it "thwarting counterpossible inference and logic" - TCIAL.
As a Lakatosian, I would like to come up with a method to incorporate these anomalies into the research programme of counterpossible logic somehow. Of course, an opponent would just create a world where anomaly incorporation did not exist. Which at the moment leaves me at a loss, but I will try to get back to this later, maybe.
The other option for a Lakatosian would be to exclude the anomalies by definition. This would be something like "there can be no worlds that defy the laws of inference" and declare that statement to be itself a law of inference. While this would clearly degenerate the counterpossible research programme, it would at least save it from direct refutation. However, this would be a bad idea. We might want to evaluate the consequences of a world where conjunction elimination does not exist
So okay. Let's go ahead and evaluate those consequences. In a world where conjunction elimination did not exist, you could not derive "Aristotle was sad" from "Aristotle and Descartes were sad". While this seems strange to us, to the Aristotle and Descartes of that world, it would be commonplace. As such, we can simply say that the laws of inference are not universal across worlds. They only apply to the worlds in which they hold. So, we can say this also holds for counter-possible rules of inference as well. As such, there is a world where no rules of inference can apply, but that does not stop us from using them elsewhere. Whether this method is incorporation or exclusion, I do not know. Perhaps a little of each. Perhaps the research programme of counterpossibles is both progressive and degenerate.
I don't think that a statement "conjunction elimination holds" is even something truth-apt, in this world or in any other. So this worry of yours looks like a total non-issue to me.
Look into Browerian intuitionism, and especially Heyting's version of it.
@@СергейМакеев-ж2н Browerian intuitionism uses conjunction elimination, so I'm not sure what your point is. I mean if we have P&Q, is it not truth apt to conclude A in Browerian intuitionism?
@@InventiveHarvest I don't mean the intuitionistic logic system (though I am a fan of that too), but Brower's and Heyting's philosophical view on the nature of logic itself. Which too is called intuitionism. (It's confusing, I know.)
They believe that there is no "one true logic" out there in the world, that logic is just a tool to solve a problem, and you are free to choose whichever tool you like. There is no property of the world itself that determines which tool is the "right" one.
And Heyting in particular said that you can even choose different logics for different problems, or invent a new logic for every new problem. There is no need for a "one size fits all" logic for all areas of life.
@@СергейМакеев-ж2н@user-qm4ev6jb7d @user-qm4ev6jb7d okay, so when I talk about worlds, possible and impossible, I am not advocating for moral realism. Worlds are basically just places for hypotheticals. All of the worlds that evaluate the hypotheticals according to classic logic are possible worlds, and all worlds that do not are impossible worlds. A world where conjunction elimination does not exist is just a world that corresponds to a system of logic without the conjunction elimination rule.
The problem is that we are trying to make rules of inference to deal with such impossible worlds. If you say that you want to use conjunction elimination in order to say when a counterpossible is vacuous or not, then you will have a problem with the world where that rule does not exist.
My solution to this is to say that inference rules to evaluate counterpossibles are not universal. This is basically an intuitionist approach to counterpossible inference.
From the 7+5 example, it seems that counter factual are often not about considering actual counter factual worlds, but ways of pointing to and describing very specific causal relationships (in a sense, how two pieces of knowledge seem to “move with each other). When stating the counter factual, the way these facts are related to each the is structurally preserved, almost formally, even though the world is not in fact possible. This also applies to all the scientific counterpossibles. When you assume, for example, the sun is made up of an ideal gas, it entails very specific simplifications of your calculations, and that’s the sort of “important part.” It’s like causality and relational it’s is a web, and when we make cojnterpossible statements, we are pointing to an independent (or can practically be treated as independent part of the web, even if it’s in fact contradictory with other parts of the web.
Commenting more for algorithm engagement than serious criticism. At 8:43, there’s a minor error: p! is the product of all positive natural numbers less than or equal to p rather than the the product of all primes less than or equal to p
It seems as though we need to distinguish between modal and epistemological statements.
Because I haven't replicated enough of the relevant experiments myself, I don't personally know that water is H2O. My knowledge of its molecular structure is entirely dependent on some level of trust in social processes to do part of the work of developing knowledge. For all I know personally, water could be have an different structure, or it could be something like an element in the sense that people imagined when they first talked about air, fire, earth, and water as elements. But with the resources of my cultural environment, I know that it's H2O.
I think the previous paragraph is correct -- even though I don't think there are any possible worlds with alt-water that replicates all observable characteristics of water (including the ones that led us to infer its molecular structure) while not having that molecular structure. What I do by eating, drinking, sweating, and metabolizing could not be exactly replicated with alt-water. There could be a world with a lot of superficial similarities, but not enough for alt-water to be water.
I think the previous two paragraphs, even if I've made mistakes in one or both, are compatible. One is epistemological. The other is modal.
It's also worth pointing out that P -> (Q | ~Q) is valid when using a paraconsistent implication! (As well as the binary case.)
It's a fallacy of relevance, and only the full, relevant implication can see that P => (Q | ~Q) is invalid.
I don't buy the idea that all conditionals are material and apparent counterexamples are to be explained as heuristics for evaluating them. Material conditionals are easy to evaluate. We need heuristics for hard problems, not easy ones. Whether Q holds on the supposition of P just is what is meant by if P then Q, at least in typical cases. So to call suppositional inferences a heuristic is making the tail wag the dog. The suppositionalist is also better able to explain the fact that we can conditionalise almost any kind of speech act, not just propositions. A conditional command is a command within the context of a supposition; it is not an implication. Likewise for other speech acts.
The concept of close and distant possible worlds can be thought of like using different camera lenses. Imagine a world where you own a full-frame camera. If you dream about using a 75mm lens to photograph your coffee cup, this dream might be considered a "closer" possible world compared to a dream where you're using a 14mm lens. The 75mm lens makes the cup seem larger and more detailed, just like a close possible world shares more similarities with your actual reality.
If you dream about owning a full-frame camera, this might be considered a "closer" possible world to reality than a dream about owning a time machine because the distance between you and the coffee cup doesn't change just because you're using a different lens or camera but not so in the advent of a radically advanced technology. If technology becomes a gold standard for conceiving possible worlds then it seems a counter possible world is interdependent on any decentralised observer’s imagination. However imagination is indexical to the technology sector. This entails corporate power within a political system based on minimal trust of imagination. I can trust my lens to inform me of a possible world but know its false due to being situated neither in my walking or dream state but within a transnational corporation as if the mind is colonised with counter possible worlds based on the kind of diet derived from the agricultural sector that is foundational for my purported walking state.
Kane, I can't keep up with the rain of hardcore philosophy you're dealing! Nobody like you on the net.
@39:45 or so, you mention the notion of “maximal but inconsistent” sets of propositions as models for impossible worlds semantics. I’m not sure what you mean. Of course I understand what you mean by inconsistent, but in the case of consistent sentences, the notion of maximality used in classical model theory for propositional logic along these lines entails that the set is deductively closed (and complete), and when the assumption of deductive closedness or completeness is applied to any set of inconsistent formulas, the resulting complete set of formulas is the set of all formulas.
You can change the language if you like from propositional logic to some other logic, with a different semantics or a different notion of deductions, or you can introduce notions of paraconsistency perhaps, but it’s because I thought that the account that you were presenting was intended to use the context of classical propositional logic; hence my confusion.
The word "maximal" is being used in a broader sense here. It's just intended to mean something like "complete", as in a complete description, so that nothing is left out. Nothing is indeterminate in virtue of a failure to specify one way or the other (though things might be specified to be indeterminate).
@@KaneB
Thanks for your response. I still find that unclear, if the context is to remain propositional logic, because if S is a deductively closed inconsistent set of formulas (i.e. a model for an inconsistent propositional theory) that is maximal in the sense that you describe and p is a propositional formula such that p is not in the set S then shouldn’t its negation, ~p, be in S? If both p and its negation are not in S, then the question “According to theory S, does p hold?” remains unanswered by the theory at hand, making the deductively closed theory S non-maximal in your sense, I’d say.
If you grant me that, then it seems to me that you’re notion of maximality entails completeness, and then inconsistency together with deductive closeness entail that for every propositional formula p both p and its negation are in S.
Summing up, it seems that this leads to having only one inconsistent model, i.e., any two inconsistent theories have the same model, making them semantically equivalent.
This leads me to propose that Timothy Williamson’s account should be replaced by accounts that used non-maximal sets of formulas for inconsistent worlds. Another approach that seems to make more sense to me is to use a hierarchy of propositional languages (or more generally, languages), in which some languages provide a way to have meta-theoretical discussions about the notions of worlds that are models of inconsistent theories for other languages lower in the hierarchy. I expect there are philosophers and mathematical logicians who have assessed this previously.
Edit: As I posted, I noticed that I’d missed your bit about specifying some things as indeterminate. It seems to me that that fits under my original comments about changing the logic system. Rather than using classical propositional logic, we can use a three-valued propositional logic system, in which the truth value “indeterminate” is “between” false and true: false < indeterminate < true; in a sense meaning that our logic system accommodates three distinct levels of ontological commitment for asserted propositional variables, and hence (by a recursive definition of satisfaction a la Tarski) for any given propositional formula. In this case I’d think that in order to use the Williamson notion of an impossible world, we must also do away with the law of contradiction in our deduction system.
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@@writerightmathnation9481 I'm not sure why we would insist on using classical propositional logic here. When I say "maximal" and "complete", I'm not using these terms in any particular formal sense. This is because different philosophers will have different views about the appropriate formal machinery. If there is a way of modelling this in classical propositional logic, that's nice; if not, so much the worse for classical propositional logic in this context -- or so, I assume, the nonvacuist would say.
@@KaneB
I do not mean to suggest that I’d “insist” on using propositional logic. But I think it would be helpful to use it as a starting place to formalize these ideas of impossible worlds for use in other logic systems as well.
Regarding non-classical logic and possible worlds semantics, I’m sure much has been done, but again, most of the non-classical systems of which I’m aware can be analyzed quite robustly using variants, or extensions, of the approaches used in the formal (and informal) study of classical logic. Indeed, one reason that I enjoy your videos so much is that I can usually see (at least, intuitively) how to formalize your approach as variants of classical logic.
The reason that I mention classical propositional logic is that you mentioned in the video that a traditional(classical?) way to model possible worlds semantics is to represent a possible world with a maximal consistent set of propositions (aka propositional formulas), and in fact that is a very classical way to transition from classical propositional semantics to a possible worlds semantics for various modal logic systems in formal logic. This in fact is a nice way to think of Kripke semantics for propositional logic, and it’s a good start for the predicate calculus as well.
If you aren’t using the term “maximal” in any formal sense, then how does one assess in a specific instance whether a given set of propositions actually is maximal inconsistent? The maximality condition isn’t related to the logic system one uses in reasoning in this impossible worlds semantics?
Does the statement “if P then Q” have any meaningful difference from the statement “A possible cause of Q is P”(the reason I say “possible” is because the first statement being true does not mean there aren’t other possible causes for Q). If there is no meaningful difference between those two statements, then in the scenario of P being impossible, the first statement would just be false.
Forgive my ignorance, but if there are no P-worlds, then wouldn’t “nothing” be true in them rather than “anything?” Or I guess the idea is that since they don’t exist, we can claim whatever we like about them without fear of being proven wrong. That does make some sense now that I think about it. Perhaps it’s also related to whether one believes in platonism, i.e., whether an impossible P-world actually exists in some intangible realm of “existence.”
Thats a quirk of the mathematics. Since proving something requires you to exhaust all options, there being no options will make something trivial to prove. You just exhaust the 0 options
Yeah, if you think about it, the statement “if P had been, Q would have been” is simply asking wether P causes Q. The whole “think of a hypothetical world” thing is just some tool we use to think about causality. If P is impossible, the first statement is false, because P does not cause Q(because P does not exist)
Hi Mister B, i have a question abt your work. Who long did the ppt take you for this? How many vids do you typically have planned at any given time? Do you have a target topic you want to reach talking about with your work ? Greetings, Kate
for Removing Nonvacuist Intuitions at 17:33 the more accurate train of logic is not that 0=1, but that the answer is in base 9 ... and the number of correct answers is still true 0
for a world in which 5 + 7 = 13 in decimal, then it would still bee more accurate to assume that the math is using different axioms for addition and equalities (thus one would not be able subtract numbers from both sides without a full list of axioms)
I dont understand the vacuoist intuition. I get that if a situtation isnt occuring in any possible world (it is impossible) then there is no way to refute the conditional. But isnt it equally true to say that there is no way to confirm it?
If this triangle was four sided it would have sides.
According to the vacuoist since there is no world in which a triangle has four sides there is no way to refute it.
But there is also no way to confirm it. So shouldnt the truth value be indeterminate as opposed to vacuously true?
Self referential pararoxes (liar paradox, quine's paradox, russel paradox, etc) imply the existence of statements that are neither true nor false. Why wouldn't counterpossibles fall into this same category?
@about 17:39, you explain the Timothy Wiliamson argument but there’s a flaw in that argument that is easy for a mathematician to spot.
The phrase “if 5+7 had been equal to 13” connotes to a mathematician that the mathematical structure providing context to this counterpossible statement is just “some other semantics for the plus symbol and the use of numerals”.
For a somewhat different example, consider the theory of abelian groups, which is usually formulated in a language that uses a plus symbol, essentially because the behavior (according to the semantics in a particular model that comprises a specific abelian group) of the binary operation is very well aligned with the behavior of the plus sign as used in ordinary addition of integers, which is the “common man” usual context for using the plus sign. A specific abelian group one might consider is the group of modulo six integer arithmetic, in which there are only six elements but each has infinitely many names. In this context, 0=6=12=…= 6k, 1=7=13=…=6k+1, …, 5=11=17=…=6k+5, and similarly for negative numbers, e.g. -2=4=…=6k-2, etc. The equal signs here do not mean equality of integers but congruence modulo 6, or, in the terminology I used above, a statement like 0=12 or 0=6 or 3=9 means that the symbol (numeral) on the left of the equal sign and the numeral on the right of the equal sign are just different names for the same mathematical object in the group under consideration. From this we DO NOT conclude that 0=1; in fact in arithmetic modulo 6, the numeral 0 is definitely not a name for the same modulo six number as is the numeral 1.
In this case, the statement 1+7=2 is evaluated to true by this semantic interpretation, because the rule for computing in the group of integers modulo 6 is “add the given numerals as integers and then compute the remainder upon division by six”. Since 8 has a remainder of 2 upon division by 6, the numerals 8 and 2 name the same element of modulo six arithmetic.
In this sense, the mathematician considering the hypothesis that the question in the quiz includes “1+7=2” would perhaps say “maybe”, because even abelian group theory isn’t the only alternative meaning for the use of a plus sign in mathematics, and event the use of numerals with a plus sign varies quite a bit depending upon the subdiscipline and the topic of a given publication, or, in the case of a quiz, the course topic.
Thus, in the scenario at hand in Timothy Williams’ argument, a mathematician just responds “the plus sign must not mean the same as usual addition for that quiz, if 5+7 was intended to be 13”. Merely the antecedent “if 5+7 had been equal to 13” is not enough information to tell a mathematician what the plus sign is being used for on the quiz in question. Thus the truth value of the counterpossible at hand cannot be determined from the available context.
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Hey just quick heads up, "square the circle" doesnt nean what u think it means, its like...
- given an idealised compass
- given a straight-edge
- u can put compass origin on intersections
Make a square of equal area to circle, circle is pi*r**2 which is transendental, lets make things tractable and assume r is integer, u only get rationals out of intersections, therefore no transendental, therefore no equal area
I had to google what that phrase meant, ngl
What do you think I think it means?
42:33 incompleteness strikes again!
3:00 Not tricky at all. There is only 1 world. There is no cottoncandy ground world.
The law of identity has you covered
"All counterpossibles are vacuously true"(X) seems, to me, to be a simple restatement of the principle of explosion(Y) - In case I'm using the wrong name here, I mean the "rule" that anything and/or everything follows from a contradiction.
Is there any particular reason to treat X and Y as different propositions?
(maybe my belief that impossibilities are contradictions is a false belief?)
Y should follow from X. If A then B is vacuously true if A is false in classical logic because you basically need that whenever A is true, so is B there are “no cases to check,” as it were. If A is true and false, the. If A then B is true for all B, by statement X. This is your principle of explosion, statement Y. So in a sense they’re not the exact same statement but Y is a trivial corralary of X.
This video makes me feel very stupid, and I like it.
In telling effectual
wait, you don't believe in paraconsistent logic?
You shouldn't take anything I say in the lecture videos as an expression of my own views, unless I explicitly state that I'm expressing my own views.
As for paraconsistent logic, no actually, I wouldn't say that I believe in it. I don't believe in any logic.
i wass nr💯 to like this video
Intuituvely I would argue for vacuism. If P then (Q and not Q) is obviously not rational and false. But this is assuming P is a rational statement. If P in itself is non rational it would make sense non rational conclusions would follow. Even trying to reason about the by definition unreasonable seems naive, especially using reasonable language :D
Keep in mind that vacuism is the position that *all* counterpossibles are vacuous, not just some. And most counterpossibles are not of the form "if P then Q and not Q". You can construct a counterpossible where the impossible part contains a logical contradiction embedded *into* a conjunction where the non-contradictory part is sufficient to entail the conclusion. Arguably this is the form of "If Hobbes squared the circle, African children wouldn't have cared.".
There's also the issue of, it is not intuitively obvious that logical possibility and metaphysical possibility are equal in size. For something to be a counterpossible, it must simply be the case that it takes the form "If P were true then Q would be true." and there are no possible worlds where P. But "there are no possible worlds where P" and "P is logically contradictory" are not trivially equivalent claims. You'd have to make a positive claim that they are.
What the hell? The 5+7 part is the most deranged thing ive ever seen
The world was destroyed yesterday, therefore I'm liking and commenting today.
heheheheh pee worlds
still better than pee pee poo poo worlds
#RM3 does not suffer this nonsense. In binary logic it's all true, but binary logic sucks. You need 3 truth values to get the full picture. The conclusion is not relevant. True => Both (or inconsistent, or unknown) is False in #RM3. 3 truth values captures the irrelevance, and makes so called "informal fallacies" into "formal fallacies". Like P => (Q | ~Q) --- so even if the moon is made of cheese, you still cannot claim it is raining in Ecuador. It could be cheesing.