One of the best examples of an equivocated syllogism I have ever seen was in a debate between Lawrence Krauss and WLC:
1. All mammals exhibit homosexual behavior
2. WLC is a mammal
I forget what the conclusion was. Probably "There is/is not a God", depending on who was making the argument. I forget. I was so busy coming up with nouns that could refer either to sets of objects or individual objects to construct similar syllogisms that I missed the debate
“ The conclusion of a deductive argument is only guaranteed because it’s trivial. But if it’s trivial, then it shouldn’t be able to tell you anything new. So if the conclusion does tell you something new, it doesn’t follow from the premises. And if it doesn’t tell you anything new, then why bother making the argument? Either way, the paradox of analysis makes logic look useless.”
Unfortunately, this seems to be the same type of equivocation that you deride in the rest of your post! “Trivial” has two different meanings that are being traded off of here. Mathematical proofs are “trivial” but not “trivial”; especially because our minds are finite and cannot know every infinite possibly true thing of a deductive system.
Furthermore, how have you reached the conclusions that you do in your post? With logic? If so, then by your own admission, it was trivial. If, on the other hand, you used logic profitably here, then there is some explaining to do. At present, you have good examples of a particular way language and argumentation can be misused, but I’m not sure if that can bear the weight of your title.
I think the section on the paradox of analysis confused a lot of commenters. It is not actually part of my argument, but some introductory exposition about the concerns that philosophers have raised for deduction (well, analysis) that motivates the subsequent discussion. Although of course the post still uses logical entailments to make its argument! I don't think it's a problem because the goal is to show that logic fails by its own lights - to undermine logic from the inside.
Doing physics with math and discovering the tensile strength you need for your bridge is using axioms to discover a useful conclusion. How do you make a distinction between “useless deduction” and this kind of useful deduction?
My head is spinning but I’m fascinated by your analysis. I’m giving it some parking space in my brain so I have it on hand when I read philosophical arguments in future.
You say that, "if the conclusion [of a deductive argument] does tell you something new, it doesn’t follow from the premises." But there seem to be counterexamples. Haven't some sound deductive arguments increased our knowledge? Like novel and important proofs in mathematics, for example? Or imagine some NASA scientist chugging through some equations to find the necessary thrust to escape Earth's gravity. Surely that could be very, very informative and useful, and yet the result of pure deduction. But, if so, then what you say is false about deduction being either trivial or invalid.
Also, doesn't your essay contain arguments that are meant to tell us something new? Many people disagree with you, after all, e.g. about whether, as you put it, "Logic is a logical fallacy." You're trying to convince them, to teach them. With deductive arguments, no? And yet you say, "if the conclusion [of a deductive argument] does tell you something new, it doesn’t follow from the premises." Does this mean that, according to you, your arguments must be invalid? That seems like a bad result, a kind of self-defeat.
When I say your essay contains arguments, I have in mind examples like this: "The conclusion of a deductive argument is only guaranteed because it’s trivial. But if it’s trivial, then it shouldn’t be able to tell you anything new. So if the conclusion does tell you something new, it doesn’t follow from the premises."
If, on the other hand, your essay does *not* in fact contain any valid arguments, then... why should we accept that your conclusions follow from your premises?
Also, you say you've "taken liberties" with your interpretation of the paradox of analysis, when you say, "the paradox of analysis, which says that a deduction can’t be both true and informative..." But, for what it's worth, I'd say this goes beyond merely taking liberties, and veers into flat-out misrepresentation of what the paradox of analysis actually is. It's a puzzle about analyses, not deductive arguments. For example: https://philpapers.org/rec/KINPAC-3
And the puzzle that besets analyses does not beset deduction generally.
Formal or propositional logic is the topic (0th order), while maths is 1st order. The overall thrust is against using "logic" as a means of promoting certainty, because that is prohibited by logic itself.
A. "All men are mortal, Socrates Is a man, therefore Socrates is mortal"
B. "From the axioms of Euclidean geometry, the Pythagorean theorem follows"
...You're basically saying B seems to have told us something new and useful, but A may or may not have, yes?
My answer would be that, on some level, B didn't tell us anything "new". The thing with B is, Euclidean geometry told us Everything all at once. Within those five little axioms, the whole story of geometry was encoded. But, finite creatures that we are, it takes us a long time to sift through a pile of infinite information, even if the physical pile is small enough to fit on a single page. Whereas, with A, we've got a tiny amount of information in a tiny pile. So A is more "trivial"
We definitely seem to be on parallel paths here. First, I must admit that my approach to formal logic was never really a good faith attempt to find real value in it, so I do not suspect that I have a strong grasp of it generally.
There are two clarifications I would need, though I suspect part of your conclusion indirectly addresses the direction you are taking as to the positive value formal logic has on offer:
1. What you mean by "epistemic value," such that formal logic generates no "surplus"
2. What you mean by "trivially true," though I will respond inferring the meaning that seems to be implied
I intuit your overall argument approximates to a qualitative version of Godel's incompleteness theorems where formal logic must choose between (internal) consistency and completeness, which to me would seem quite valuable. I also suspect that, once we are allowed to leave the confines of strict, formal commitments, we can more readily explore how an internal lack of surplus value, the local triviality of any truths "generated," and as you put it, just moving around uncertainties, can synthetically generate surplus, nontrivial value and improved confidences.
For example, we discovered some of our intuitions lead us astray by proving that we unknowingly inhere certain contradictory implications. Likewise, formal logic enabled computation, which while annoyingly treated as if does more than ALSO just move around uncertainties, is nevertheless a very efficient way to do so.
Of primary interest to me is how to treat triviality, and relatedly, significance and relevance. At a glance, I would intuit that the internal "triviality" of logical progression gives way to what utility such progressions can contribute when synthesized with other processes, at which point what was a "trivial step" at one level takes on "significance" at the next, because the success of that step has significant effects. Still, this significance remains abstract and local, whereas general relevance relies once again at other levels not reachable by structural extensions, even at the new, synthetic level. Another way we might put this is as a new instance of triviality, potentially of a different kind.
Once we take some set of relevantly interactive processes, each with its own set of local uncertainties, significant elements, and trivialities, suddenly we have also introduced a new type of uncertainty: whether or not we included all and only those relevant elements, raising questions of potentially arbitrary selections, what we should consider to be "sufficient" or "necessary" conditions when at this level of synthesis, and if our models or methods seek to judge relevance by what is "coherence," then we also inhere the problem of what constitutes "signal" versus "noise."
As an extension of the points you are making here, I think it is reasonable to suspect that much of the formal and pragmatic sciences have fallen prey to a subtle trap. One thing that mimics coherence quite well is "adherence" to a paradigm. If, in the name of fighting pseudoscience, biases, pop science pressures, ect., you collectively conform to an intuition of "what noise looks like," then you've introduced all those things you are trying to keep out: triviality, arbitrariness and noise, because they don't come in predictable shapes and sizes or from specific personality types. This move always presumes more consistency and completeness than is justifiable... it "moves around uncertainties" and then declares "that which is arbitrary noise to also be trivializable," effectively ignoring what is collectively inconvenient, not by individual or even collective intent, but simply by the powers of motivated reasoning that they presume, in action, to be less susceptible to... despite all evidence to the contrary.
To be fair, though, this is more joint critique than judgment, as I would probably do the same in their position. But I am not in their position, and so I intend to make some noise :)
Also, in the broadest sense, one cannot be a physicalist, a computationalist, a Bayesian idealist, and still use terms like "trivial," "arbitrary" and "noise" as if they are real things, because physicalism can only recognize everything as information (signal), and if noise is a threshold effect relative to some computationalist capacity, then Bayesian idealism eats itself in dramatic fashion by compounding uncertainty across every prior. I'd be willing to bet someone else's money, it would eat itself back into a coin flip.
To be clear, I am not against physicalists, computationalists or bayesian idealists. I'm just not going to pretend you can be them all at once, presume other positions are biased, and settle for "less biased" and call it a day. No one is new to what lazy gatekeeping looks like. It's practically the old normal now.
What a wonderful article! I have a lot to think about from reading it, but I wanted to share one initial thought, less as contradictory confrontation, and more shuffling around uncertainties. The sentence "Meaning, after all, is the bridge between words and the objects to which they refer" struck me as missing the reality that words do not always refer to objects. I would instead propose the consideration of an analogous construction: "Words are the bridge between meaning and the things/entities/phenomena which eschew it".
Take "love" as an example. Love, as a phenomenon, is real: even if one were to argue that love does not have a primordial source within the ontic, it is still the name given to refer to a force which does influence the physical world around us. One can perform household chores in inspiration of love for their partner in such a manner that is distinct from how that individual would manifest their environment to their bespoke wishes.
One can argue that love is an ontic process perceived as transcendent by the limitations of our mind, but inarguably the meaning that that sensation can have is beyond the scope of science as we currently understand it to uncover. If, then, that mode of uncovering is foreclosed, but the glacial tip still juts out, so to speak, than it would be denying The Real to claim that the whole structure is illusory.
Thus, it does not appear to me that meaning is the mediation between the word and the Real, but rather that the word mediates meanings and makes visible what, without language, is a differential significance in the sensory. I believe this is most obvious in the extrasemiotic significance that the word can inspire: "Grandma's ring", even among those who share a mutual understanding of whom 'Grandma' and which particular 'ring' is referred, will conjure in the mind different surplus personal significances.
If meaning were to be the mediator, than the accretion of meaning would have to correspond to some likewise change in the Real or in words.
One of the best examples of an equivocated syllogism I have ever seen was in a debate between Lawrence Krauss and WLC:
1. All mammals exhibit homosexual behavior
2. WLC is a mammal
I forget what the conclusion was. Probably "There is/is not a God", depending on who was making the argument. I forget. I was so busy coming up with nouns that could refer either to sets of objects or individual objects to construct similar syllogisms that I missed the debate
Very good analysis!
“ The conclusion of a deductive argument is only guaranteed because it’s trivial. But if it’s trivial, then it shouldn’t be able to tell you anything new. So if the conclusion does tell you something new, it doesn’t follow from the premises. And if it doesn’t tell you anything new, then why bother making the argument? Either way, the paradox of analysis makes logic look useless.”
Unfortunately, this seems to be the same type of equivocation that you deride in the rest of your post! “Trivial” has two different meanings that are being traded off of here. Mathematical proofs are “trivial” but not “trivial”; especially because our minds are finite and cannot know every infinite possibly true thing of a deductive system.
Furthermore, how have you reached the conclusions that you do in your post? With logic? If so, then by your own admission, it was trivial. If, on the other hand, you used logic profitably here, then there is some explaining to do. At present, you have good examples of a particular way language and argumentation can be misused, but I’m not sure if that can bear the weight of your title.
I think the section on the paradox of analysis confused a lot of commenters. It is not actually part of my argument, but some introductory exposition about the concerns that philosophers have raised for deduction (well, analysis) that motivates the subsequent discussion. Although of course the post still uses logical entailments to make its argument! I don't think it's a problem because the goal is to show that logic fails by its own lights - to undermine logic from the inside.
Doing physics with math and discovering the tensile strength you need for your bridge is using axioms to discover a useful conclusion. How do you make a distinction between “useless deduction” and this kind of useful deduction?
My head is spinning but I’m fascinated by your analysis. I’m giving it some parking space in my brain so I have it on hand when I read philosophical arguments in future.
You say that, "if the conclusion [of a deductive argument] does tell you something new, it doesn’t follow from the premises." But there seem to be counterexamples. Haven't some sound deductive arguments increased our knowledge? Like novel and important proofs in mathematics, for example? Or imagine some NASA scientist chugging through some equations to find the necessary thrust to escape Earth's gravity. Surely that could be very, very informative and useful, and yet the result of pure deduction. But, if so, then what you say is false about deduction being either trivial or invalid.
Also, doesn't your essay contain arguments that are meant to tell us something new? Many people disagree with you, after all, e.g. about whether, as you put it, "Logic is a logical fallacy." You're trying to convince them, to teach them. With deductive arguments, no? And yet you say, "if the conclusion [of a deductive argument] does tell you something new, it doesn’t follow from the premises." Does this mean that, according to you, your arguments must be invalid? That seems like a bad result, a kind of self-defeat.
When I say your essay contains arguments, I have in mind examples like this: "The conclusion of a deductive argument is only guaranteed because it’s trivial. But if it’s trivial, then it shouldn’t be able to tell you anything new. So if the conclusion does tell you something new, it doesn’t follow from the premises."
If, on the other hand, your essay does *not* in fact contain any valid arguments, then... why should we accept that your conclusions follow from your premises?
Also, you say you've "taken liberties" with your interpretation of the paradox of analysis, when you say, "the paradox of analysis, which says that a deduction can’t be both true and informative..." But, for what it's worth, I'd say this goes beyond merely taking liberties, and veers into flat-out misrepresentation of what the paradox of analysis actually is. It's a puzzle about analyses, not deductive arguments. For example: https://philpapers.org/rec/KINPAC-3
And the puzzle that besets analyses does not beset deduction generally.
Formal or propositional logic is the topic (0th order), while maths is 1st order. The overall thrust is against using "logic" as a means of promoting certainty, because that is prohibited by logic itself.
If I understand your comment correctly...
A. "All men are mortal, Socrates Is a man, therefore Socrates is mortal"
B. "From the axioms of Euclidean geometry, the Pythagorean theorem follows"
...You're basically saying B seems to have told us something new and useful, but A may or may not have, yes?
My answer would be that, on some level, B didn't tell us anything "new". The thing with B is, Euclidean geometry told us Everything all at once. Within those five little axioms, the whole story of geometry was encoded. But, finite creatures that we are, it takes us a long time to sift through a pile of infinite information, even if the physical pile is small enough to fit on a single page. Whereas, with A, we've got a tiny amount of information in a tiny pile. So A is more "trivial"
We definitely seem to be on parallel paths here. First, I must admit that my approach to formal logic was never really a good faith attempt to find real value in it, so I do not suspect that I have a strong grasp of it generally.
There are two clarifications I would need, though I suspect part of your conclusion indirectly addresses the direction you are taking as to the positive value formal logic has on offer:
1. What you mean by "epistemic value," such that formal logic generates no "surplus"
2. What you mean by "trivially true," though I will respond inferring the meaning that seems to be implied
I intuit your overall argument approximates to a qualitative version of Godel's incompleteness theorems where formal logic must choose between (internal) consistency and completeness, which to me would seem quite valuable. I also suspect that, once we are allowed to leave the confines of strict, formal commitments, we can more readily explore how an internal lack of surplus value, the local triviality of any truths "generated," and as you put it, just moving around uncertainties, can synthetically generate surplus, nontrivial value and improved confidences.
For example, we discovered some of our intuitions lead us astray by proving that we unknowingly inhere certain contradictory implications. Likewise, formal logic enabled computation, which while annoyingly treated as if does more than ALSO just move around uncertainties, is nevertheless a very efficient way to do so.
Of primary interest to me is how to treat triviality, and relatedly, significance and relevance. At a glance, I would intuit that the internal "triviality" of logical progression gives way to what utility such progressions can contribute when synthesized with other processes, at which point what was a "trivial step" at one level takes on "significance" at the next, because the success of that step has significant effects. Still, this significance remains abstract and local, whereas general relevance relies once again at other levels not reachable by structural extensions, even at the new, synthetic level. Another way we might put this is as a new instance of triviality, potentially of a different kind.
Once we take some set of relevantly interactive processes, each with its own set of local uncertainties, significant elements, and trivialities, suddenly we have also introduced a new type of uncertainty: whether or not we included all and only those relevant elements, raising questions of potentially arbitrary selections, what we should consider to be "sufficient" or "necessary" conditions when at this level of synthesis, and if our models or methods seek to judge relevance by what is "coherence," then we also inhere the problem of what constitutes "signal" versus "noise."
As an extension of the points you are making here, I think it is reasonable to suspect that much of the formal and pragmatic sciences have fallen prey to a subtle trap. One thing that mimics coherence quite well is "adherence" to a paradigm. If, in the name of fighting pseudoscience, biases, pop science pressures, ect., you collectively conform to an intuition of "what noise looks like," then you've introduced all those things you are trying to keep out: triviality, arbitrariness and noise, because they don't come in predictable shapes and sizes or from specific personality types. This move always presumes more consistency and completeness than is justifiable... it "moves around uncertainties" and then declares "that which is arbitrary noise to also be trivializable," effectively ignoring what is collectively inconvenient, not by individual or even collective intent, but simply by the powers of motivated reasoning that they presume, in action, to be less susceptible to... despite all evidence to the contrary.
To be fair, though, this is more joint critique than judgment, as I would probably do the same in their position. But I am not in their position, and so I intend to make some noise :)
Also, in the broadest sense, one cannot be a physicalist, a computationalist, a Bayesian idealist, and still use terms like "trivial," "arbitrary" and "noise" as if they are real things, because physicalism can only recognize everything as information (signal), and if noise is a threshold effect relative to some computationalist capacity, then Bayesian idealism eats itself in dramatic fashion by compounding uncertainty across every prior. I'd be willing to bet someone else's money, it would eat itself back into a coin flip.
To be clear, I am not against physicalists, computationalists or bayesian idealists. I'm just not going to pretend you can be them all at once, presume other positions are biased, and settle for "less biased" and call it a day. No one is new to what lazy gatekeeping looks like. It's practically the old normal now.
Great article, thank you. This is the level of questioning in philosophy that I aspire to
What a wonderful article! I have a lot to think about from reading it, but I wanted to share one initial thought, less as contradictory confrontation, and more shuffling around uncertainties. The sentence "Meaning, after all, is the bridge between words and the objects to which they refer" struck me as missing the reality that words do not always refer to objects. I would instead propose the consideration of an analogous construction: "Words are the bridge between meaning and the things/entities/phenomena which eschew it".
Take "love" as an example. Love, as a phenomenon, is real: even if one were to argue that love does not have a primordial source within the ontic, it is still the name given to refer to a force which does influence the physical world around us. One can perform household chores in inspiration of love for their partner in such a manner that is distinct from how that individual would manifest their environment to their bespoke wishes.
One can argue that love is an ontic process perceived as transcendent by the limitations of our mind, but inarguably the meaning that that sensation can have is beyond the scope of science as we currently understand it to uncover. If, then, that mode of uncovering is foreclosed, but the glacial tip still juts out, so to speak, than it would be denying The Real to claim that the whole structure is illusory.
Thus, it does not appear to me that meaning is the mediation between the word and the Real, but rather that the word mediates meanings and makes visible what, without language, is a differential significance in the sensory. I believe this is most obvious in the extrasemiotic significance that the word can inspire: "Grandma's ring", even among those who share a mutual understanding of whom 'Grandma' and which particular 'ring' is referred, will conjure in the mind different surplus personal significances.
If meaning were to be the mediator, than the accretion of meaning would have to correspond to some likewise change in the Real or in words.