Metaphysics, Metacognition, Language and Number

by Mark English

The logical positivists took a very hard anti-metaphysical line. They were right, in my view, to see traditional metaphysics as being futile and pointless. The essential problem with metaphysics is epistemic. How (given a basically scientific view of the world) can purely metaphysical statements be justified? Rudolf Carnap and most of his Vienna Circle colleagues didn’t think they could and consequently saw no place for metaphysics as a serious discipline.

There is no denying that fundamental, foundational and relational questions arise naturally in the course of scientific and other forms of rigorous inquiry. These kinds of questions are not only worthwhile, they are necessary and inescapable, and to call them philosophical or (in certain cases) metaphysical is not out of line with common usage. Problems arise, however, when philosophical or metaphysical thinking becomes detached from empirical reality and begins to feed on itself.

In the 1940’s and ’50s, Carnap articulated a nuanced account of ontological claims in the context of mathematical and scientific inquiry. He saw such claims as being either trivially true or false (if considered within the theoretical framework in question) or nonsensical (if not). The former were associated with “internal” questions, the latter with “external” questions. Internal questions are asked with a particular framework in mind. Do numbers exist? Within the framework of arithmetic, (trivially) yes. But do numbers really exist in some absolute sense? The question, arguably, is meaningless.

This approach works for formal disciplines and strictly scientific theoretical concepts but the sciences are not entirely formal. They have their origin in our interactions with, and natural curiosity about, the world. It is a mistake to imagine that we are ever entirely locked into specific and rigid linguistic or theoretical frameworks.

Ordinary thinking is an element of our engagement with the world and is never entirely mechanical or formal. It is not formal because interpretation of one kind or another is always involved, in the sciences and elsewhere. And it is holistic in the sense of not being comprised of discrete levels or self-contained modules. Levels and modules are useful abstractions in dealing with human cognition. And, to the extent that they are useful, you could say that they are real, they exist. My point is simply that they are not discrete or self-contained. Not only are various parts of the brain interconnected in complex ways, the broader physical, social and cultural matrix within which neural processing occurs and upon which it depends is also holistic and massively interconnected. A broader context, in other words, always obtrudes.

An ability to conceptualize and deal in a practical way with a wide range of contingencies involves special forms of thinking – or meta-thinking. ‘Metacognition’ is a term applied in the area of cognitive neuroscience to a form (or forms) of higher-order thinking exclusive to humans and strongly associated with the prefrontal cortex. It involves both monitoring (involving an awareness of sensory input and stored information) and control (deploying this information for decision-making and intelligent action). My focus here is on aspects of meta-thinking which relate to language and number.

Alfred Tarski developed the notion of metalanguage, though he was concerned mainly with formal rather than natural languages. Karl Popper explicitly drew on Tarski’s concept of metalanguage to defend a form of the correspondence theory of truth. The linguist Roman Jakobson appealed to the same basic idea when, late in his career, he outlined what he saw as the functions of language. One of these was the metalingual (or metalinguistic) function. It applies when a language is used to talk about itself.

The notion of metalinguistic awareness relates closely to metacognition and is often utilized in explaining such phenomena as code-switching and language alternation. But metalinguistic awareness also applies to phenomena which occur in strictly monolingual environments. As noted above, languages are routinely used in a reflexive way (i.e. to refer to themselves). A speaker’s awareness of implied (as distinct from literal) meaning also requires a certain level of metalinguistic awareness; likewise using and understanding metaphors. Using and understanding irony requires a relatively high level of metalinguistic awareness. Sarcasm is less subtle than irony but provides a clearer illustration that what is literally being said is not always what is actually being said, the intended sense being (in the case of sarcasm) the converse of the literal sense.

The general idea that a broader context always obtrudes applies not just to ordinary life and language use but also to specialized scientific and scholarly work. No significant area of study is self-contained. Not even formal disciplines, such as arithmetic.

Gödel’s work demonstrated the limitations of formal axiomatic systems. He showed that no such system is capable of proving all truths about the arithmetic of natural numbers. He also demonstrated that no formal system which is complex enough to model basic arithmetic can prove its own consistency.

Formal systems then (at least those beyond a certain level of complexity) are not self-contained, not sufficient unto themselves. They are necessarily situated in – and in a real sense are dependent on – a broader context. And any expanded system is dependent on a yet broader context in the same way.

Gödel was a Platonist and saw his incompleteness theorems as vindicating his position on the power of the human mind. The main lesson I take from his work, however, is that productive thinking is necessarily contingent rather than self-contained; that it necessarily engages with a wider world.

What this wider world consists in or of is open to debate. It comprises seemingly very different kinds of things and/or processes: the processes studied by mathematicians; the physical processes studied by physicists and biologists; social and cultural processes; etc..

But on what basis – other than practicality or convenience – do we draw dividing lines between these different kinds of process and, by extension, between disciplines?

Because of the problems of justifying metaphysical statements I prefer to remain metaphysically agnostic and avoid making claims about the nature of these various processes which go beyond common sense, common usage and the relevant disciplines. Neither ordinary thinking nor rigorous intellectual inquiry requires an explicit metaphysical foundation. Effective thinking, speaking and research do have prerequisites, but such a foundation is not one of them.

Sure, language and our natural habits of thought constantly throw up or betray largely implicit metaphysical assumptions or commitments. This is something to be aware and wary of, however, not something which should form a basis or foundation for serious metaphysical claims or systematizing.

The philosophical arguments against realism with which I am most familiar (including Carnap’s view) have, as I see it, a fatal flaw. They give too great a prominence to language and fail to give sufficient recognition to non-linguistic aspects of human thought and understanding.

In respect of the existence or non-existence of entities postulated by scientific theories, Carnap’s approach works well enough because the theories in question are clearly identifiable and distinguishable one from another. If you move beyond particular theories, however, and focus on more mundane concepts which we can approach from many directions and in many ways, there is no single language or system or theory to which we can appeal. This applies to concrete things that we might touch or eat or bump into and also to more complex or abstract things like an ambush or a stock market correction or a political movement or the passing of a bill into law.

Even something like the concept of number can be approached and conceptualized in different ways: using set theory, for example; or the lambda calculus developed by Alonzo Church. Alternatively, numbers and counting can be dealt with via psychology or via linguistics (where the focus is on the ways people calculate and conceptualize number in the course of ordinary social life).

Though there are certainly links between language and counting etc., counting and number clearly take us beyond the bounds of natural language and linguistic processing. It is notable in this connection that savants with advanced computational abilities may be linguistically impaired, just as many linguistically gifted people can’t handle mathematics.

And what are we to make of birds that keep track of the comings and goings of their potential prey by counting and remembering how many entered or exited the burrow they are spying on? These predators would not be much interested in questions about the concept of number, but their counting abilities derive from a pattern of neural processing which necessarily represents or instantiates the concept in some form. Arguably, some such primitive, pre-linguistic and pre-theoretical notion of number underlies even our most sophisticated mathematical ideas and capacities.

23 Comments »

  1. And what are we to make of birds that keep track of the comings and goings of their potential prey by counting and remembering how many entered or exited the burrow they are spying on?

    I’m inclined to think it a mistake to read too much into this. Counting is just a kind of measuring — a way of getting information about the environment. You don’t really have a concept of number, until you use numbers without engaging in counting.

    Overall, I thought this was a pretty good essay.

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    • Neil

      “I’m inclined to think it a mistake to read too much into this [birds counting].”

      Am I reading too much into it?

      I always like to take things back to basics and it just seems intrinsically interesting that you have birds apparently able to do something that seems to involve addition (and subtraction). I am not saying that these cognitive processes involve conscious awareness of the calculations involved, but these are processes which plausibly bear some relation to our own counting abilities and so could be seen to represent some kind of precursor or prerequisite for more advanced arithmetical abilities.

      The concept of number derives from counting (objects or iterations), I take it. So for the concept (or proto-concept) to be instantiated you need at the very least to have a world with things in it, and/or within which things happen (and happen again); and you need to have some observer taking note.

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      • Am I reading too much into it?

        Yes, I think you are.

        A child can count on his fingers without needing a concept of numbers. You can perhaps say that it depends on a concept of quantity, but that’s not the same as a concept of numbers.

        It is one thing to say “two apples plus three apples gives you five apples.” It is a different thing to say “2+3=5.” Only in that second case, have you made numbers into abstract entities in their own right.

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        • Neil

          “A child can count on his fingers without needing a concept of numbers. You can perhaps say that it depends on a concept of quantity, but that’s not the same as a concept of numbers.”

          I am using the phrase “concept of number” to mean something different from what you mean by “concept of numbers,” I think.

          Also, finger counting involves a degree of abstraction. You are not just counting fingers, you are using the fingers *as symbols*. It is not a great leap from finger counting to Roman numerals.

          “It is one thing to say “two apples plus three apples gives you five apples.” It is a different thing to say “2+3=5.” Only in that second case, have you made numbers into abstract entities in their own right.”

          Yes, there is a distinction between counting objects in the world and doing abstract mathematical calculations. But you seem to want to restrict the concept of number to abstract mathematics consciously pursued as such. Such an approach, it seems to me, is unnecessarily restrictive and fails to do justice to the concept. For one thing, “two”, “three” and “five” are *number* words, are they not? Some kind of number concept is obviously in play here.

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          • I am using the phrase “concept of number” to mean something different from what you mean by “concept of numbers,” I think.

            Yes, I agree. But if you are using “number” to refer to something very simple, then why bring it up? Why should we even care whether birds have something that simple?

            For one thing, “two”, “three” and “five” are *number* words, are they not?

            A child can count on fingers before learning number words. Counting, at a basic level, is a simple sequencing operation, much like “eeny meeny miny moe” or like playing hopscotch or like knitting. The birds ability to build a robust nest seems far more impressive.

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  2. “The philosophical arguments against realism with which I am most familiar (including Carnap’s view) have, as I see it, a fatal flaw. They give too great a prominence to language and fail to give sufficient recognition to non-linguistic aspects of human thought and understanding.”

    But semiotics (not semiology) begins with more than sufficient recognition of non-linguistic aspects of human thought and understanding. And while Pierce hoped that this would develop a logic predicated on a Moderate Realism, in fact there is a powerful argument derivable from semiotics against realism (the indefinite interconnection between signs leading to a multiplication of possible interpretations). And indeed interpretation seems to me the real problem here, however modeled.

    My own sense of it is that knowledge works something in the manner of choice as described by Thomas Aquinas – as a resting point rather than a certitude.

    “But on what basis – other than practicality or convenience – do we draw dividing lines between these different kinds of process and, by extension, between disciplines?” Practicality and convenience may be the best we can hope for.

    Still, I must acknowledge that this has been your most generous and yet strongest paper on these issues.

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    • ejwinner

      Thanks.

      “My own sense of it is that knowledge works something in the manner of choice as described by Thomas Aquinas – as a resting point rather than a certitude.”

      Choice? That sounds like we might unilaterally decide what is the case rather than determining (or trying to determine) what is the case by observation, experiment, scholarly activity, etc.. Interpretation is always an issue but I suspect we have different views on its significance, ramifications, etc..

      Certitude may well be out of reach. I am comfortable with that.

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  3. Very clear. A few minor points:

    Some metacognitive faculties are shared with other apes eg

    https://www.ncbi.nlm.nih.gov/pubmed/30068995

    “…counting and number clearly take us beyond the bounds of natural language and linguistic processing…”

    https://www.ncbi.nlm.nih.gov/pubmed/28744252

    The approximate number system (ANS; Dehaene, 1997; Halberda et al., 2008). The ANS is thought to be present from birth (Izard et al., 2009), functional in highly educated and uneducated people (Pica et al., 2004; Nys et al., 2013), and non-human animals (Cantlon and Brannon, 2006;Agrillo et al., 2008) and hence independent of language. Similar
    to math, the ANS relies on regions of parietal cortex…

    But, as very basic informational processing, just because different brain regions are involved in music, math and speech, doesn’t exclude them of all being language in nature. Your mention of the lambda calculus reminds me of “it from bit” – language for everything..

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    • David

      Thanks. And thanks for the links.

      Those monkeys that “knew when they knew” the correct response were certainly engaged in some level of metacognition.

      On the other issue you raise, you say that the evidence suggests that the ANS and maths are independent of language (I agree), but then you seem to want to extend language to cover these things (and more). Why not just say that mathematical processing is independent of *natural* language processing? Broader conceptions of language take in other systems, but I am unclear how far you can usefully push this.

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  4. You’ve made many of these sorts of anti-metaphysical and Realist statements before, but what I miss are any serious arguments for them. Until I see some real engagement with the arguments of anti-realists and those like myself who embrace a pluralist metaphysics, all I’m left with are your assertions and an account of some of the personality traits that explain your embrace of them, in the genealogical sense of ‘explain’.

    It would be refreshing if you actually engaged with the arguments of your opponents, many of which I’ve made myself, in essays here in the Agora, as well as in peer-reviewed work, but which one also finds in the work of Quine, Davidson, Putnam, and others. After all arguments, not theses or positions, are the primary business of philosophy.

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    • Dan

      I deliberately tried to keep my claims modest in his piece and as far as possible non-metaphysical (or metaphysically agnostic).

      At the heart of what I am saying is this claim: “It is a mistake to imagine that we are ever entirely locked into specific and rigid linguistic or theoretical frameworks.” Such a view is in accordance (I would say) both with ordinary experience and with what we have learned in scientific contexts about human cognition and metacognition. I specifically highlighted the notion of metalinguistic awareness.

      I also mentioned Gödel’s theorems as possibly providing support for my basic claims.

      If I were to do a piece on the philosophical issues to which you allude I would have to engage directly with the post-Carnap philosophical literature on these topics. As you know, my interests and expertise (such as it is) do not extend to this daunting and very extensive body of work. Popper’s “The Myth of the Framework” could be a possible starting point for me, however.

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  5. (1) It might interest you to know that there’s a family of metaphilosophical positions (articulated by Amie Thomasson, Guiseppina D’Oro, and others) claiming that metaphysical arguments should be understood as metalinguistic negotiation. The idea (in their eyes but my words) is that arguments that were traditionally seen as arguments about what or how things really are should be seen rather as arguments about what things should be counted as, or how things should be conceptualized. They have several ways of responding to the complaint that this turns “substantive” claims into “mere ways of speaking.” But one might also point to Quine’s arguments in favor of being less sanguine about our ability to clearly and cleanly distinguish (i) learning to conceptualize in a certain way from (ii) learning about that which is conceptualized in that way.

    (2) You might find inspiration in Kant, who argued that our mathematical knowledge has its origin in our preconceptual, embodied point-of-view-ish-ness. He argued that our competence with counting, which grounds our most sophisticated mathematical knowledge, emerges from a nonconceptual acquaintance with spatiotemporal sameness and difference. Of course he recognizes that full-blown mathematical understanding involves conceptual articulation. But no one did more to upset the rationalist assumption that mathematics could be unspooled by a disembodied mind from a handful of innate concepts.

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    • Animal Symbolicum

      Thanks for your suggestions. Amie Thomasson’s book Ontology Made Easy looks interesting. She draws heavily on Carnap, I see. And Giuseppina D’Oro defends a pre-Davidsonian position, drawing on Collingwood. I am wary of Collingwood’s idealism, however (as I am of Kant’s).

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  6. Hi Mark: This is an excellent piece, in my opinion. It is grounded in your own research, but it is still very relevant to us today.

    So here is a classical reply. Nothing can be both red and green all over at the same time. This is a true proposition, but it is not based upon experience. Nothing could verify or falsify it. Yet it is not trivial or meaningless or even tautological. Does that not open the door to a sort of “metaphysics” that you would wish to prohibit or otherwise deny any credibility?

    Alan

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    • Thanks Alan.

      You are getting at some deep issues in respect of logic, thought and language. You can call them metaphysical if you like. My approach can accommodate this.

      I made the point that important questions about fundamental issues arise naturally in the course of various forms of intellectual inquiry and to call these questions philosophical and in certain cases metaphysical is quite appropriate.

      Investigations into the logic of ordinary thinking, language, formal logical systems and the philosophy of logic can all certainly fit within the scope of what I am talking about.

      I would be inclined to see examples like the one you offer as indicating something about the nature of our understanding of the logical structure of reality.

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      • Your phrase “the logical structure of reality” surprises me. You seem to be endorsing the idea that there is such a thing. But isn’t it central to positivism (including logical positivism) that there is no logical structure of reality? I think that view is correct. Logic operates only in the linguistic domain.

        The red/green story is one of the points that turned Wittgenstein away from logical positivism. But he also denied that it led to any sort of metaphysics. Here’s one passage:

        Let us take the assertion: “No object is red and green simultaneously”. Do I mean by this that I have never hitherto seen such an object? Obviously not. I mean: “I cannot ever see such an object”, “Red and green cannot be in the same place”.  Here I would simply ask: What does the word “can” mean here? The word “can” is obviously a grammatical (logical) concept, not a factual one. Assume now that the assertion “An object cannot be red and green” is a synthetic judgement, and the words “cannot” mean logical impossibility. Then, since a proposition is the negation of its negation, there must also be the proposition “An object can be red and green”. This proposition would be likewise synthetic. As a synthetic proposition it has sense, and that means that the state of affairs it depicts can exist. So, if “cannot” means logical impossibility, we come to the conclusion that the impossible is possible after all. The only way out which remains for Husserl is for him to explain that there is a third sort of possibility. To this I would reply: it is impossible to invent words, but I cannot think of any thoughts to go with them. (Quoted in Kenny, Wittgenstein, 89.)

        It seems that at that stage (around 1929) he hadn’t yet found his own “third possibility”, namely the theory of language as a form of social practice.

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  7. I’ll mention Rico the border collie again,
    https://www.researchgate.net/publication/8515648_Word_Learning_in_a_Domestic_Dog_Evidence_for_Fast_Mapping

    who could infer that a novel word referred to the novel item among the collection where he already knew the names of all the other objects. Mark has already mentioned the birds counting the comings and goings of prey into a burrow – ie they infer if 2 in, 1 out, then 1 must still remain to come out. I think it is Lakos and Nunez who explain that Modus Ponens is a formalisation of the preverbal concept of nested container-ship (A reliably contains B; if A then B). Lakos and Nunez also report that “at about seven months, babies can recognize the numerical equivalence between arrays of objects and drumbeats of the same number” ie small number exist abstractly (I believe there are similar results for other animals).

    All this is to say that we can naturalize logical space, including providing it with a success semantics.

    I was first inclined to wonder if Wittgenstein was taking the piss regarding deuteranopia (see also the McCulloch illusion).

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  8. Alan

    “Your phrase “the logical structure of reality” surprises me. You seem to be endorsing the idea that there is such a thing. But isn’t it central to positivism (including logical positivism) that there is no logical structure of reality?”

    The full phrase, by the way, was “… *our understanding of* the logical structure of reality…” So my implicit claim would be slightly different from what you imply.

    Leaving aside the historical question, you seem to be assuming that I am a positivist or logical positivist. I endorse the view (held by the logical positivists) that metaphysics can not be a serious, stand-alone discipline. And I am sympathetic to other aspects of logical empiricism. But I am not endorsing everything they said or believed.

    “Logic operates only in the linguistic domain.”

    Natural languages and other basic aspects and modes of human cognition incorporate implicit or explicit logics which arguably derive (at least in part) from experiences of one kind or another.

    What is more, animals routinely do clever things without the benefit of language. I always liked that story of Chrysippus’s dog. Whether you interpret it as disjunctive syllogistic reasoning or probabilistic reasoning based on cognitive maps, this sort of thing certainly appears to involve something analogous to human reasoning (even if it is unconscious). Arguably our basic reasoning capacities derive ultimately from various sorts of non-linguistic or pre-linguistic cognition.

    I take it that David Duffy, with his canine and human baby examples, sees these things in a similar way to the way I see them.

    With your (or LW’s or Husserl’s) red/green story, I see Wittgenstein’s point about “can” but I am not sure (even leaving aside perceptual complications relating to color) that this example needs to be seen as being entirely linguistic. Think of a similar type of example: Schrödinger’s cat. Could a cat actually be in a superposition of states (dead/alive)? This question is about how the world is, not just about language.

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    • “Logic” as I understand it requires assertions and denials, conjunctions and disjunctions, conditionals, inferences, objections, and other suchlike linguistic formations. There are no such things in the non-linguistic world. Hence my view.

      Do these logical structures “derive (at least in part) from experiences of one kind or another”? No, there is no experience of a disjunction or of an inference or of a counter-argument. Experience is just not like that. What would such things look like or feel like?

      Do concepts derive from “experiences of one kind or another”? No. That conceptual empiricism is what LW was trying to refute. After the red/green argument he went on to the failure of ostensive definition argument. To my mind these were big advances in philosophy. The outcome was neither “realism” nor “anti-realism” but something else again. It had a three-part structure involving language as a form of social practice lying between experience and human needs.

      What about intelligent apparently non-linguistic animals? I’ll defer to the experts such as David on that.

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      • Hi Alan,

        I am a little surprised that there is such as a “non-linguistic world” because everything that we know exists has been described, but I guess I shouldn’t be.

        Everything that I as a human being am aware of is arguably due to its logical structure. Otherwise I would not be aware of it. Mathematics does a good job of describing the entire observable universe, hence the Mathematical Universe Hypothesis. The unobservable universe, Dark Matter and Dark Energy?, can not yet be modeled by maths or be described by ordinary language. Unobservable everyday things such as feelings are not mathematical or empirical, but are still exquisitely logical – my West Highland terrier lets me know when he is hungry or in need of a bathroom break. When I say squirrel, he looks up into the trees. If our non-linguistic communications were not logical, I could never have a clue about what he is signaling, and vice versa.

        The MUH and panpsychism’ are attempts at referring to the logic in everything, organic and inorganic. Timing and structure are essential to everything. When we and all living creatures ‘look’ at it, it makes sense, it’s logical.

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  9. Hi Mark,

    I think you are correct, as I understand you. Purely metaphysical statements cannot be verified, but they could be the stuff of dreams: Justice loves beauty and begets truth.

    A true sentence ‘is not the truth’. We constantly utter conflicting true statements. I am therefore very curious, are you planning to deal with the political implications of our profound human abilities and limitations – our contradictions?

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  10. Alan

    There is obviously a big gulf between your view on these matters and mine. Comments automatically close after 10 days so looks like we won’t be able to pursue these matters here.

    You wrote:

    ” “Logic” as I understand it requires assertions and denials, conjunctions and disjunctions, conditionals, inferences, objections, and other suchlike linguistic formations. There are no such things in the non-linguistic world. Hence my view.”

    You can define logic in various ways. As a (normative) academic discipline, for example. But inference (which you mention) relates to non-linguistic behavior as well as linguistic behavior. You are probably thinking of deductive inference (monotonic). But non-monotonic logics deal with everyday reasoning. We routinely draw inferences which are not articulated but which manifest themselves in our behavior. And similar patterns of behavior are also evident in non-linguistic animals.

    I am curious about how our thinking patterns (including “logic”) actually developed in an evolutionary sense and also in terms of ontogenesis. The parallels between our basic thinking patterns and that of non-linguistic animals cannot just be ignored. I can’t see the motivation for restricting the notion of logic in the way you are doing.

    “Do these logical structures “derive (at least in part) from experiences of one kind or another”? No, there is no experience of a disjunction or of an inference or of a counter-argument. Experience is just not like that. What would such things look like or feel like?”

    I claimed that our logical structures *derive from* experiences, not that they *are* experience(s).

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