Rational Human Mindedness and The Hierarchy of Increasingly Rich Formal Systems.

Robert Hanna

Kurt Gödel, circa 1931, and the first page of the revolutionary article that explicitly demonstrated the incompleteness of mathematical logic and also implicitly vindicates human rationality.

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You can also download and read or share a .pdf of the complete text of this essay HERE.

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Rational Human Mindedness and The Hierarchy of Increasingly Rich Formal Systems

Ever since I wrote a brace of books during the 00s called Kant, Science, and Human Nature (Hanna, 2006a) and Rationality and Logic (Hanna, 2006b), I’ve been deeply interested in the thesis that there are necessary and indeed essential connections between rational human mindedness and formal systems.

What are these connections?

In Rationality and Logic, and in a later book from the 10s, Cognition, Content, and the A Priori, I argued that every actual and possible logic is cognitively constructed by rational (human) animals, and that logic itself is the set of categorically normative, innately specified first principles of human theoretical rationality — a universal a priori minimally non-contradictory proto-logic — when taken together with all the supplementary humanly-constructed ceteris paribus principles of an open-ended plurality of logical systems, just as morality is the set of categorically normative, innately specified first principles of human practical rationality — a universal a priori dignitarian proto-morality — when taken together with all the supplementary humanly-constructed ceteris paribus principles of an open-ended plurality of moral systems (the morality-of-logic thesis) (Hanna, 2006b, 2006c, 2015: ch. 5).

But in this essay, I want to pursue another route to an even broader yet essentially similar conclusion, that includes mathematical formal systems and also any physical system or physical theory that presupposes or uses basic arithmetic.

What I’ve called weak transcendental idealism says that (i) necessarily, the basic metaphysical and ontological structures of the manifestly real world conform to the basic innate structures of our rational human cognitive capacities, especially including consciousness and self-consciousness, (ii) necessarily, if the manifestly real world exists, then if conscious and self-conscious rational human cognizers were to exist, they would be able to cognize that world veridically to some salient extent, which in turn is fully consistent with conscious and self-conscious rational human cognizers not actually existing at any given time, and (iii) the existence of the manifestly real world necessitates the real possibility of conscious and self-conscious rational human cognizers, but not their actual existence at any given time (Hanna, 2015: section 7.3).

Nevertheless, as a matter of contingent fact, some conscious and self-conscious rational human cognizers do actually exist now, simply because they do actually think and can also know that the statement “Necessarily, if I think, then I exist” is true a priori: for example, the writer of this sentence and all sapient readers of it.

Weak transcendental idealism is philosophically justified by inference-to-the-best-explanation, because it’s is the best overall explanation of skepticism-resistant human knowledge, especially a priori knowledge (Hanna, 2015: chs. 6–8).

Weak transcendental idealism is also necessarily equivalent with what I’ve called the moderate anthropic principle for cosmology, which says that our real possibility as conscious and self-conscious rational human cognizers is built into the nomological structure of the cosmos from the Big Bang forward, although, again, this does not entail that we must actually exist at some time, even though, as a matter of contingent fact, we do actually exist now (Hanna, 2021a, 2022: section 4.4).

Relatedly, I also hold a thesis I’ll call potentiality panpsychism, which says that necessarily, all parts of the cosmos are either (i) actually organismic minded systems, (ii) actual societies of organismic minded systems, (iii) actually or at least potentially, proper parts of organismic minded systems, or (iv) actually or at least potentially constitutive necessary conditions for organismic minded systems.

Potentiality panpsychism is weaker than classical panpsychism or panexperientialism, since potentiality panpsychism doesn’t require that everything in the cosmos actually has consciousness, proto-consciousness, or experiences; but at the same time potentiality panpsychism is also strong enough to entail the neo-organicist worldview, provided we also add to potentiality panpsychism the thesis that all organisms, even unicellular ones, are at least proto-conscious.

In turn, the neo-organicist worldview says that (i) the cosmos is essentially processual, purposive, and self-organizing, and (ii) there’s a single, unbroken metaphysical continuity between The Big Bang Singularity, temporally asymmetric/unidirectional non-equilibrium negentropic thermodynamic matter/energy flows, organismic life, conscious minded animals of all kinds including conscious minded human animals, human free agency, human rationality, and human dignity (Torday, Miller Jr, and Hanna, 2020; Hanna, 2022).

Now, there’s a hierarchy of formal systems, running from decidable formal systems to incomplete formal systems, that (i) mirrors their degree of semantic and structural richness from least rich to most rich and also (ii) is necessarily and indeed essentially connected with weak transcendental idealism, the moderate anthropic principle for cosmology, potentiality panpsychism, and the neo-organicist worldview.

Let’s begin with elementary logic (Mates, 1972), and more specifically with that fragment of elementary logic known as sentential logic, aka SL.

SL is consistent (i.e., it contains no contradictions), sound (i.e., all provable sentences or theorems are true sentences), complete (i.e., all true sentences are provable sentences or theorems), and decidable (i.e., there’s a computable procedure for determining the truth or falsity of any sentence).

And this is all because SL is simply the logic of boolean (binary-coded) truth-functions.

Moreover, SL holds even if there are no objects in the world, i.e., even for empty domains, since all that matters is the truth or falsity of the sentences, without any appeal to semantic devices for reference to objects, one-place properties, or many-place relations .

From a broadly Kantian point of view, all true sentences (tautologies) of SL are analytically true according to Kant’s “supreme principle,” or necessary and sufficient contradiction-criterion, for analyticity: a sentence is analytically true if and only if its denial entails a contradiction (Kant, 1997: pp. 279–281, A150–153, B189–193; Hanna, 2001: ch. 3, 2015: ch. 4).

Now, the next formal system in our hierarchy, monadic predicate logic, aka MPL, another fragment of elementary logic, with unary first-order quantification into one-place predicates, is also consistent, sound, complete, and decidable, for essentially the same reasons that SL is consistent, sound, complete, and decidable (Boolos and Jeffrey, 1989: ch. 25); and just like SL, MPL also holds even if there are no objects in the world, i.e., even for empty domains.

But MPL can also be supplemented by decomposable intensions assigned to its one-place predicate-terms, which allows for sentences to be analytically true according to Kant’s sufficient but not necessary containment-criterion or identity-criterion for analyticity: a sentence of categorical subject-predicate form is analytically true if either (i) its predicate-intension is contained in its subject-intension or (ii) its predicate-intension is identical to its subject-intension (Kant, 1997: pp. 141–142, A6–7/B10–11; Hanna, 2001: ch. 3).

Let’s call this intensional-analyticity-supplemented version of MPL, MPL*.

MPL* is equivalent to Kant’s pure general logic (Kant, 1997: pp. 194–195, A52–55/B76–79; Hanna, 2015: ch. 4, 2021b).

And, just like SL and MPL, MPL* or pure general logic also holds even if there are no objects in the world, i.e., even for empty domains.

Now, polyadic predicate logic with identity, aka PPLI, which is elementary logic itself, with multiple first-order quantification into many-place predicates, is consistent, sound, complete — and therefore all truths of PPLI are provable sentences or theorems — but not decidable (Church, 1936).

Correspondingly, in my opinion, there are two independently sufficient reasons why PPLI/elementary logic isn’t decidable, even though it’s still complete.

First, truth-functionality alone isn’t sufficient to determine the truth or falsity of every sentence in PPLI/elementary logic: given the law of (token-token) identity as a law of logic,

(x) x=x

one of whose substitution-instances is

a=a

at least one object has to exist in the world, namely the object named by “a”; and, given the truth of any relational sentence containing a two-place relational predicate with two distinct terms naming different objects, for example,

R2ab

at least two objects have to exist in the world, and so-on for all n-adic relational predicates.

In short, in PPLI/elementary logic, domains have to be populated: so, necessarily, it has existential commitment.

For Kant, existential commitment, which is guaranteed only by intuitional cognition, not conceptual cognition, determines the difference between analyticity and syntheticity: so, from a broadly Kantian point of view, PPLI/elementary logic is a synthetic logic, not an analytic logic, and therefore it falls outside pure general logic (Hanna, 2001: chs. 4–5).

Second, in order to do proofs beyond decidability in PPLI/elementary logic, both rules and also rule-following are needed, and, as it turns out, rule-following is an irreducibly normative activity that presupposes conscious and self-conscious rational cognizers, as members of a community of language-users, whenever it exceeds mere boolean computability, as the later Wittgenstein’s and also Saul Kripke’s or “Kripkenstein’s” solution to the rule-following paradox shows (Kripke, 1982; Hanna, 2006b: pp. 158–168, 2021c: ch. XII).

The rule-following paradox has two versions: (i) no rule can be applied without requiring another rule to interpret the first rule, and so-on, which entails a vicious regress of rules (Kant also explicitly recognized this problem at 1997: pp. 268–269, A132–134/B171–174) (ii) given only a finite and denumerable set of inputs (arguments) and outputs (values) to a given function, i.e., given only a partial function, nevertheless an indeterminately large number of different rules can describe the same set of inputs/outputs and also diverge on subsequent inputs/outputs, so therefore the different rules will determine different complete functions, even given the same partial functions, and there’s no purely mechanical or naturalistic way to avoid this underdetermination of the rules by those partial functions.

Moreover, according to later Wittgenstein and “Kripkenstein” alike — and I strongly agree — in order to fix the interpretation and applications of a rule, language-using communities of conscious and self-conscious rational human cognizers are needed in order to figure out “how to go on” and therefore also to decide what counts as correct, as opposed to incorrect, applications of the rule (Hanna, 2006b: 158–168, 2021c: ch. XIII).

Anticipating later Wittgenstein and “Kripkenstein” by roughly 200 years, Kant calls this very same ability “so-called mother-wit” or Mutterwitz, and in turn this ability is essentially bound up with the innate conscious and self-conscious rational human capacity for judgment, or what he calls “the natural power of judgment” or natürlicher Urteilskraft (Kant, 1997: pp. 268–269, A133–134/B172–174), not only for “determining” judgment, which subsumes individual objects under given general concepts, but also and especially for “reflecting” judgment, which projects from individual objects to general concepts, either inductively or abductively (Kant, 2000; Hanna, 2017).

Moreover, once we have conscious and self-conscious rational human cognizers in play as members of language-using communities, then, as per the above, in order to explain skepticism-resistant a priori knowledge, weak transcendental idealism is required and therefore also the moderate anthropic principle for cosmology is required, since they’re necessarily equivalent (Hanna, 2015: chs. 6–8, 2021a, 2022: section 4.4).

Next in the hierarchy of richness, there’s a formal system of arithmetic, primitive recursive arithmetic, aka PRA (Skolem, 1967), that’s weaker than full-dress Peano Arithmetic, because, unlike PA, PRA contains no quantifiers, that’s also consistent, sound, complete, and decidable, if we extend decidability from mere boolean computability to all recursive functions, as per The Church-Turing Thesis, which says that necessarily, all computable functions are recursive functions and conversely (Turing, 1936/1937; Church, 1937).

Like PPLI/elementary logic, from a broadly Kantian point of view, PRA is also synthetic, not analytic, since it necessarily requires populated domains and existential commitment to numbers or sets.

From a broadly Kantian point of view, then, when we advance from SL to PRA, we’ve advanced from analytic (mere boolean) computability to synthetic (Church-Turing) computability.

Significantly, PRA is also subject to the rule-following paradox, as per later Wittgenstein and “Kripkenstein,” hence rule-following in PRA is also irreducibly normative and presupposes conscious and self-conscious rational human cognizers, and therefore also invokes weak transcendental idealism and the moderate anthropic principle for cosmology.

But, now moving on to the richest formal system in our hierarchy, Peano Arithmetic, aka PA — which consists of any logical system essentially like that presented in Whitehead’s and Russell’s Principia Mathematica (Whitehead and Russell, 1962), plus the basic Peano Axioms for arithmetic, for example: (i) 0 is a number, (ii) the successor of any number is a number, (iii) no two numbers have the same successor, (iv) 0 is not the successor of any number, and (v) any property which belongs to 0, and also to the successor of every number which has the property, belongs to all the numbers — is consistent, sound, and incomplete (if consistent), and contains not only undecidable, but also unprovable true sentences (Gödel, 1967).

This is of course because Gödel showed in 1931 that there are true but undecidable, unprovable sentences in PA when we map the truth-definition for PA into PA itself, using the formal method of assigning self-referring Gödel-numbers to sentences (the first incompleteness theorem); moreover, this proof requires non-denumerably infinite or transfinite numbers or sets, as per Georg Cantor’s argument for the existence of the transfinite domain (Cantor, 1891, 2019; Gödel, 1967).

So, in addition to the fact that any system richer than mere boolean computable systems like SL requires irreducible rule-following normativity and therefore presupposes conscious and self-conscious rational human cognizers as members of language-using communities, hence also invokes weak transcendental idealism and the moderate anthropic principle for cosmology, the necessarily populated domains and existential commitments of PA (guaranteeing syntheticity) also exceed denumerable infinity and therefore also exceed not only mere boolean computability, but also all computability that satisfies the Church-Turing thesis (Turing, 1936/1937; Church, 1937), and therefore is non-mechanical or organic, since Turing-computability is a necessary condition of formal mechanicity and natural mechanicity alike (Hanna, 2018: ch. 2, 2022: section 2.4).

Now, because mapping the truth-definition for PA into PA itself, entails the existence of undecidable, unprovable true sentences, then, assuming consistency, every formal system that’s at least as rich as PA cannot demonstrate its own consistency (the second incompleteness theorem), hence it requires that its consistency and the truth of its sentences be established outside that system itself, for example, as Gödel himself held, by acts of mathematical intuition that constitute a priori mathematical knowledge, and therefore by conscious and self-conscious rational human cognizers (Gödel, 1967; Tait, 2010).

Therefore, the existence of true but undecidable, unprovable sentences in PA entails (i) the first and second incompleteness theorems, (ii) the necessity of populated domains and existential commitment (hence syntheticity), (iii) non-denumerably infinite or transfinite domains, and (iv) rule-following for the provable but undecidable sentences, all of which jointly entail or at least strongly suggest — in the particular case of what secures consistency and the truth of undecidable and unprovable mathematical axioms, as per Gödel’s own view — (v) conscious and self-conscious rational human cognizers as members of language-using communities, hence also jointly necessitate (vi) weak transcendental idealism and (vii) the moderate anthropic principle for cosmology, and in turn jointly entail (viii) Turing-incomputability and (ix) anti-mechanicity or organicity, therefore also jointly entail (x) the neo-organicist worldview, if we add the thesis of potentiality panpsychism to weak transcendental idealism and the moderate anthropic principle for cosmology.

Finally, then, it follows that any formal system that’s rich enough to contain the Peano axioms for PA, and any physical system or physical theory that presupposes or uses PA, will also have all ten of these fundamentally important features.

Or in other words, every formal system and physical system or physical theory that’s semantically and structurally rich enough to provide an even minimally adequate model or theoretical representation of the manifestly real world in its logical and/or mathematical specific character, will be necessarily and indeed essentially connected with rational human mindedness.[i]

NOTE

[i] I’m grateful to Scott Heftler for thought-provoking conversations on and around the topics of this essay.

BIBLIOGRAPHY

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(Cantor, 2019). Cantor, G. “A Translation of G. Cantor’s ‘Ueber eine elementare Frage der Mannigfaltigkeitslehre’.” Trans. P.P. Jones et al. 23 August. Available online HERE.

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