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Joan Moschovakis' SEoP article "Intuitionistic Logic" claims:

"Although intuitionistic analysis conflicts with classical analysis, intuitionistic Heyting arithmetic is a subsystem of classical Peano arithmetic. It follows that intuitionistic propositional logic is a proper subsystem of classical propositional logic, and pure intuitionistic predicate logic is a proper subsystem of pure classical predicate logic."

What is the meaning of 'pure' in this quote, as well as this answer to the question Equivalence between fragments of intuitionistic and classical logics? (Another related response to the question Does the intuitionistic logic contain classical (in a sense)? but I think it only applies to propositional/predicate logic.) Can we generalize Moschovakis' statement to "intuitionistic logic is a proper subsystem of classical logic"? Or is the jump from first to second-order logic (e.g, characterizing $\mathbb{R}$) where the subsystem relation breaks down? Something related to the 'pure' property?

Tankut Beygu
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shea
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    I would interpret "pure" to mean atomized in this context, so in its simplest form. I don't think the word pure carries any significance here and you can just ignore it in the sentence without any loss. – CyclotomicField Jun 01 '24 at 23:02
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    @CyclotomicField That's not quite true, "pure" has a technical meaning here. It means that we're looking at the "axiom-free" fragment - that is, "pure intuitionistic logic is contained in pure classical logic" is shorthand for "every intuitionistic tautology is a classical tautology." – Noah Schweber Jun 02 '24 at 03:22

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By designating intuitionistic predicate logic in the given context with "pure," Joan Moschovakis indicates that it does not include any additional non-logical axioms (arithmetic, set-theoretical etc.) and the vocabulary items accordingly needed (i.e., term-forming functions and individual constants; viz., individual variables are the only terms).

Thus, with respect to the axiomatisation owing mainly to Stephen Kleene (see his Introduction to Metamathematics), the pure definition of intuitionistic predicate (quantified) calculus, IQC consists of the inference rules R1-R3 and the axiom schemas Ax1-Ax12, and that of intuitionistic propositional calculus, IPC, consists only of propositional letters as atomic formulas, the inference rule R1 and the axiom schemas Ax1–Ax10:

Rules of Inference

R1. Modus Ponens: From $A$ and $A\rightarrow B$, conclude $B$.

R2. Universal Generalisation: From $C\rightarrow A(x)$, where $x$ does not occur free in $C$, conclude $C \rightarrow\forall xA(x)$.

R3. Existential Introduction: From $A(t)\rightarrow C$, where $x$ does not occur free in $C$, conclude $\exists xA(x)\rightarrow C$.

Logical Axiom Schemas

Ax1. $A\rightarrow (B\rightarrow A)$

Ax2. $(A\rightarrow B)\rightarrow((A\rightarrow(B\rightarrow C))\rightarrow (A\rightarrow C))$

Ax3. $A\rightarrow(B\rightarrow A\wedge B)$

Ax4. $A\wedge B\rightarrow A$

Ax5. $A\wedge B\rightarrow B$

Ax6. $A\rightarrow A\vee B$

Ax7. $B\rightarrow A\vee B$

Ax8. $(A\rightarrow C)\rightarrow((B\rightarrow C)\rightarrow(A\vee B\rightarrow C))$

Ax9. $(A\rightarrow B)\rightarrow((A\rightarrow\neg B)\rightarrow\neg A)$

Ax10. $\neg A\rightarrow(A\rightarrow B)$

Ax11. $\forall xA(x)\rightarrow A(t)$

Ax12. $A(t)\rightarrow\exists xA(x)$

Hence, the following, where $0$ is an individual constant, $'$ is a unary function symbol, $+$ and $\cdot$ are binary function symbols, all of which are familiar to us from Peano Arithmetic, are left out:

Induction Schema

Ax13. $A(0)\land\forall x(A(x)\rightarrow A(x'))\rightarrow\forall xA(x)$

Arithmetic Axioms

Ax14. $(a'=b')\rightarrow(a=b)$

Ax15. $\neg(a'=0)$

Ax16. $(a= b)\rightarrow((a=c)\rightarrow(b=c))$

Ax17. $(a=b)\rightarrow(a'=b')$

Ax18. $(a+0)=a$

Ax19. $(a+b')=(a+b)'$

Ax20. $(a\cdot 0)=0$

Ax21. $(a\cdot b')=(a\cdot b)+a$

Tankut Beygu
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