## Abstract

In the first part, we discussed the reasons that have led Prawitz to the adoption of the theory of grounds, as well as how this theory appears in the writings that Prawitz has so far devoted to his ground-theoretic approach.

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## Notes

- 1.
The restriction to the first-order does not disregard the great interest of an approach that includes expansions of languages of grounding with operational symbols for primitive operations related to logical constants of higher order than the first. This is a point that we will leave out for mere reasons of space, and that could be fruitfully developed later. In a sense, our discussion will be equivalent to a formal framework for the analysis of

*first-order inferential validity*; hence, to authorize expansions with operational symbols for primitive operations on grounds would be equivalent to a formal framework for the analysis of*inferential validity of**n*-*th order*—namely of*inferential validity as such*. - 2.
We have omitted a few details to avoid lengthening our discussion. These details are not essential to a general understanding of the formal development proposed here, but for completeness, we outline them in this footnote.

First, consider the case of operations on grounds of second level

*f*that*vacuously*bind*ξ*^{α}, for*α*occurring in the domain of some entry of their operational type. This requires the ground-variables of the linguistic expression of an operation on grounds of first-level*h*to be considered as having an index (other than that concerning the entry of the operational type where the ground-variable occurs), and that*f*bind ground-variables*of a specific index**i*(other than that concerning the entry of the operational type where the ground-variable occurs). The binding is then vacuous when*ξ*^{α}has in*h*an index*j*≠*i*. The definition of identity requires a slight modification to take into account vacuous binding of ground-variables. For example, \(\rightarrow I \xi ^{\alpha }_2(\rightarrow I \xi ^{\alpha }_1(\xi ^{\alpha }_1))\) must not be identical—nor equivalent—to \(\rightarrow I \xi ^{\alpha }_1(\rightarrow I \xi ^{\alpha }_2(\xi ^{\alpha }_1))\), although both are grounds for ⊢*α*→ (*α*→*α*). This can be satisfied by requiring that for two operations on grounds of second level to be identical, there must be a “structure-preserving” re-indexing of the ground-variables such that all ground-variables have the same indices in the two cases—e.g. \(\rightarrow I \xi ^{\alpha }_2(\rightarrow I \xi ^{\alpha }_1(\xi ^{\alpha }_1))\) and \(\rightarrow I \xi ^{\alpha }_3(\rightarrow I \xi ^{\alpha }_1(\xi ^{\alpha }_1))\) can be re-indexed to \(\rightarrow I \xi ^{\alpha }_4(\rightarrow I \xi ^{\alpha }_1(\xi ^{\alpha }_1))\).We have also not defined what we can call

*variables-variants*, which are important when considering nested compositions of operations that bind individual variables. The definition is as follows in the first-level case (the second-level case is analogous): given a*B*-operation on grounds \(f([ \underline {x}_1, \ldots ,\underline {x}_{n + 1} -\underline {x}] \\underline {y}, \xi ^{\alpha _1( \underline {x}_1)}, \ldots , \xi ^{\alpha _n( \underline {x}_n)})\) of operational type \(\alpha _1( \underline {x}_1), \ldots , \alpha _n( \underline {x}_n) \rhd \beta ( \underline {x}_{n + 1})\), let \( \underline {w}\) be a sub-sequence of \( \underline {x}_1, \ldots ,\underline {x}_n\). A \( \underline {w}\)-variant of this operation on grounds is a total constructive function$$\displaystyle \begin{aligned} h([[\underline{x}_1, \ldots , \underline{x}_{n + 1} - \underline{x}] - \underline{w}] \ \underline{y}, \xi^{\alpha_1(\underline{x}_1)}, \ldots , \xi^{\alpha_m(\underline{x}_n)})\end{aligned}$$such that, for every \( \underline {k}\) of the same length as \([[ \underline {x}_1, \ldots ,\underline {x}_{n + 1} -\underline {x}] -\underline {w}] \\underline {y}\), for every

*g*_{i}ground on*B*for \(\vdash \alpha _i( \underline {x}_i)[ \underline {k}/[ \underline {x}_i -\underline {x}] -\underline {w}]\) (*i*≤*n*),$$\displaystyle \begin{aligned} f(\underline{k}, g_1, \ldots , g_m)\end{aligned}$$is a ground on

*B*for \(\vdash \beta ( \underline {x}_{n +1})[ \underline {k}/ \underline {x}_{n + 1} -\underline {w}]\). The \( \underline {w}\)-variant is again an operation on grounds of operational type \(\alpha _1( \underline {x}_1), \ldots , \alpha _m( \underline {x}_n) \rhd \beta ( \underline {x}_{n + 1})\), except instead of grounds for the entries of the operational type after replacement of all the free individual variables with (names of) individuals, it can take as values operations on grounds for these very entries where some individual variables may remain free, depending on whether \( \underline {w}\) contains some of the individual variables that occur free in this entry. When defining equations of variable-variants, we perform the same steps as in the case of defining equations for open grounds, except that (names of) individuals are replaced in the arguments that the operation to be defined takes as values. - 3.
We have therefore a distinction between the ground-variables used to speak about operations on grounds, and those that occur in languages of grounding. The former should be expressed with a distinct notation, such as Id(

*ε*^{α}), or simply a typed “hole” Id(−^{α}). However, to avoid overburdening the notation, we have omitted this distinction. - 4.
\( \underline {y}\) is the sequence of the individual variables of the operational type of

*den*^{∗}(*ϕ*) in the case when*den*^{∗}(*ϕ*) is an operation on grounds of first level, and of the individual variables of the co-domains of the operational type of*den*^{∗}(*ϕ*) in the case when*den*^{∗}(*ϕ*) is an operation on grounds of second level. - 5.
Theorem 40 leads to a seemingly interesting observation. Given any language of grounding Λ on an atomic base

*B*on a background language*L*, suppose that it is always possible to associate to Λ, via Curry-Howard isomorphism, a formal system Σ_{Λ}on*L*which includes the atomic system of*B*. We indicate the usual conservativeness of theories—extended to admissibility of rules under substitution—as*conservativeness on provability*, and the conservativeness of Definition 39 as*conservativeness on denotation*. The following applies. Given a language of grounding Λ_{1}on an atomic base*B*, and given a non-primitive expansion Λ_{2}of Λ_{1}, there exists a denotation function*den*^{∗}of the elements of the alphabet of Λ_{2}such that Λ_{2}is a conservative expansion on the denotation of Λ_{1}with respect to*den*^{∗}if, and only if, \(\Sigma _{\Lambda _2}\) is provability conservative with respect to \(\Sigma _{\Lambda _1}\). The left-right direction of the equivalence is a rather immediate consequence of Theorem 40; it is sufficient to choose as*den*^{∗}the function that associates to each non-primitive operational symbol*F*of Λ_{2}a coding of the derivation of the operational type of*F*in \(\Sigma _{\Lambda _1}\). The left-right direction is obvious in the case of closed terms, and in the case of open terms that correspond to the denotation of primitive operational symbols that do not bind individual or ground-variables. In the case instead of non-primitive operational symbols that bind individual or ground-variables, we know that each of their applications corresponds to a derivation in \(\Sigma _{\Lambda _1}\), which guarantees admissibility under substitutions in \(\Sigma _{\Lambda _1}\).

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Piccolomini dÁragona, A. (2023). Languages of Grounding. In: Prawitz's Epistemic Grounding. Synthese Library, vol 469. Springer, Cham. https://doi.org/10.1007/978-3-031-20294-0_5

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