A Classical Linear $λ$-Calculus based on Contraposition
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917245266952192 |
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| author | Barenbaum, Pablo Bonelli, Eduardo Lerena, Leopoldo |
| author_facet | Barenbaum, Pablo Bonelli, Eduardo Lerena, Leopoldo |
| contents | We present a novel linear $λ$-calculus for Classical Multiplicative Exponential Linear Logic (\MELL) along the lines of the propositions-as-types paradigm. Starting from the standard term assignment for Intuitionistic Multiplicative Linear Logic (IMLL), we observe that if we incorporate linear negation, its involutive nature implies that both $A\multimap B$ and $B^\perp\multimap A^\perp$ should have the same proofs. The introduction of a linear modus tollens rule, stating that from $B^\perp\multimap A^\perp$ and $A$ we may conclude $B$, allows one to recover classical MLL. Furthermore, a term assignment for this elimination rule, {the study of proof normalization in a $λ$-calculus with this elimination rule} prompts us to define the novel notion of contra-substitution $t \{ a \backslash\!\backslash s \}$. Introduced alongside linear substitution, contra-substitution denotes the term that results from "grabbing" the unique occurrence of $a$ in $t$ and "pulling" from it, in order to turn the term $t$ inside out (much like a sock) and then replacing $a$ with $s$. We call the one-sided natural deduction presentation of classical MLL, the $λ_{\rm MLL}$-calculus. Guided by the behavior of contra-substitution in the presence of the exponentials, we extend it to a similar presentation for MELL. We prove that this calculus is sound and complete with respect to MELL and that it satisfies the standard properties of a typed programming language: subject reduction, confluence and strong normalization. Moreover, we show that several well-known term assignments for classical logic can be encoded in $λ_{\rm MLL}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_02822 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Classical Linear $λ$-Calculus based on Contraposition Barenbaum, Pablo Bonelli, Eduardo Lerena, Leopoldo Logic in Computer Science We present a novel linear $λ$-calculus for Classical Multiplicative Exponential Linear Logic (\MELL) along the lines of the propositions-as-types paradigm. Starting from the standard term assignment for Intuitionistic Multiplicative Linear Logic (IMLL), we observe that if we incorporate linear negation, its involutive nature implies that both $A\multimap B$ and $B^\perp\multimap A^\perp$ should have the same proofs. The introduction of a linear modus tollens rule, stating that from $B^\perp\multimap A^\perp$ and $A$ we may conclude $B$, allows one to recover classical MLL. Furthermore, a term assignment for this elimination rule, {the study of proof normalization in a $λ$-calculus with this elimination rule} prompts us to define the novel notion of contra-substitution $t \{ a \backslash\!\backslash s \}$. Introduced alongside linear substitution, contra-substitution denotes the term that results from "grabbing" the unique occurrence of $a$ in $t$ and "pulling" from it, in order to turn the term $t$ inside out (much like a sock) and then replacing $a$ with $s$. We call the one-sided natural deduction presentation of classical MLL, the $λ_{\rm MLL}$-calculus. Guided by the behavior of contra-substitution in the presence of the exponentials, we extend it to a similar presentation for MELL. We prove that this calculus is sound and complete with respect to MELL and that it satisfies the standard properties of a typed programming language: subject reduction, confluence and strong normalization. Moreover, we show that several well-known term assignments for classical logic can be encoded in $λ_{\rm MLL}$. |
| title | A Classical Linear $λ$-Calculus based on Contraposition |
| topic | Logic in Computer Science |
| url | https://arxiv.org/abs/2602.02822 |