We examine the Or-to-If inference (also known as the Direct Argument), which infers \(\lnot \varphi \rightarrow \psi \) from \(\varphi \vee \psi \) . While this inference is valid in classical logic, it has sparked considerable debate in conditional logic. We advocate for a pluralistic conception of validity, arguing that our intuitions about validity are inherently ambiguous and can be systematically analyzed through distinct theoretical notions. The various notions of validity proposed arise from different treatments of premises (as beliefs, assertions, or assumptions), which yields three distinct forms of validity. Together with the differentiation between indicative and subjunctive conditionals, our semantic framework ultimately generates six variants of Or-to-If. We also show how the three notions of validity can be unified into one notion, by treating a logical consequence as a 4-ary relation, or by expressing the differently treated premises explicitly in the object language. This refined approach enables a precise characterization of the specific conditions under which Or-to-If holds or fails. While the paper focuses primarily on Or-to-If, the proposed pluralistic framework shows promise in reconciling the often-observed divergence between formal logical systems and concrete reasoning practices in natural language.

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Six Faces of Or-to-If

  • Xuefeng Wen

摘要

We examine the Or-to-If inference (also known as the Direct Argument), which infers \(\lnot \varphi \rightarrow \psi \) from \(\varphi \vee \psi \) . While this inference is valid in classical logic, it has sparked considerable debate in conditional logic. We advocate for a pluralistic conception of validity, arguing that our intuitions about validity are inherently ambiguous and can be systematically analyzed through distinct theoretical notions. The various notions of validity proposed arise from different treatments of premises (as beliefs, assertions, or assumptions), which yields three distinct forms of validity. Together with the differentiation between indicative and subjunctive conditionals, our semantic framework ultimately generates six variants of Or-to-If. We also show how the three notions of validity can be unified into one notion, by treating a logical consequence as a 4-ary relation, or by expressing the differently treated premises explicitly in the object language. This refined approach enables a precise characterization of the specific conditions under which Or-to-If holds or fails. While the paper focuses primarily on Or-to-If, the proposed pluralistic framework shows promise in reconciling the often-observed divergence between formal logical systems and concrete reasoning practices in natural language.