Notre Dame Journal of Formal Logic

Syntax and Semantics of the Logic $\mathcal{L}^\lambda_{\omega\omega}$

Carsten Butz


In this paper we study the logic $\mathcal{L}^\lambda_{\omega\omega}$, which is first-order logic extended by quantification over functions (but not over relations). We give the syntax of the logic as well as the semantics in Heyting categories with exponentials. Embedding the generic model of a theory into a Grothendieck topos yields completeness of $\mathcal{L}^\lambda_{\omega\omega}$ with respect to models in Grothendieck toposes, which can be sharpened to completeness with respect to Heyting-valued models. The logic $\mathcal{L}^\lambda_{\omega\omega}$ is the strongest for which Heyting-valued completeness is known. Finally, we relate the logic to locally connected geometric morphisms between toposes.

Article information

Notre Dame J. Formal Logic, Volume 38, Number 3 (1997), 374-384.

First available in Project Euclid: 12 December 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03G30: Categorical logic, topoi [See also 18B25, 18C05, 18C10]
Secondary: 03C75: Other infinitary logic


Butz, Carsten. Syntax and Semantics of the Logic $\mathcal{L}^\lambda_{\omega\omega}$. Notre Dame J. Formal Logic 38 (1997), no. 3, 374--384. doi:10.1305/ndjfl/1039700744.

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