Abstract
We first introduce the syntax and semantics of FOL, then we look at its proof theory. We extend the natural deduction calculus for propositional logic to that for FOL. We then proceed with a similar treatment for sequent calculus for FOL. Furthermore, this time we will consider how to improve a proof calculus, which leads to several variants of LK. Finally, we demonstrate Gentzen’s main theorem, the cut-elimination theorem for sequent calculus, which is the longest and arguably the most complex proof of this book. In doing so, we also introduce important proof techniques such as proof by induction.
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Notes
- 1.
Coincidentally, \(\times \) in type signatures can be understood as conjunction and \(\rightarrow \) as implication. And \(A \wedge B \rightarrow C\) is equivalent to \(A \rightarrow B \rightarrow C\). We will come back to such correspondences at the end of this book.
- 2.
This is just an intermediate step. We will simplify it further soon.
- 3.
This does not contradict with the completeness proof, which only guarantees that there is a derivation.
- 4.
The word “detour” on Page 32 refers to some proof steps in natural deduction that introduces a formula via an introduction rule and then eliminates the formula via an elimination rule. Such “detours” do not necessarily contribute to longer proofs. In fact, in sequent calculus, a detour in the form of the cut rule often shortens proofs.
- 5.
Fun fact: the German mathematician Gauss (1777–1855) found this equation for \(n = 100\) when he was seven years old.
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Hou, Z. (2021). First-Order Logic. In: Fundamentals of Logic and Computation. Texts in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-030-87882-5_2
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DOI: https://doi.org/10.1007/978-3-030-87882-5_2
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