This paper is about the elementary metatheory of equational predicate logic—that is, first‐order logic that uses Leibniz's substitution of ‘equals for equals’ as a primary rule of inference. We present here two different ‘faithful’ formalizations of this logic and prove that both versions are sound and complete. A by‐product of this study is a proof that the ‘full’ Leibniz rule is strictly stronger than the ‘no‐capture’ Leibniz rule. We obtain this result here for a complete Logic. We also show that under some reasonable conditions, propositional Leibniz, no‐capture Leibniz, and a full‐capture version are all equivalent, provided that the latter is restricted to act on universally valid premises whenever capture is allowed.

It is the object of this paper to study the topological properties of finite graphs that can be embedded in the n -dimensional integral lattice (denoted N n ). The basic notion of deletability of a node is first introduced. A node is deletable with respect to a graph if certain computable conditions are satisfied on its neighborhood. An equivalence relation on graphs called reducibility and denoted by “∼” is then defined in terms of deletability, and it is shown that (a) most important topological properties of the graph (homotogy, homology, and cohomology groups) are ∼-invariants; (b) for graphs embedded in N 3 , different knot types belong to different ∼-equivalence classes; (c) it is decidable whether two graphs are reducible to each other in N 2 but this problem is undecidable in N n for n ≥ 4. Finally, it is shown that two different methods of approximating an n -dimensional closed manifold with boundary by a graph of the type studied in this paper lead to graphs whose corresponding homology groups are isomorphic.