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The fine structure of Peircean ligatures and lines of identity

Abstract Lines of identity in Peirce's existential graphs (beta) are logically complex structures that comprise both identity and existential quantification. Yet geometrically they are simple: linear continua that cannot have “furcations” or cross “cuts.” By contrast Peirce's “ligatures” are geometrically complex: they can both have furcations and cross cuts. Logically they involve not only identity and existential quantification but also negation. Moreover, Peirce makes clear that ligatures are composed of lines of identity by virtue of the fact that such lines can be “connected” with one another and can “abut upon” one another at a cut. This paper shows in logical detail how ligatures are composed and how they relate to identity, existential quantification, and negation. In so doing, it makes use of Peirce's non-standard account of the linear continuum, according to which, when a linear continuum is separated into two parts, (1) the parts are symmetric rather than (as the standard account of Dedekind holds) asymmetric, and (2) the one point at which separation occurs actually becomes two points, each of which is a Doppelgänger of the other.