/sci/ - Science & Math

Jean-Yves Girard recently has caused controvery for criticizing predicate logic for assuming the existence of "properties of individuals" like [math]P(t)[/math] where [math]P[/math] is a property and [math]t[/math] is an individual.
Traditional predicate logic operates under the assumption that we can make statements about individuals having certain properties, such as "Socrates is mortal" (where "mortal" is a property and "Socrates" is an individual).

Girard argues that we cannot speak of abstract properties like "blue" without relating them to specific objects or situations, such as "the sky is blue." In this sense, he takes a more realistic approach to logic, focusing on the relationships between objects and their properties rather than treating properties as standalone entities.

To address this issue, Girard proposes replacing the idea of individuals with linear propositions (e.g. [math]T, U, V[/math]), which can be thought of as atomic formulas that do not reference any specific objects.
Equality between individuals can then be expressed as linear equivalence between these linear propositions. This approach requires linearity, as Girard demonstrates that without it, we can only have at most two "individuals," which is not sufficient for a general logical framework.

In summary, Jean-Yves Girard criticizes predicate logic for its unrealistic treatment of properties and individuals. He suggests that a more natural approach to logic involves employing linearity, as seen in linear logic, which allows for a better understanding of the relationships between objects and their properties.
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