|Lambda Calculus, philosophy: The lambda calculus provides a way to avoid problems related to paradoxes, since, unlike the quantification of predicate logic, it does not make any existence assumptions. Where the quantification (Ex)(Fx) is translated colloquially as "There is an x with the property F" (in short "Something is F"), the translation of the corresponding form in the Lambda calculus is "An x, so that...". See also 2nd order logic._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
Rules/composition/composition rules/syntax/Bigelow/Pargetter: one can also go the other way and want to simplify the rules. That is what the
λ-categorical language/Lambda calculus/Lambda notation/Lambda abstraction/Bigelow/Pargetter does: (see also Cresswell I and II, as well as Montague).
For example: Negation: surprisingly, one can assign a referent to it and keep it thus out of the rules:
Vs: we then have another referential layer in the theory.
Negation: we can assign a set theoretical symbol that represents the value "true" or "false".
((s) Truth value/Frege/(s): assigns a referent to the negation, a "thing": "the false".
Bigelow/Pargetter: then we have a judgement function that assigns the semantic value (or referent) V(a) to a symbol a.
1: be "true".
0: be "false".
Definition semantic value: (the negation V(a)) is then the function ω~, so that
ω ~ (1) = 0 ω ~ (0) = 1
is appropriate for compound expressions (internal/external negation, conjunction, etc.)
Lambda categorical language/λ/Lambda/Rules/Bigelow/Pargetter: such languages have extremely few composition rules.
We have more referring symbols for this.
Realism: would describe this as ontologically honest.
Semantics/Bigelow/Pargetter: but the realist does not have to commit himself to one semantics instead of another. The semantics does not decide upon ontology._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.
J. Bigelow, R. Pargetter
Science and Necessity Cambridge 1990