1. From Propositional Logic to Quantified Structures
Historical Materialist Hook
[RESEARCH REQUIRED: The introduction of quantified structures and predicates is situated within the broader materialist history of science. Utilize Boris Hessen’s The Social and Economic Roots of Newton’s Principia to establish how highly abstract scientific and logical theories are driven directly by the technological and economic needs of the capitalist class.]
Core Technical Synthesis
[RESEARCH REQUIRED: Explain the transition from simple sentence letters to the internal architecture of atomic sentences. Introduce the recursive syntax of universal (∀) and existential (∃) quantifiers, alongside the distinct roles of constants, predicates, and arity.]
Sandbox Exploration
Experiment with quantified syntax before attempting the problem sets. Note the difference between a constant a and a variable x.
Guided Example: Arity and Quantifiers
Translating “Everything is connected”: Let Cxy = x is connected to y.
Translation: ∀x ∀y Cxy
Problem Set
Exercise 1.1
Define a domain of discourse and two predicates (one 1-place, one 2-place). Write a first-order formula that utilizes both universal and existential quantifiers.
Dictionary Reference Map
- For quantified syntax: forall x Calgary (Magnus et al. 2025, 172-195)
- For computational structures: Logic in Computer Science (Huth & Ryan 2004, 93-109)