4. Model Checking and Counter-Examples
Historical Materialist Hook
[RESEARCH REQUIRED: Discuss the pragmatic need to invalidate arguments systematically without exhaustive verification. Connect the mathematical counter-model to the stress-testing of economic projections.]
Core Technical Synthesis
[RESEARCH REQUIRED: In FOL, truth is relative to the model. Using computational counter-modeling, explain how to explicitly invalidate arguments by constructing custom domains where premises evaluate to true while the conclusion evaluates to false. Reinforce the distinction between syntactic provability and semantic validity.]
Guided Example: Invalidating an Argument
Argument: ∃x Fx, ∃x Gx \(\vDash\) ∃x (Fx /\ Gx)
Counter-Model: Domain: {1, 2} Extension of F: {1} Extension of G: {2}
Premise 1 is True (1 is F). Premise 2 is True (2 is G). Conclusion is False (Nothing is both F and G). The argument is invalid.
Problem Set
Exercise 4.1
Construct a counter-model to prove the following argument is invalid: ∀x (Fx \/ Gx) \(\vDash\) ∀x Fx \/ ∀x Gx.
1. ∀x (Fx \/ Gx)
2. ∀x Fx \/ ∀x GxDictionary Reference Map
- For model checking theory: Logic in Computer Science (Huth & Ryan 2004, 128-140)
- For invalidating arguments: forall x Calgary (Magnus et al. 2025, 246-265)