Lambda notation is of importance since lambda reduction, which is the process to apply lambda notation and get new FOL expressions from where the formal parameter variables are bound to specified terms.
$$\lambda x.P(x)(A)\rightarrow P(A)$$
The lambda reduction can apply to one lambda notation multiple times. 
$$
\lambda x.\lambda y.Near(x,y)(Bacaro)\rightarrow \lambda y.Near(Bacaro,y)
\\
\lambda y.Near(Bacaro,y)(Centro)\rightarrow \ Near(Bacaro,Centro)
$$


Lambda Reduction

Lambda Notation can provide a way to abstract from fully specified FOL formulas in a way that will be particularly useful for semantic analysis.
$$\lambda x.P(x)$$


University of Colorado at Boulder

First-Order Logic (FOL) is a flexible, well-understood, and computationally tractable meaning representation language.


First-Order Logic

An on-going but a helpful book resource about NLP
https://web.stanford.edu/~jurafsky/slp3/

Speech and Language Processing (3rd ed. draft) 

Lambda Notation


One FOL formula also has a truth value.

Formulas involving ∃ (Existential quantification) are true if a substitution of terms for variables results in a formula that is true in the model. 

Formulas involving ∀(Universal quantification) must be true under all possible substitutions.


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