In the same way that Datalog emerges from logic programming as a deductive database language, such formulas can support a very expressive framework for constraint deductive databases, allowing hypothetical queries and universal quantifications.
For example, the formulas of a deductive system may be seen as objects of a category whose arrows f : A → B are deductions of B from A. In fact, this point of view is important in theoretical computer science, where formulas are thought of as types and deductions as operations.
When mathematicians and many philosophers engage in deductive reasoning, they occasionally invoke formulas in a formal language to help disambiguate, or otherwise clarify what they mean.
The formula ψ X,Y) is a deductive generalization of ϕ(X), if it holds in Σ that the less general ϕ implies the more general ψ where for the free variables X (the ones that occur in ϕ and possibly in ψ) the universal closure and for free variables Y (the ones that occur in ϕ only) the existential closure is taken: Σ ⊨ ∀ X ∃ Y ( ϕ ( X ) → ψ ( X, Y ) ).
As such, it has no meaning, or perhaps better, the meaning of its formulas is given by the deductive system and the semantics.
Exactly the same is true of the set of formulae that are derivable in a formalized deductive calculus.
The story breaks from the established formula: there is no puzzle to be solved, no dazzling display of deductive genius.
If one builds one's deductive calculus with care, one will be able to convince oneself that all the formulae derivable in the calculus are logical truths.
In the present experiment, we can interpret this activation as coming from deductive mental operations during the application of the syntactic rules underlying the formation of hierarchical formulae in the process of the verification of whether a given string of symbols represents a correct formula or not.
And excels at deductive reasoning.
It is deductive and dogmatic.
