Proposed Restriction on Universal Instantiation (Natural Deduction)
I recommend the adhering to constraint on global instantiation : UI might not be made use of to present new variables. The variable defined need to be an "old" variable, i.e. it has to currently have actually been presented by either an energetic property or by existential instantiation (E - removal). (An energetic property is one that has actually not been shut - off or released by what I call a verdict declaration.)
The key advantage of this constraint is to remove the demand to take into consideration dependences amongst variables that are developed with existential instantiation. Such factors to consider never ever appear ahead up in "real mathematics." The constraint I recommend appears to be implied also in one of the most strenuous mathematical evidence, otherwise in typical official logic.
With this constraint in position, dependences amongst variables require not be taken into consideration due to the fact that the only variables on which you can after that make global generalizations are those presented in a property that has actually ultimately been shut - off or released by what I call a verdict declaration. The verdict declaration and also succeeding declarations that are stemmed from it as well as additionally describe these variables can constantly be globally generalised no matter the visibility of any kind of various other variables that were presented by E - removal.
I have actually applied this constraint in my evidence checker/editor (readily available free at http://www.dcproof.com). Experiment with it. Attempt to "break it" if you can. Resolve the quick tutorial to find out the system.
An additional advantage, from an instructional viewpoint, has actually been the removal from my program of numerous what have to have been perplexing advising messages concerning dependences amongst variables. It additionally streamlines the needs for global generalizations (A - intro).
Adhering to is a recap of the regulations of reasoning I make use of in my system that pertain to the handling of free variables :
Free variables might be presented just using a property (presumption) or by existential requirements (E - removal).
Existential requirements (E - removal) permits an extra free variable to be defined for any kind of energetic, existentially evaluated declaration.
Universal requirements (A - removal) permits any kind of free variable presented by an energetic property or any kind of algebraic expression in such free variables to be defined for any kind of energetic, globally evaluated declaration.
Existential generalization (E - intro) might be related to any kind of free variable or any kind of algebraic expression in several free - variables that is located in any kind of energetic declaration.
Universal generalizations (A - intro) might be related to any kind of free variable that (a) is located in an energetic declaration, and also (b) is not described by any kind of energetic property, and also (c) was not presented by existential requirements (E - removal). (There are no factors to consider offered for any kind of dependences amongst variables.)
Free variables presented after a property declaration and also prior to the equivalent verdict declaration might not show up because verdict declaration.
In my program, I make use of shade - coding absolutely free variables. Environment-friendly shows that a Universal Generalization is feasible. Red shows that a Universal Generalization is not feasible. When first presented, all free variables are red. A free variable that was presented by a property is transformed to environment-friendly when that property is shut - off (or released) by a verdict declaration. It is properly reestablished, transforming back to red if it is described in succeeding property.
Your remarks would certainly be valued.
You can constantly add some tautology as a presumption (claim $x \rightarrow x$) and afterwards later on release it, so it feels like your constraint stands.
Examine "Structural Proof Theory", a publication by von Plato and also Negri, which reviews such concerns ; it additionally funds an evidence editor PESCA by Aarne Ranta.