About
This tool autocompletes justifications within the given firstorder natural deduction.
This tool does not...

automatically find proofs or insert helpful formulas;

enable you to input or verify your own justifications [but it does list all candidates for justifications, if you're using the same conventions].
In short, this tool enables you to write down your (valid) deductions quickly.
It can also be used as a deduction verifier, but this is not really the intended use case.
Everything else mentioned is mostly doable, but there are other tools for those purposes.
Syntax
The following connectives and quantifiers are available:
 Symbol  In text 
Logical NOT  ¬  ¬  ~      
Logical AND  ∧  /\  &  &&  *  .  
Logical OR  ∨  \/  \/  +      , 
Conditional  →  >  >  =>    
Biconditional  ↔  <>  =  <>  <=>   
Falsum  ⊥  #  __     
Universal quantifier  ∀  A  @     
Existential quantifier  ∃  E      
Relation  A—Z  A—Z      
Variable/Constant  a—z  a—z      
Deduction (Fitchstyle) is a list of lines, where each line is a block of 's (vertical lines) followed by a formula.
If A is an asummption, use L instead of the rightmost .
For example:
L ¬¬A
 A
Gentzenstyle deductions are also supported, but only as an output.
Help and conventions
Note that relational symbols (including propositions) are limited in length to one character.
Letters 'A' and 'E' can be used for propositions (0ary relations), but it might be confusing.
However, this tool will correctly detemine whether Ax
is universal quantification or an atomic formula, assuming there are no typos in the starting formula.
You should put brackets everywhere, except when it doesn't matter.
The only predefined precedence is that all unary operators (including quantifiers) have higher priority than binary operators.
Conventions for quantifiers:
 The main restriction is that you do not use the same letters for variables and (pseudo)constants.
Which letters you use for what purpose is up to you.
For example, a valid proof of ∀x Px from the assumption ∀x ¬¬Px would go as follows:
∀x ¬¬Px; ¬¬Pc; Pc; ∀x Px.
This restriction is not enforced (not abiding it will not cause a syntax error).
For example, if you have Pt ∧ Px, you can deduce ∃x (Px ∧ Px).
However, if you have Px ∧ Pt, since you are trying to introduce x, and the first occurence of x in
the original formula Px ∧ Pt is Px, this situation is interpreted as replacing x with x.
With this intepretation, the second occurence of Px causes an error (it differs from Pt).
A good way to solve this would be to explicitly ask the user what term is being replaced in order to formulate a correct justification,
but making the user care about justifications is exactly what we're trying to avoid.
 Introducing ∀, specifically ∀x F(x), can be done given a formula F such that F is exactly like F(x),
except that all free occurences of some letter t (which may or may not actually appear in F) within F
are replaced with x.
There are the usual restrictions:
 t should not be mentioned in any assumption, unless it belongs to an already completed subproof
 F should not contain a quantifier over t whose scope contains x an occurence of x (e.g. ∀tQx)

Introducing ∃, specifically ∃x F(x), can be done given a formula F such that F is exactly like F(x),
except that all zero or more occurences of some letter t (which may or may not actually appear in F) within F
are replaced with x.
There are the usual restrictions:
 F should not contain a quantifier over t whose scope contains an occurence of x (e.g. ∀tQx)
 Eliminating ∀, specifically ∀x F(x), is done by inserting any formula F such that
all free occurences of x (which may or may not actually appear in F(x)) within F
are replaced with t.
There are the usual restrictions:
 F should not contain a quantifier over t whose scope contains x an occurence of x (e.g. ∀tQx)
 Eliminating ∃, specifically ∃x F(x), is done by inserting a subproof beginning with a formula F such that
all free occurences of some letter t (which may or may not actually appear in F) within F
are replaced with x.
The subproof should and with a formula not containing an occurence of t.
There are the usual restrictions:
 t should not be mentioned in any preceding line, unless it belongs to an already completed subproof
 F should not contain a quantifier over t whose scope contains x an occurence of x (e.g. ∀tQx)
Rules for logical connectives are mostly standard.
There are some additional conventions regarding justifications that probably won't matter such as:
 when introducing ∧ from lines numbered a, b; the order will be a, b if a < b, otherwise b, a
 when eliminating → from lines a: F → G, b: F; the order will be a, b regardless of whether a < b
 when introducting ⊥ from a: X, b: ¬X; the order will be a, b
 similarly for other connectives
Author
Luka Mikec, nameDOTsurnameATgmailDOTcom, and insert 1 before AT