Convert string which illustrates some list into pandas dataframe?

Looks like a case for regular expressions.

import re
from ast import literal_eval
import pandas as pd

s = ('[[[1],1,¬q,"A",[]],[[2],2,p→q,"A",[]],[[3],3,p,"A",[]],'
     '[[2,3],4,q,"→E",[2,3]],[[1,2,3],5,q∧ ¬q,"∧I",[1,4]],[[1,2],6,¬p,"¬I",[3,5]]]')

rows = []

# split at ',' followed by two closing ]]
for x in re.split(r"(?<=\]\]),", s[1:-1]):
    
    # split at ',' after closing ] OR between '"' and opening [ 
    left, middle, right = re.split(r"(?<=\]),(?=\d)|(?<=\"),(?=\[)", x[1:-1])

    # split the middle part at ','
    middle = middle.split(",")
    
    rows.append([literal_eval(left), *middle, literal_eval(right)])
    


df = pd.DataFrame(rows, columns=['Assumptions', 'Index', 'Proposition', 'Premisses', 'Rule'])
df["Index"] = df.Index.astype(int)
df["Premisses"] = df.Premisses.str.strip('"')

Result:

  Assumptions  Index Proposition Premisses    Rule
0         [1]      1          ¬q         A      []
1         [2]      2         p→q         A      []
2         [3]      3           p         A      []
3      [2, 3]      4           q        →E  [2, 3]
4   [1, 2, 3]      5       q∧ ¬q        ∧I  [1, 4]
5      [1, 2]      6          ¬p        ¬I  [3, 5]

This solution does not require Regex

import pandas as pd

s = '[[[1],1,¬q,"A",[]],[[2],2,p→q,"A",[]],[[3],3,p,"A",[]],[[2,3],4,q,"→E",[2,3]],[[1,2,3],5,q∧ ¬q,"∧I",[1,4]],[[1,2],6,¬p,"¬I",[3,5]]]'
s1 = s[2:-2].split('],[')
result = []

for item in s1:
    lst = []
    group = False
    temp = ''

    for e in item:
        #extract group with square brackets `[ ]`
        if e == '[':
            group = True
            temp = ''
        elif e == ']' and temp[-1] != ']':
            temp += e
            lst.append(temp)
            group = False
            temp = ''

        if group == True:
            temp += e
        elif e != ']':
            #below non-group, to extract string between commas `,`
            if e == ',' and temp != '':
                lst.append(temp)
                temp = ''
            elif e != ',' and e != '"':
                temp += e

    result.append(lst)

df = pd.DataFrame(result, columns=['Assumptions', 'Index', 'Proposition', 'Premisses', 'Rule'])
print(df)

  Assumptions Index Proposition Premisses   Rule
0         [1]     1          ¬q         A     []
1         [2]     2         p→q         A     []
2         [3]     3           p         A     []
3       [2,3]     4           q        →E  [2,3]
4     [1,2,3]     5       q∧ ¬q        ∧I  [1,4]
5       [1,2]     6          ¬p        ¬I  [3,5]