• closes BUG (string dtype): logical operation with bool and string failing #60234
• Tests added and passed if fixing a bug or adding a new feature
• All code checks passed.
• Added type annotations to new arguments/methods/functions.
• Added an entry in the latest doc/source/whatsnew/v3.0.0.rst file if fixing a bug or adding a new feature.

Using logical operators (e.g., |, &) on non-boolean data, where this data should be cast to bool, works for most types (e.g., float, strings). However, these operations fail with pyarrow-backed strings and numpy-backed strings.

This PR fixes the issues with pyarrow-backed string arrays by casting them into boolean arrays when they are used with logical operators. The newly implemented helper functions convert_string_to_boolean_array and cast_for_logical perform the casting, while the ARROW_LOGICAL_FUNCS dictionary has been modified to use these helper functions in the process of performing logical operations (see pandas/core/arrays/arrow/array.py).

This PR fixes the issues with numpy-backed string arrays by casting them into boolean arrays whenever they are used with boolean arrays in logical operations. This is done in the logical_op function (see pandas/core/ops/array_ops.py).