diff --git a/Library/Homebrew/cli/args.rb b/Library/Homebrew/cli/args.rb
index 5a313b5249..709a84e77f 100644
--- a/Library/Homebrew/cli/args.rb
+++ b/Library/Homebrew/cli/args.rb
@@ -1,4 +1,4 @@
-# typed: true # rubocop:disable Sorbet/StrictSigil
+# typed: strict
 # frozen_string_literal: true
 
 require "ostruct"
@@ -6,6 +6,17 @@ require "ostruct"
 module Homebrew
   module CLI
     class Args < OpenStruct
+      # FIXME: Enable cop again when https://github.com/sorbet/sorbet/issues/3532 is fixed.
+      # rubocop:disable Style/MutableConstant
+      # Represents a processed option. The array elements are:
+      #   0: short option name (e.g. "-d")
+      #   1: long option name (e.g. "--debug")
+      #   2: option description (e.g. "Print debugging information")
+      #   3: whether the option is hidden
+      OptionsType = T.type_alias { T::Array[[String, T.nilable(String), String, T::Boolean]] }
+      # rubocop:enable Style/MutableConstant
+
+      sig { returns(T::Array[String]) }
       attr_reader :options_only, :flags_only
 
       # undefine tap to allow --tap argument
@@ -17,10 +28,12 @@ module Homebrew
 
         super
 
-        @processed_options = []
-        @options_only = []
-        @flags_only = []
-        @cask_options = false
+        @cli_args = T.let(nil, T.nilable(T::Array[String]))
+        @processed_options = T.let([], OptionsType)
+        @options_only = T.let([], T::Array[String])
+        @flags_only = T.let([], T::Array[String])
+        @cask_options = T.let(false, T::Boolean)
+        @table = T.let({}, T::Hash[Symbol, T.untyped])
 
         # Can set these because they will be overwritten by freeze_named_args!
         # (whereas other values below will only be overwritten if passed).
@@ -28,10 +41,12 @@ module Homebrew
         self[:remaining] = []
       end
 
+      sig { params(remaining_args: T::Array[T.any(T::Array[String], String)]).void }
       def freeze_remaining_args!(remaining_args)
         self[:remaining] = remaining_args.freeze
       end
 
+      sig { params(named_args: T::Array[String], cask_options: T::Boolean, without_api: T::Boolean).void }
       def freeze_named_args!(named_args, cask_options:, without_api:)
         options = {}
         options[:force_bottle] = true if self[:force_bottle?]
@@ -46,6 +61,7 @@ module Homebrew
         )
       end
 
+      sig { params(processed_options: OptionsType).void }
       def freeze_processed_options!(processed_options)
         # Reset cache values reliant on processed_options
         @cli_args = nil
@@ -53,8 +69,8 @@ module Homebrew
         @processed_options += processed_options
         @processed_options.freeze
 
-        @options_only = cli_args.select { |a| a.start_with?("-") }.freeze
-        @flags_only = cli_args.select { |a| a.start_with?("--") }.freeze
+        @options_only = cli_args.select { _1.start_with?("-") }.freeze
+        @flags_only = cli_args.select { _1.start_with?("--") }.freeze
       end
 
       sig { returns(NamedArgs) }
@@ -63,10 +79,10 @@ module Homebrew
         self[:named]
       end
 
-      def no_named?
-        named.blank?
-      end
+      sig { returns(T::Boolean) }
+      def no_named? = named.blank?
 
+      sig { returns(T::Array[String]) }
       def build_from_source_formulae
         if build_from_source? || self[:HEAD?] || self[:build_bottle?]
           named.to_formulae.map(&:full_name)
@@ -75,6 +91,7 @@ module Homebrew
         end
       end
 
+      sig { returns(T::Array[String]) }
       def include_test_formulae
         if include_test?
           named.to_formulae.map(&:full_name)
@@ -83,6 +100,7 @@ module Homebrew
         end
       end
 
+      sig { params(name: String).returns(T.nilable(String)) }
       def value(name)
         arg_prefix = "--#{name}="
         flag_with_value = flags_only.find { |arg| arg.start_with?(arg_prefix) }
@@ -96,6 +114,7 @@ module Homebrew
         Context::ContextStruct.new(debug: debug?, quiet: quiet?, verbose: verbose?)
       end
 
+      sig { returns(T.nilable(Symbol)) }
       def only_formula_or_cask
         if formula? && !cask?
           :formula
@@ -141,11 +160,13 @@ module Homebrew
 
       private
 
+      sig { params(option: String).returns(String) }
       def option_to_name(option)
         option.sub(/\A--?/, "")
               .tr("-", "_")
       end
 
+      sig { returns(T::Array[String]) }
       def cli_args
         return @cli_args if @cli_args
 
@@ -165,10 +186,12 @@ module Homebrew
         @cli_args.freeze
       end
 
-      def respond_to_missing?(method_name, *)
+      sig { params(method_name: Symbol, _include_private: T::Boolean).returns(T::Boolean) }
+      def respond_to_missing?(method_name, _include_private = false)
         @table.key?(method_name)
       end
 
+      sig { params(method_name: Symbol, args: T.untyped).returns(T.untyped) }
       def method_missing(method_name, *args)
         return_value = super
 
diff --git a/Library/Homebrew/cli/parser.rb b/Library/Homebrew/cli/parser.rb
index 4ac4625171..79db0ede47 100644
--- a/Library/Homebrew/cli/parser.rb
+++ b/Library/Homebrew/cli/parser.rb
@@ -16,7 +16,6 @@ module Homebrew
       # FIXME: Enable cop again when https://github.com/sorbet/sorbet/issues/3532 is fixed.
       # rubocop:disable Style/MutableConstant
       ArgType = T.type_alias { T.any(NilClass, Symbol, T::Array[String], T::Array[Symbol]) }
-      OptionsType = T.type_alias { T::Array[[String, T.nilable(String), T.nilable(String), String, T::Boolean]] }
       # rubocop:enable Style/MutableConstant
       HIDDEN_DESC_PLACEHOLDER = "@@HIDDEN@@"
       SYMBOL_TO_USAGE_MAPPING = T.let({
@@ -25,7 +24,7 @@ module Homebrew
       }.freeze, T::Hash[Symbol, String])
       private_constant :ArgType, :HIDDEN_DESC_PLACEHOLDER, :SYMBOL_TO_USAGE_MAPPING
 
-      sig { returns(OptionsType) }
+      sig { returns(Args::OptionsType) }
       attr_reader :processed_options
 
       sig { returns(T::Boolean) }
@@ -177,7 +176,7 @@ module Homebrew
         @constraints = T.let([], T::Array[[String, String]])
         @conflicts = T.let([], T::Array[T::Array[String]])
         @switch_sources = T.let({}, T::Hash[String, Symbol])
-        @processed_options = T.let([], OptionsType)
+        @processed_options = T.let([], Args::OptionsType)
         @non_global_processed_options = T.let([], T::Array[[String, ArgType]])
         @named_args_type = T.let(nil, T.nilable(ArgType))
         @max_named_args = T.let(nil, T.nilable(Integer))
@@ -671,7 +670,7 @@ module Homebrew
       def process_option(*args, type:, hidden: false)
         option, = @parser.make_switch(args)
         @processed_options.reject! { |existing| existing.second == option.long.first } if option.long.first.present?
-        @processed_options << [option.short.first, option.long.first, option.arg, option.desc.first, hidden]
+        @processed_options << [option.short.first, option.long.first, option.desc.first, hidden]
 
         args.pop # last argument is the description
         if type == :switch
diff --git a/Library/Homebrew/commands.rb b/Library/Homebrew/commands.rb
index bd15e09d2b..bff04823ef 100644
--- a/Library/Homebrew/commands.rb
+++ b/Library/Homebrew/commands.rb
@@ -199,7 +199,7 @@ module Commands
     return if path.blank?
 
     if (cmd_parser = Homebrew::CLI::Parser.from_cmd_path(path))
-      cmd_parser.processed_options.filter_map do |short, long, _, desc, hidden|
+      cmd_parser.processed_options.filter_map do |short, long, desc, hidden|
         next if hidden
 
         [long || short, desc]
diff --git a/Library/Homebrew/manpages.rb b/Library/Homebrew/manpages.rb
index f49184cf12..3b3048017f 100644
--- a/Library/Homebrew/manpages.rb
+++ b/Library/Homebrew/manpages.rb
@@ -96,9 +96,10 @@ module Homebrew
       man_page_lines.compact.join("\n")
     end
 
+    sig { params(cmd_parser: CLI::Parser).returns(T::Array[String]) }
     def self.cmd_parser_manpage_lines(cmd_parser)
       lines = [format_usage_banner(cmd_parser.usage_banner_text)]
-      lines += cmd_parser.processed_options.filter_map do |short, long, _, desc, hidden|
+      lines += cmd_parser.processed_options.filter_map do |short, long, desc, hidden|
         next if hidden
 
         if long.present?