Почему пути включения наследуются даже для частных ссылок

У меня есть простой проект, состоящий из 2 "библиотек" a и b и исполняемого файла c. Я установил include path для библиотеки a (PUBLIC) в X (для целей тестирования такого пути на самом деле нет).
Теперь b связывается в частном порядке с a и c связывается с b. Я ожидаю, что X не будет в include path при компиляции c.cpp, потому что" он не должен знать о a, так как он находится в частном порядке в b", но, по-видимому, он здесь.

Итак, мой вопрос-намеренное поведение или нет и почему.

Мой код:

A.cpp

void a() {}

B.cpp

void b() {}

C.cpp

int main() {}

CMakeLists.txt

cmake_minimum_required(VERSION 2.8)

add_library(a a.cpp)
target_include_directories(a PUBLIC X)

add_library(b b.cpp)
target_link_libraries(b PRIVATE a)

add_executable(c c.cpp)
target_link_libraries(c PUBLIC b)

(подробный) журнал компиляции:

 /usr/local/Cellar/cmake/3.8.1/bin/cmake -H/Users/alexeyd/test/dir -B/Users/alexeyd/test/dir/build --check-build-system CMakeFiles/Makefile.cmake 0
/Applications/Xcode.app/Contents/Developer/usr/bin/make -f CMakeFiles/Makefile2 c
/usr/local/Cellar/cmake/3.8.1/bin/cmake -H/Users/alexeyd/test/dir -B/Users/alexeyd/test/dir/build --check-build-system CMakeFiles/Makefile.cmake 0
/usr/local/Cellar/cmake/3.8.1/bin/cmake -E cmake_progress_start /Users/alexeyd/test/dir/build/CMakeFiles 6
/Applications/Xcode.app/Contents/Developer/usr/bin/make -f CMakeFiles/Makefile2 CMakeFiles/c.dir/all
/Applications/Xcode.app/Contents/Developer/usr/bin/make -f CMakeFiles/a.dir/build.make CMakeFiles/a.dir/depend
cd /Users/alexeyd/test/dir/build && /usr/local/Cellar/cmake/3.8.1/bin/cmake -E cmake_depends "Unix Makefiles" /Users/alexeyd/test/dir /Users/alexeyd/test/dir /Users/alexeyd/test/dir/build /Users/alexeyd/test/dir/build /Users/alexeyd/test/dir/build/CMakeFiles/a.dir/DependInfo.cmake --color=
Dependee "/Users/alexeyd/test/dir/build/CMakeFiles/a.dir/DependInfo.cmake" is newer than depender "/Users/alexeyd/test/dir/build/CMakeFiles/a.dir/depend.internal".
Dependee "/Users/alexeyd/test/dir/build/CMakeFiles/CMakeDirectoryInformation.cmake" is newer than depender "/Users/alexeyd/test/dir/build/CMakeFiles/a.dir/depend.internal".
Scanning dependencies of target a
/Applications/Xcode.app/Contents/Developer/usr/bin/make -f CMakeFiles/a.dir/build.make CMakeFiles/a.dir/build
[ 16%] Building CXX object CMakeFiles/a.dir/a.cpp.o
/Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/bin/c++   -I/Users/alexeyd/test/dir/X   -o CMakeFiles/a.dir/a.cpp.o -c /Users/alexeyd/test/dir/a.cpp
[ 33%] Linking CXX static library liba.a
/usr/local/Cellar/cmake/3.8.1/bin/cmake -P CMakeFiles/a.dir/cmake_clean_target.cmake
/usr/local/Cellar/cmake/3.8.1/bin/cmake -E cmake_link_script CMakeFiles/a.dir/link.txt --verbose=1
/Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/bin/ar qc liba.a  CMakeFiles/a.dir/a.cpp.o
/Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/bin/ranlib liba.a
[ 33%] Built target a
/Applications/Xcode.app/Contents/Developer/usr/bin/make -f CMakeFiles/b.dir/build.make CMakeFiles/b.dir/depend
cd /Users/alexeyd/test/dir/build && /usr/local/Cellar/cmake/3.8.1/bin/cmake -E cmake_depends "Unix Makefiles" /Users/alexeyd/test/dir /Users/alexeyd/test/dir /Users/alexeyd/test/dir/build /Users/alexeyd/test/dir/build /Users/alexeyd/test/dir/build/CMakeFiles/b.dir/DependInfo.cmake --color=
Dependee "/Users/alexeyd/test/dir/build/CMakeFiles/b.dir/DependInfo.cmake" is newer than depender "/Users/alexeyd/test/dir/build/CMakeFiles/b.dir/depend.internal".
Dependee "/Users/alexeyd/test/dir/build/CMakeFiles/CMakeDirectoryInformation.cmake" is newer than depender "/Users/alexeyd/test/dir/build/CMakeFiles/b.dir/depend.internal".
Scanning dependencies of target b
/Applications/Xcode.app/Contents/Developer/usr/bin/make -f CMakeFiles/b.dir/build.make CMakeFiles/b.dir/build
[ 50%] Building CXX object CMakeFiles/b.dir/b.cpp.o
/Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/bin/c++   -I/Users/alexeyd/test/dir/X   -o CMakeFiles/b.dir/b.cpp.o -c /Users/alexeyd/test/dir/b.cpp
[ 66%] Linking CXX static library libb.a
/usr/local/Cellar/cmake/3.8.1/bin/cmake -P CMakeFiles/b.dir/cmake_clean_target.cmake
/usr/local/Cellar/cmake/3.8.1/bin/cmake -E cmake_link_script CMakeFiles/b.dir/link.txt --verbose=1
/Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/bin/ar qc libb.a  CMakeFiles/b.dir/b.cpp.o
/Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/bin/ranlib libb.a
[ 66%] Built target b
/Applications/Xcode.app/Contents/Developer/usr/bin/make -f CMakeFiles/c.dir/build.make CMakeFiles/c.dir/depend
cd /Users/alexeyd/test/dir/build && /usr/local/Cellar/cmake/3.8.1/bin/cmake -E cmake_depends "Unix Makefiles" /Users/alexeyd/test/dir /Users/alexeyd/test/dir /Users/alexeyd/test/dir/build /Users/alexeyd/test/dir/build /Users/alexeyd/test/dir/build/CMakeFiles/c.dir/DependInfo.cmake --color=
Dependee "/Users/alexeyd/test/dir/build/CMakeFiles/c.dir/DependInfo.cmake" is newer than depender "/Users/alexeyd/test/dir/build/CMakeFiles/c.dir/depend.internal".
Dependee "/Users/alexeyd/test/dir/build/CMakeFiles/CMakeDirectoryInformation.cmake" is newer than depender "/Users/alexeyd/test/dir/build/CMakeFiles/c.dir/depend.internal".
Scanning dependencies of target c
/Applications/Xcode.app/Contents/Developer/usr/bin/make -f CMakeFiles/c.dir/build.make CMakeFiles/c.dir/build
[ 83%] Building CXX object CMakeFiles/c.dir/c.cpp.o
/Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/bin/c++   -I/Users/alexeyd/test/dir/X   -o CMakeFiles/c.dir/c.cpp.o -c /Users/alexeyd/test/dir/c.cpp
[100%] Linking CXX executable c
/usr/local/Cellar/cmake/3.8.1/bin/cmake -E cmake_link_script CMakeFiles/c.dir/link.txt --verbose=1
/Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/bin/c++   -Wl,-search_paths_first -Wl,-headerpad_max_install_names  CMakeFiles/c.dir/c.cpp.o  -o c libb.a liba.a
[100%] Built target c
/usr/local/Cellar/cmake/3.8.1/bin/cmake -E cmake_progress_start /Users/alexeyd/test/dir/build/CMakeFiles 0

Интересная часть журнала:

[ 83%] Building CXX object CMakeFiles/c.dir/c.cpp.o 
/Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/bin/c++   -I/Users/alexeyd/test/dir/X   -o CMakeFiles/c.dir/c.cpp.o -c /Users/alexeyd/test/dir/c.cpp

Я использую CMake 3.8.1