Machine learning image training set data as whole object but finding part of object
Train with the same kinds of inputs as you wish the network to recognise. There is no easy way to add logical combinations that you think should help the network, because deep neural networks, despite the hype as the best thing to hit AI, are essentially just really dumb complex statistical analysers.

Provided you supply training images, the CNN will learn how to statistically combine components to decide if overall an image is a chair or not. For instance if the image is of a person wearing a padded jacket, perhaps that will look a little like the back of a chair. The NN might output a 0.75 confidence that a back of a chair is in the image. If you manually combine decisions from components, you will need to figure out from that whether to say there is a chair in the image. If you leave it to the network, the CNN itself will combine this data with the additional detection there is a 0.2 confidence that the person's legs are chair legs (and they are in the wrong relative place) and decide - hopefully - that there is no chair.

You don't need to explicitly label components of objects for the NN to learn that there are components and what they look like. The outputs of Deep Dream and other CNN visualisations demonstrate that internally the CNN builds up its own structures that represent parts of objects. They are not necessarily the same parts as a person would focus on, but by allowing the training process to choose, you should be able to  get a good recognition rate.

So, if you give the network a lot of chair parts to recognise separately, it will get better at labelling those chair parts accurately. But that will not really help the network recognise a chair. In fact it could be worse at the main task if the training images are too different from images it is expected to recognise. Plus, it will be your task to then decide how to re-combine the classes to detect the chair later.

There probably are architectures and image-processing pipelines that can be used to combine logical knowledge of parts and their poses into whole objects. It is something I very much expect to be an active area of research in computer vision systems. However, these approaches will not use a simple CNN and altering training data as the solution.