In this work, we present an experimental technique for producing polarized computer generated holograms. We provide background in the propagation of light through dielectric media for use when discussing the spatial light modulator’s (SLM) ability to display holograms. We also cover Jones vector analysis and Fourier optics to understand the encoded holograms. Using these concepts, we conduct experiments verifying the ability of the SLM to produce polarized holograms. The size of and location at which we observe these holograms are established experimentally, as well as the large degree of flexibility we have in observing holograms as a result of using an SLM. The focus then shifts, and the polarization of computer generated holograms is explored. We report that we can successfully polarize a computer generated hologram, and that we can program the polarization state through the addition of a constant phase shift on the SLM. We expand on the ability to polarize holograms and explore polarization logic operations. The functionality of our polarization logic operation is verified by encoding a polarized “Cheshire Cat” hologram onto the SLM. We construct the “Cheshire Cat” from the superposition of two nearly identical holograms. This results in either the addition or subtraction of the identical complex amplitude components of the individual holograms, depending on the analyzer axis. This method provides a proof of concept of the development of polarization logic operation, and expands the possibilities of optical logic.