For alkyl groups attached to the phenol ring, there are two main points that your reasoning does not take account of -
- Inductive effect is distance dependent. As the distance increases, its effect becomes very small.
- Even if the group were near, the general trend is that hyperconjugation has stronger effect than inductive effect.
You can refer the following links for an explanation on hyperconjugation -
More the number of hydrogen atoms attached to the benzyl carbon, more is the number of hyperconjugative structures possible and more is the withdrawal of positive charge (or equivalently, donation of negative charge) into the benzene ring.
Phenol forms phenoxide ion on losing H+ and the phenoxide tries to delocalize its negative charge into the ring. If hyperconjugation already donates electrons into the ring, the extent of delocalization decreases, making phenoxide less stable and hence phenol less acidic.
As a result, less the number of benzylic hydrogen, more is the stability of phenoxide and more the acidity. That explains the order S > R > Q > P.
NOTE : Hyperconjugation, like resonance (mesomeric effect), operates only at the ortho and para positions. In case the same groups were attached at meta, then hyperconjugation would have no role and your order based on inductive effect would be correct.