What did the Buddha mean by the four types of reasoning in his advice to the Kalamas?
From Help! The Kalama Sutta, Help! by Buddhadasa Bhikkhu:


  
  Ma takkahetu: Don’t believe something solely on the grounds of logical reasoning (takka). Logic is merely one branch of knowledge
  that people use to try to figure out the truth. Takka or Logic is not
  infallible. If its data or inferences are incorrect, it can go wrong.
  Ma nayahetu: Don’t believe or accept something merely because it appears correct on the grounds of Naya or what is now called
  "philosophy." In Thailand, we translate the Western term philosophy as
  prajna. Our Indian friends cannot accept this because "naya" is just a
  point of view or opinion; it isn’t the supreme understanding properly
  referred to as panya or prajna. Naya or nayaya is merely a method of
  deductive reasoning based on hypotheses or assumptions. Such reasoning
  can err when the method or hypothesis is inappropriate.
  Ma akaraparivitakkena: Don’t believe or accept something simply because of superficial thinking, that is, because it appeals to what
  we nowadays call "common sense," which is merely snap judgments based
  on one’s tendencies of thought. We like to use this approach so much
  that it becomes habitual. Some careless and boastful philosophers rely
  on such common sense a great deal and consider themselves clever.
  Ma ditthinijjhanakkhantiya: Don’t believe accept something to be true merely because it agrees or fits with one’s preconceived opinions
  and theories. Personal views can be wrong and our methods of
  experiment and verification may be inadequate, neither of which lead
  us to the truth. This approach may seem similar to the scientific
  method, but can never actually be scientific, as its proofs and
  experiments are inadequate.