What did the Buddha mean by the four types of reasoning in his advice to the Kalamas? In AN 3:65, while advising the Kalamas on how to choose among a variety of contradictory belief systems, the Buddha mentions ten inadequate criteria for truth:


  Come, Kalamas. Do not go by oral tradition, by lineage of teaching, by
  hearsay, by a collection of texts, by logic, by inferential
  reasoning, by reasoned cogitation, by the acceptance of a view after
  pondering it, by the seeming competence of a speaker, or because you
  think, 'The ascetic is our teacher.'


I am interested in the four criteria that seem to relate to verbal reasoning (an alternative translation in parentheses):


Logic (surmise)
Inferential reasoning (axiom)
Reasoned cogitation (specious reasoning)
Acceptance of a view after pondering it (bias towards a notion that has been pondered)


Bikkhu Bodhi has a footnote that, much to my consternation, says of these criteria, 


  [They are] four types of reasoning recognized by thinkers in the Buddha's age; their differences need not detain us here.


I understand the main point being made (that we should go by direct experience), but I am curious about the precise meaning of these four criteria. I am looking for an answer that explains the differences between these four types of reasoning by providing an example of each one from everyday life. Bonus for any relevant historical context.
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.