What did the Buddha mean by the four types of reasoning in his advice to the Kalamas?

Question

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.

Answer 1

There are 3 types of teachers:

• The traditionalists (anussavikā) - who base their teaching on tradition - e.g. my ancestors believed in this so I am also believing it. Generally superstitious ideas stem from this logic.
• The rationalists and speculators [metaphysicians] (takki vimamsi) - who base their teaching in faith-based thinking and reasoning - e.g. because God created animals for us to eat, it is OK to kill them for food.
• The experientialists - having directly known the Dharma for themselves (sāmam yeva dhammam abhinnāya) - e.g. when I concentrate by gazing on a coloured disk I get a blissful feeling hence I have merged with God/Brahma.

The 4 items mentioned below fall under the 2nd category, "by the way of reasoning".

• Logic (surmise) - if you do not hold something it falls to the ground, therefore there must be a tortoise who lifts the earth so it does not fall
• Inferential reasoning (axiom) - God has a plan, me losing my job may be part of this plan
• Reasoned cogitation (specious reasoning) - when I say this mantra it always rained, therefore saying this mantra makes rain
• Acceptance of a view after pondering it (bias towards a notion that has been pondered) - one needs an outlet for stress, when one enjoys oneself by partying and drinking the stress goes away, the best thing to do when one is stressed is to party and get drunk

The other answer also has a good summary of what these items mean.

Answer 2

From Help! The Kalama Sutta, Help! by Buddhadasa Bhikkhu:

5. 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.

6. 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.

7. 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.

8. 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.