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In a standard syllogism, the two premises of the argument are independent of each other, so all extrinsic warrant being equal, the argument will prove the conclusion to be probable if and only if...

P(Premise 1) * P(Premise 2) > 50%

If the probabilities of each premise happen to be equal, this turns out to be about 71% certainty in each. This calculation obviously turns out much higher for arguments with more than two premises.

But in modus tollendo tollens arguments, "not Q" is far from independent of "P implies Q." Taking a mathematical argument for God's existence (the X would be replaced with some particular degree)...

1) If atheism is true, then the world would not be built on mathematical structures of complexity greater than degree X,

2) The world is built on mathematical structures of complexity greater than degree X,

3) Therefore, atheism is false.

Suppose the opponent agrees with premise 1, with the ~71% certainty required for each premise of the other syllogism. In this argument, in contrast to the other syllogism, premise 2 is not independent of premise 1. If we bring to this argument a 50% prior probability that atheism is true (at least methodologically to keep an open mind), then there seems to be a 50% chance that premise 1 provides a defeater for premise 2, so that the argument given for premise 2 would have to yield much higher certainty than 71%. Am I thinking about this in the right way? Is it impossible (or improper) to evaluate the premises of a modus tollendo tollens argument independently of each other? What are the probability requirements for a good modus tollendo tollens argument?

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    Can you explain why you think modus tollens depends on probably at all? The standard logic forms for deductive arguments are not generally probabilistic.
    – virmaior
    Commented Jun 4, 2016 at 4:45
  • The logical structure isn't probabilistic, that's right; if the premises are true, then the conclusion necessarily follows. Probabilities come in when we assess the plausibility of the premises. The more certain one is of the premises, the higher the minimum bound of certainty he or she must assign to the conclusion in his or her noetic structure.
    – DicePower
    Commented Jun 4, 2016 at 5:23
  • If you want to include probabilistic considerations you will have to use a different kind of logic. Classical logic has only true and false, nothing in between. Take a look at this en.wikipedia.org/wiki/Probabilistic_logic
    – E...
    Commented Jun 4, 2016 at 8:19
  • In my mind, this issue sounds like, "Billy: Hey, if we add your 20 brownies to my 40 brownies, will we have enough food for everyone? Joey: Brownies can't be added! Addition is an arithmetic operator and only works on numbers!" Is there really a non-pedantic concern in assessing the epistemic probability of premises in arguments that take this form (even if we have to step beyond classical logic to do so)?
    – DicePower
    Commented Jun 5, 2016 at 4:33
  • Pedantic. That's not a nice way to reply to someone who is trying to help. Anyway, I didn't say you can't step beyond classical logic. You can. And in fact it has been done. This is what I'm pointing out to you, that probabilistic reasoning requires suitable logic and you should look into that.
    – E...
    Commented Jun 5, 2016 at 18:58

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There are many issues with several terms. Disregarding them, no, in "modus tollendo tollens" arguments, a higher probability of certainty is not required. As the first premise is conditional, it has nothing to do with the fact whether atheism is true, but what would follow if atheism is true. So, even with a prior probability of 1/2 that atheism is true, you can adjust probabilities of other part of the premise 1 so that the individual probabilities still come to around 71%.

What is important to keep in mind, however, is that for second premise, the truth must not be based on truth of conclusion. If that is the case, then the argument will be circular and thus unconvincing. Non circularity ensures required independence.

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  • Comments are not for extended discussion; this conversation has been moved to chat.
    – user2953
    Commented Apr 26, 2018 at 16:57
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Your wording has many issues. Certainty cannot be less than 100%. 100% expresses certainty and the answer must be absolute. There is no middle ground or exception. By definition alone certainty expresses the given answer is impossible to be false. I am certain that dogs are not reptiles. I am certain that triangles have three sides.

To say 71 % certain makes no sense. Percentages are possibilites the answer is correct. Science uses percentages where there is no certainty. The inference at hand expresses if the premise is absolutely true not if it is uncertain.

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  • How about calling it a 71% likelihood? There must be some way of expressing this acceptably.
    – Mark Andrews
    Commented Apr 23, 2018 at 0:44
  • Yes likihood is similar to expressing a percentage. The main point is neither term expresses a certainty. X is either certain or X is not certain. There is no middle ground.
    – Logikal
    Commented Apr 23, 2018 at 2:45

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