How the Distinction between Deductive vs. Inductive Arguments Can Mask Uncertainty

Everyone who has taken a philosophy 101 class has learned the distinction between deductive and inductive arguments. It goes like this. Only deductive arguments may be valid; an argument is valid if and only if the truth of its premises guarantees the truth of its premises. Otherwise, the argument is invalid. If an argument is both valid and contains all true premises, then the argument is sound.

Not all invalid arguments are worthless, however, and the concept of an inductive argument shows why. An (inductive) argument is (inductively) strong if and only if (1) it is invalid; and (2) its premises confer a high probability upon its conclusion. In order to reinforce (2), some inductive logic textbooks will place the word “probably” inside the conclusions of inductive arguments.

Notice that validity is a binary, all-or-nothing affair. Just as one cannot be “sort of pregnant,” an argument cannot be “somewhat” valid. In contrast, inductive strength is a matter of degree. Inductively strong arguments confer a high probability on their conclusion, whereas weak arguments don’t.

So far, so good. It seems to me, however, that this distinction can sometimes mask the fact that uncertainty is often present in “real world” deductive arguments.

Consider the following argument (Deductive Argument 1 or DA1):

(1) If A, then B.

(2) A.

(3) Therefore, B.

Question: What is the probability of B?

Since DA1 is a valid argument, we know that if (1) and (2) are true, (3) has to be true.  So the probability of B conditional upon A is 1. In symbols, Pr(B|A)=1. This is they key “insight,” if you will, of learning that DA1 is valid.

But we want to know the unconditional probability of B, Pr(B), not the contribution made to Pr(B) by the probability of B conditional upon A, Pr(B|A). So, again, what is the contribution made to Pr(B) by A itself? Answer: Pr(A). Suppose A is true by definition. In that case, its probability is 1 and so Pr(B)=1. Now suppose the probability of A is 50%. In that case, Pr(B)=0.5. The probability calculus implies, when Pr(B|A)=1, that the contribution to Pr(B) made by A alone will always equal Pr(A), so long as Pr(A) is not zero, which in “real world” problems is usually the case.

In his book, Objecting to God, Colin Howson makes a similar point and then writes this.

“A possibly more surprising feature of the logic of probability is that it subsumes the logic of conjecture and refutation. It tells us that if evidence is inconsistent with a hypothesis under test, i.e., if it is refuting evidence, then that evidence reduces the probability of a hypothesis to zero. The formal theory of probability implies that if H entails that E is not true then Prob(E|H)=0 so long as Prob(H) is nonzero, as in all applications it will be. Looking at Bayes’s Theorem, we see that if Prob(H) is nonzero and Prob(E|H)=0 then Prob(H|E)=0. Thus the logic of probability subsumes the deductive logic of refutation as a special case. What is more interesting is the vastly more extensive territory outside the relatively small and safe domain where deductive logic can play its protective role.”

Consider William Lane Craig’s version of the fine-tuning argument which goes like this.

1. The fine-tuning of the universe’s initial conditions is either the result of chance, necessity or design. (Premise)
2. It is not the result of chance or necessity. (Premise)
3. Therefore, it is the result of design. (From 1 and 2)

This argument is clearly valid. We want to know the probability of (3). As in the case of DA1, the probability of (3) will depend upon the probability of (2). If we have a very weak degree of belief that (2) is true, say we think Pr(2)=0.25, then, by itself, this argument only warrants the belief Pr(3)=0.25.

* Thanks to Robert Greg Cavin for helpful comments on an earlier version of this post.

About Jeffery Jay Lowder

Jeffery Jay Lowder is President Emeritus of Internet Infidels, Inc., which he co-founded in 1995. He is also co-editor of the book, The Empty Tomb: Jesus Beyond the Grave.

  • Victor Reppert

    I shouldn’t wonder if this isn’t just an instance of the total evidence problem. It’s not possible, in the course of considering any particular piece of evidence, to consider all the other relevant evidence, so we have to look at evidence one piece at a time. Hence, any argument, however strong, is can be outweighed by other evidence. I certainly wouldn’t want to deny this, though it does seem to be denied from time to time with respect to the argument from evil.

    • Jeffery Jay Lowder


      Interesting comment. I’m not sure if I think this is just an instance of the total evidence problem or not. For example, suppose the probability of “fine-tuning” conditional upon theism is 10%, whereas on naturalism it is 0.01%. By itself, that fact would be evidence favoring theism over naturalism, since it increases the ratio of the probability of theism to the probability of naturalism.

      But, just to continue the thought experiment, let’s assume that “fine-tuning” is the only evidence relevant to God’s existence, so there is no worry about other evidence potentially outweighing the evidence of “fine-tuning.” Even if there is no other relevant evidence, I think the point of my OP still applies, since there can still be uncertainty about the “fact” of “fine-tuning” for reasons wholly apart from whether is any any other, outweighing evidence. For example, we may just lack absolute certainty about whether there really is “fine-tuning.” Or, if there is “fine-tuning,” we may have uncertainty about whether it is the product of chance, necessity, or design.

      I think if I were to bracket my point under well known issues in inductive logic, I would categorize it as the problem of uncertain evidence (cf. Jeffrey conditionalization). Your thoughts?

  • Doug Benscoter

    Jeff, as I learned while taking logic as an undergraduate philosophy major (eight years ago! Time sure flies by), plausibly true inductive arguments are said to be cogent. Craig’s version of the fine-tuning argument is logically valid, as you say, but since it makes use of probability in its premises, you might call it a cogent argument, as opposed to a sound one. That’s all technical-speak, though. The debate should really center around whether (2) is more plausible than its negation.

    • Jeffery Jay Lowder

      Doug Benscoter

      Two comments.

      1. Inductive logic as a discipline does not seem to have the same degree of maturity as its sibling, deductive logic. Terminology is a case in point. For example, Wesley Salmon’s textbook, Logic, uses the term “inductive correctness.” Also, IIRC, William Gustason, in his book Reasoning from Evidence, refers to “inductive strength.”

      2. To borrow a metaphor from physics and calculus, inductive strength has vector-like properties. One is direction and the other is magnitude. The “direction” refers to whether the evidence increases the probability of a hypothesis, decreases its probability, or has no effect (because it is irrelevant). The “magnitude” refers to how much of an effect, if any, it has. So, if “whether (2) is more plausible than its negation” means “direction,” which I think it does, then I must disagree. We also need to consider the “magnitude” or amount of support the evidence provides. This is especially important when dealing with things like the fallacy of understated evidence, which applies to most versions of the fine-tuning argument, including Craig’s.

      • Doug Benscoter

        Jeff, I confess that the fine-tuning argument has never been my strong suite. My expertise (if you can call it that) is in the defense of the Aristotelian-Thomistic arguments. I’ll simply allow philosophers with more knowledge of the fine-tuning argument, such as yourself, Craig and Collins to debate its merit.

        • Jeffery Jay Lowder

          Thanks. AT arguments have never been my strong suite. Someday I need to read Feser’s book…

  • Doug Benscoter

    Also (and this is slightly off-topic), I’ve been trying to find some common ground between theists and atheists. Would you agree with the following argument?

    1. Whatever exhibits regularity is not the result of chance alone. (Premise)
    2. The laws of nature exhibit regularity. (Premise)
    3. Therefore, the laws of nature are not the result of chance alone. (From 1 and 2)

    This would lead us to the conclusion that the laws of nature are the result of someone’s (design) or something’s (necessity) providence. (You can choose another word other than “providence,” if you feel it’s a theologically-loaded term.)

    • Jeffery Jay Lowder

      Doug, I’m not sure what to make of that argument. I’m not sure I understand precisely what you have in mind by “exhibit regularity.” Based on an intuitive understanding of “exhibit regularity,” I’m inclined to say that (2) is true by definition. Regarding (1), I don’t know if I agree or not; I need to think about this. Regardless, I think I agree with (3), subject to the caveat that I don’t believe there is a multiverse.

      I guess we could create a dilemma. If there is a multiverse, then the “laws of multi-nature” (read: the laws governing the multiverse) are not the result of chance alone. If our universe is the only universe which exists, then the laws of nature are not the result of chance alone. In either case, multiverse or not, the most fundamental laws of nature (or multi-nature) are not the result of chance alone. The key assumption I am making here is that it is incoherent to say that the most fundamental laws of nature (or of multi-nature) could be the result of chance.

      (The reason for the dilemma is that, on the assumption there is a multiverse, then the laws of each individual universe (including ours) could be the result of chance alone.)

      • Doug Benscoter

        A thing exhibits regularity if it happens over and over again. I would think that the mechanism that produces a hypothetical multiverse would itself exhibit regularity. Even chaos is intelligible, and intelligibility presupposes order. It’s not as if such a mechanism would violate any laws of logic (to use another example).

        • Jeffery Jay Lowder

          Again, I don’t believe there is a multiverse, so please keep that in mind as you read my comments.

          Here is one scenario (“Multiverse Cosmology 1″ or MC1):

          1. There exists a multiverse.

          2. The multiverse exists eternally, so there is no mechanism which produced it.

          3. Our universe began to exist as a “baby universe” inside the multiverse.

          4. The mechanism which produces “baby universes,” including our universe, is governed by natural laws (“laws of multi-nature” in my last comment).

          5. The laws of multi-nature are physically necessary–not the result of chance or design.

          • Doug Benscoter

            In a hypothetical multiverse (which I recognize now that you don’t believe in) that exists from eternity past, it wouldn’t have a mechanism responsible for its origination, but wouldn’t it still have a mechanism that sustains its existence? After all, assuming the contingency of the multiverse (or universe, if there’s only one), it’s logically possible for it to cease existing.

          • Jeffery Jay Lowder

            Is it logically possible for the universe (or multiverse, if there is one) to cease existing? Maybe, maybe not. I don’t know. (I doubt that anyone can really conceive of absolute nothing, but I’m not prepared to argue that inconceivability -> impossibility.) So, let’s assume that it is possible.

            Now what? I don’t see the need for a mechanism to sustain its existence. Here I want to quote something from Stephen Law’s review of Bede Rundle’s book, Why There is Something Rather Than Nothing:

            Nor does the universe require God as a sustaining cause. Such a cause is needed when there is a disintegrating factor to be countered or inhibited. In the absence of such factors, a persisting state requires no explanation. “If something is still around after many years, this may well be remarkable, but that will be because it has somehow, against the odds, survived threats to its integrity. If there are no such
            threats, there is nothing to explain.“(p. 91)

            What threats could there be against the very existence of the universe or the multiverse? On the assumption that naturalism is true, it seems that no such threats exist.

            Now, I’m just “thinking out loud here,” i.e., not making an argument I want to stake my reputation on, but it seems to me that one could make a Bayesian argument against theism based upon the continuing existence of the universe. It would go like this.

            Given that the universe exists, the fact that it continues to exist is more probable on naturalism than it is on theism. Metaphysical naturalism entails the causal closure of the physical spatiotemporal universe, i.e., nothing that is neither a part nor product of the universe can affect it. So metaphysical naturalism conjoined with the existence of the universe entails that there are no threats to the universe’s continued existence. Theism, on the other hand, is the belief that a perfect supernatural person (God) exists who created the universe. Any being powerful enough to create the universe (or multiverse) would be powerful enough to destroy it. No matter how unlikely, theism entails a metaphysical threat to the universe’s ongoing existence, a threat which is literally impossible if naturalism is true. Thus, the continued existence of the universe is more probable on naturalism than it is on theism, and so is some evidence against theism and for naturalism.


          • Doug Benscoter

            That’s an interesting argument, but I think there’s too much counter-evidence. Rundle’s comment reminds of the objection that Newtonian physics constitutes a disproof, or otherwise undermines, the argument from motion. However, I think this objection is based on a misinterpretation of both Newtonian physics and Aristotelian metaphysics. What Newton postulates is that a thing doesn’t need a sustaining cause of its motion if it meets two criteria: a) that it is moving in a straight line; and b) that it moves in a completely empty vacuum. This is obviously not the kind of motion (change) that Aristotle had in mind when he conceived of the Unmoved Mover as the sustaining cause of all dynamic entities. An acorn that becomes an oak tree does not move in a straight line, and it is certainly not moving in an empty vacuum. Rather, it requires sustaining causes of its motion (e.g. water, sunlight and soil) to continue moving. If at any point these sustaining causes are removed, then the acorn will cease moving. Granted, I’m now focusing on the motion of the universe and not the universe’s existence per se, but I find the two objections analogous.

            There’s a lot to be said about the argument from motion and, while we can discuss that if you’d like, I also don’t want to derail your post (which is about the universe’s fine-tuning).

            As for the argument you present, I’m not sure how on naturalism the non-existence of the universe is any less possible. It seems to me that it’s metaphysically possible for something to cease to exist without having anything outside of its closed system affect it. In any case, I don’t want to put too much stress on the contingency of the universe. As far as I’m concerned, it’s a tangential issue.

          • Jeffery Jay Lowder

            I appreciate your willingness to avoid derailing my post, but don’t worry about it. The fine-tune aspect of the OP was just an example. The main point of my OP was about uncertainty in the premises of deductive arguments. So, I’m happy to discuss these issues with you here.

            When you write, “That’s an interesting argument, but I think there’s too much counter-evidence.” What is the counter-evidence?

          • Doug Benscoter

            The counter-evidence I had in mind involves the argument from motion. It also involves two other things I neglected to mention. According to Thomism, God is being itself subsisting. This means that anything’s existence implies that God exists, and that distinct entities merely participate in God’s being. Also, if God is loving, as classical theists maintain, then God’s love for his creation would mean that he would have no desire to utterly destroy it. I realize these are controversial claims, which is why I mentioned the argument from motion.

          • Doug Benscoter

            Let’s take, then, the argument from motion:

            1. Evident to the senses is motion. (Premise)

            2. Everything in motion has its motion sustained by another. (Premise)

            3. Either an Unmoved Mover exists, or else there is an infinite regress of sustaining movers. (From 1 and 2)

            4. There cannot be an infinite regress of sustaining movers. (Premise)

            5. Therefore, and Unmoved Mover exists. (From 3 and 4)

            The Unmoved Mover must be immutable, since if it were in motion, then it would be moved by another, which is contradictory. Moreover, the Unmoved Mover must be pure actuality, since potentiality entails mutability. The Unmoved Mover must also be eternal, since to cease to exist would constitute a change. The Unmoved Mover must also be unique. If there were more than one Unmoved Mover, then there would be distinctions between them. However, to be distinct from actuality is to be non-actuality, which is to say that it doesn’t exist. Other entities are distinct since they exhibit potentiality, even though they participate in the actuality of the Unmoved Mover. Next, the Unmoved Mover must be immaterial, since physical bodies have to the potential to change. Finally, the Unmoved Mover must be very powerful, if not omnipotent, in order to be the cause of all dynamic entities.

            If this analysis is correct, then the argument from motion constitutes a sound argument for God’s existence. Do you disagree with any of the argument’s premises?

          • Jeffery Jay Lowder

            It’s been awhile since I’ve thought about these arguments, but I’m assuming the point of (1) is that there are things in motion. Correct? (I’m not sure why the emphasis on “evident to the senses” was necessary.)

            Regardless, I don’t see any reason to think (4) is true. What’s the supporting argument for (4)?

          • Doug Benscoter

            With respect to (1), “evident to the senses” just means that we observe that things change. I suppose it could just say: Things are in motion. I have no objection to that modification.

            I support of premise (4), I’d point to the impossibility of forming an actual infinite by successive addition whenever one begins counting. Even granting that the universe is infinite in its past, it is still composed of finite intervals. At each finite interval, the regress of sustaining causes begins anew. For example, at 4 pm, the regress of sustaining causes begins at 0, 1, 2, 3 . . . n. It is impossible to form an actual infinite in this case, since it is always and indefinitely possible to count another number before arriving at infinity. Make sense?

          • Ryan McCarthy

            Even agreeing with premises 1 – 4 would not be sufficient to grant theism. Each statement might be necessary for theists, but that set is not sufficient since I think theism entails other additional properties, such as God being self aware, morally perfect, etc. However I can’t see any reason a priori to reject the idea that there could be a non personal being that is pure act, so even if I granted 1 – 4 I would not have reason to reject atheism and embrace theism.

            Additionally I think the point for immateriality needs a defense. Surely material objects of every day experience are in potency, but is it necessary that for an object to have matter that it be in potency? I’m not so sure. It seems to me that an atheist can simply say that it is epistemically possible that there be a material being that is pure actuality. We could of course say such an idea is implausible since it violates our every day experience, but an immaterial being seems intrinsically far more implausible.

          • Doug Benscoter

            Hi Ryan,

            I’m willing to grant for the sake of argument that the First Way (argument from motion) doesn’t in and of itself establish the existence of a personal God. However, it would be a strange form of atheism that accepts the existence of an immutable, purely actual, unique, very powerful, eternal and immaterial Unmoved Mover.

            With respect to the Unmoved Mover’s immateriality, physical things are by definition extended in space. Extension, if it is limited, is itself a potentiality, whereas the Unmoved Mover is pure actuality.

          • GGDFan777

            “Regardless, I don’t see any reason to think (4) is true. What’s the supporting argument for (4)?”

            Here is an argument in support of 4:

            1. If an infinite regress of sustaining movers is possible, than an actual infinity is possible
            2. An actual infinity is not possible.
            3. Therefore an infinite regress of sustaining movers is not possible.

            I shall assume only premiss 2 will be challenged. In defense of premiss 2 I would like to point to an article by philosopher Casper Storm Hansen of the University of Aberdeen called “New Zeno and Actual Infinity” which can be found here:


            In 1964 José Benardete invented the “New Zeno Paradox” about an infinity of gods trying to prevent a traveler from reaching his destination. In this paper it is argued,contra Priest and Yablo, that the paradox must be re-solved by rejecting the possibility of actual infinity. Further, it is shown that this paradox has the same logical
            form as Yablo’s Paradox. It is suggested that constructivism can serve as the basis of a common solution to New Zeno and the paradoxes of truth, and a constructivist interpretation of Kripke’s theory of truth is given.

            - GGDFan777

          • Jeffery Jay Lowder

            The paper looks interesting. I’ve seen arguments against an actual infinite before (primarily those used by W.L. Craig) and have found the refutations offered by Q. Smith and others to be decisive. Without having read Hansen’s paper, which I don’t have the time read now, I’ll assume for the sake of argument that his paper avoids the problems in Craig’s arguments against an actual infinite.

            We can then apply the central point of my OP to your argument. Let’s assume that (1) is absolutely certain, so its epistemic probability is 1. What about (2)? Prior to reading the paper, I may think (2) is very doubtful. For the sake of illustration only, say I estimate my degree of belief in (2) to be 0.1. Then my degree of belief in (3) is also going to be 0.1. In other words, for me, the epistemic probability that (3) is false would be 0.9.

            This type of outcome can be extended to almost any value for the probability of (2). So long as we regard (1) as absolutely certain and assume that the probability of (2) is not absolutely zero, then the probability of (3) will always equal the probability of (2).

          • Doug Benscoter

            By the way, part of the reason I bring this up is that if I were to hypothetically abandon my adherence to classical theism, I would most likely become a pantheist. The order exhibited throughout the universe is awe-inspiring, as I’m sure you’d agree, and studies have shown the benefits of prayer and meditation. Anyway, I don’t want to hijack your post, so I’ll leave any further autobiographical comments aside.

          • Jeffery Jay Lowder

            I agree with you about the evidence for the benefits of prayer and meditation. I suspect that many atheists (myself included) probably miss out on those benefits by not engaging in some sort of secular meditation.

  • Bradley Bowen

    This is very interesting stuff. But I think it is a bit more complicated than you make it seem.

    “Now suppose the probability of A is 50%. In that case, Pr(B)=0.5″

    A few complications:

    (1) What if the other premise is not certain, but only probable to some degree?
    Since both premises are required in order to infer the conclusion, if the probability of the other premise was also 0.5, then the probability of the conclusion would also be reduced (to 0.25).

    (2) Other arguments may provide additional support for the truth of the conclusion.

    Suppose we are talking about the existence of God. There is more than one argument for the existence of God. So, if the modus ponens argument you mention only makes the probability of the existence of God 0.25, there might be another deductive argument that provides additional independent reason in support of the existence of God. If the other argument (considered by itself) also makes the probability of God 0.25, then since we have two separate arguments which provide independent reasons for the existence of God, the overall probability of God, considering both arguments would have to be something greater than 0.25.

    (3) Other arguments might provide additional reason for the falsehood of the conclusion.

    If there is a modus ponens argument for the negation of B, and if the premises of that argument are certain, then that would reduce the probability of B to 0. Similarly, if the premises of a modus ponens argument for the negation of B were merely probable (say .5 probability), then we would have counterbalancing arguments for and against B. Would that then make the overall probability of B = 0.5?