Tuesday, October 4, 2011

Conditional Statements and God

A common way of understanding conditionals (in the form of “if A, then C”) is that one accepts a conditional in the case that he believes Not-A or C, but not both. The following would constitute examples of such.
A and C
If Prince William is alive, then I am currently employed.
Should be believed by premise? Yes.
A and not-C
If peanut butter is a peanut product, then the Allies lost World War II.
Should be believed by premise? No.
Not-A and C
If water is an abstract object, then George W. Bush was the most recent Republican president.
Should be believed by premise? No.
Not-A and Not-C
If a man is poor, then he is rich.
Should be believed by premise? Yes.
This last one is so counterintuitive. But consider the postulated scenario: you deny the antecedent, which means you think the man is not poor. But further, you deny the consequent, which means you think the man is not rich. Since you cannot derive a conclusion by denying the antecedent, you must deny the consequent. A denial of the consequent is just to say “the man is not rich.” Yet this results in the conclusion being “therefore, he is not poor.” This is perfectly consistent with your beliefs about the antecedent and consequent. “Now wait a minute!” one may object, “modus ponens is just as valid, and that would get us the following:”
1. If a man is poor, then he is rich.
2. A man is poor.
3. Therefore, he is rich.
This is flatly contradictory. But then notice what is being done: the affirmations are not not-A and not-C, but rather A and C. But since you, the objector, have pointed out A and C are contradictory, you do not believe A and C. Rather, you reject one or the other. But in any case, this is not an objection against not-A and not-C.
This has bearing on philosophical arguments for God. For this means one ought to accept every material conditional that fits the following states of belief for the one who is considering them: (A and C) and (Not-A and Not-C). Consider the first premise of the moral argument:
4. If God does not exist, objective moral values do not exist.
If one believes “God does not exist” and “objective moral values do not exist,” she should accept this material conditional. If she believes “God exists” and “objective moral values exist,” she ought to accept this. If she attempts to discredit the premise even while holding these states of beliefs, it is maintained that she does not understand the conditional.
If she believes “God does not exist” and “objective moral values exist,” or “God exists” and “objective moral values do not exist,” then she may rightfully reject the conditional. Therefore, we can see the only atheists/agnostics/skeptics that should be disputing this premise are the ones who believe objective moral values exist, but God does not exist. Hence, we can see it’s not open to the objector to deny both premises of the argument by saying they are false.
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  1. Randy,

    Does this happen in response to the moral argument? The only responses I've heard are to say that yes we can have objective morality without God or deny premise (2) that objective moral values do in fact exist.

  2. I thought that material conditionals are considered true in the case of not-A and C. And they're only false in the case of A and not-C. Am I missing something?

  3. Hi Mike. Sometimes, though not on the professional level, people will try to deny both premises of the moral argument. But that is not open to them. These rules for conditionals prevent that. They may express objections lodged against either or both objections, but they could not then claim to actually believe both objections hold.

  4. Hi Kief, I will tackle these in reverse order: propositions only being false in the case of A and not-C results in us thinking the conditionals are true in all three of the other combinations. This seems odd. Next, in thinking about the conjunctive state of belief in both not-A and C, let us take our example:

    1. If water is an abstract object, then George W. Bush was the most recent Republican president.

    As it so happens, I do not believe water is an abstract object. But I do believe GWB was the most recent Republican president. So, what are my options here, if I am to accept the conditional?

    2. Water is not an abstract object.

    3. Therefore, GWB was the most recent Republican president.

    (3) plainly does not follow from (1-2), even if the conclusion is true. So let us then suppose:

    4. Water is an abstract object.

    Now (3) does follow from (1) and (4), but (4) is just to affirm the antecedent, which by our parameters we are not doing! So then let's try:

    5. GWB is the most recent Republican president.

    6. Therefore, water is an abstract object.

    But (6) doesn't follow from (1) and (5). So then we can try:

    7. GWB is not the most recent Republican president.

    8. Therefore, water is not an abstract object.

    But in that case, we violate both of our parameters!

    All that said, I am open to correction, as usual. I will even edit the top of the article should this be shown to be false. :)

  5. Alex Pruss has something about this over on his blog. In the comments section, he says: "1. The queen won't invite me for dinner tonight.
    2. So: The queen won't invite me for dinner tonight or I will eat dinner in my pajamas.
    3. So: If the queen invites me for dinner tonight, I will eat dinner in my pajamas."
    (2) is disjunction introduction, and any disjunction is true so long as at least one of its disjuncts are true (which [1] is). So it seems that the move from (2) to (3) is justified. But what mystifies me is that it seems the move is justified only because he believes not-A, not because he believes not-A and C. I wonder if he is saying the account suggests that the consequent simply doesn't matter. But then the only way to get a valid conclusion is by denying the consequent. Interesting stuff to be sure.

  6. What would a fellow read (book/website/etc) to learn a bit more about conditionals?

    Thanks in advance :)

  7. Hi Aaron C thanks for the comment! Specifically, I can recomment "Philosophical Foundations for a Christian Worldview," by JP Moreland and William Lane Craig. There's generally also books on logic. Any introductory text will contain a discussion on conditionals.

  8. Aaron C,

    Check here, too:


    In general the Stanford Encyclopedia of Philosophy is a wonderful resource.

  9. Agreed. Also, I wish to point out I can indeed spell "recommend." :>

  10. I understand the ridiculousness of the ft case for the material conditional (as well as the ridiculousness of the truth-functional interpretation of the material conditional in general). I was just pointing out that it is in fact the case that the conditional is true in the ft case in truth-functional logic.

    The best sense I can make of it is that the material conditional isn't quite truth-functional, which results in these absurdities. I was just surprised to see you denying that the ft case is true given that it's a fundamental rule in truth-functional logic.

  11. Kief,

    The material conditional (or material implication) is certainly truth-functional. A glance at a truth table will show that a material conditional is true in every case except when the antecedent is true and the consequent is false.

    Now, the English conditional may in fact not be a material conditional. (I am no fan of the material conditional.) The material conditional is apt to validate certain inferences which we are inclined not to want to make.


    1.(A → B) and (C → D)
    2.∴ (A → D) or (C → B)

    Now, (1) – (2) follows according to the standard truth tables, but it seems that it seems that (2) should not follow from (1); indeed, that it would be absurd to infer (2) from (1). In words, (1) – (2) says, in effect, that if Keif is in Belfast he is in Northern Ireland, and if Keif is in Canberra he is in Australia. Hence, it is the case either that if Keif is in Belfast he is in Australia, or that if Keif is in Canberra he is in Northern Ireland.

    Take another example.

    1.¬ (A → B)
    2.∴ A

    In words, if it is not the case that if there is a benevolent god, then the prayers of evil people will be answered. Hence, there is a god. Again, if the English conditional is material, then we would have to infer there is a god from premise. However, even if there is a god, it certainly doesn't follow that there is given the premise.

  12. Sorry, what I meant was that the english conditional isn't quite truth-functional. By that I mean the material conditional doesn't fully capture the english conditional.

    But my confusion was that I don't get why Randy is making the conditional false in the ft case. He seems to have the material conditional in mind here, in which case ft should make it true. If he doesn't have the material conditional in mind, then I don't know what he means at all really.


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