What Good are Logical Entailments?

What Good Are Logical Entailments? A logical entailment can be viewed as a necessary consequence of a given premise or argument. It can often be expressed by a strict if-then conditional; i.e. If I do this, then he will do that. For a specific instance, take the argument that if Calvinism is true, then God is the author of sin.[1] When this statement is being made, it is a rather informal way of stating that Calvinism’s being true logically entails the fact that God is the author of sin.

There is yet some confusion, however, over legitimate uses of logical entailments and how they may be applied to an opposing position. That is to say, within a debate there are several different ways the entailment is applied. First, we should examine how not to use these in debate or discussion.

We cannot and should not use the entailment to claim this is the position of our opponents. Rephrased: simply because a position held X logically entails another position Y, it does not follow that everyone who holds X holds Y to be true. Perhaps they are unaware X entails Y, or perhaps they disagree X entails Y. In any case, especially if Y is an unfavorable view, it is uncharitable to ascribe to them a view which they probably do not hold. Thus, to use my above example, it would be unfair of me (in most situations) to claim, “Jim is a Calvinist. Calvinism entails God is the author of sin. It is obvious Jim should be ashamed that he thinks God is the author of sin.”

So when should we bring up logical entailments? I think we can and should bring them up in any examination of a teaching, doctrine, or philosophical claim. While we need not accuse our opponent of believing any number of false doctrines or other claims, we should always be evaluating the consequences of a belief or proposition. The proper way to do this is to examine the overall claim (“Calvinism is true”) and follow the chain of logical progression, eventually building to the overall consequence to be avoided (“God is the author of sin”). Here is a relatively easy example:

1. Calvinism is true.

2. If Calvinism is true, then man cannot help but to choose sin.

3. If Calvinism is true, then the world is exactly as God chose it to be.

4. Whatever God chooses he wants for his glory (analysis of [1]).

5. Therefore, man cannot help but to choose to sin (from [1-2]).

6. Therefore, the world is exactly as God chose it to be (from [1, 3]).

7. Therefore, God chose man to be unable not to sin for his glory (analysis of [1, 4-6]).

8. If man is unable not to sin, then he must be determined to sin.

9. If man is determined to sin, then it is because of man or God.

10. Man could not start this causal chain.

11. Therefore, man is determined to sin (from [1-2, 5, 8]).

12. Therefore, God determined man to sin (from [9-10]).

From this, we can conclude “If Calvinism is true, then God determined man to sin (same content as God being the author of sin).” (If P, then Q; If Q, then R; If R, then S or T; Not-S; If R, then T=If P, then T, or PàT)

While I won’t argue for its truth or falsehood here, this is perfectly valid. Further, the way one would develop this is by expanding the argument into a situation where accepting the entailments is not realistic for the opponent or hearers. For example:

13. If Calvinism is true, then God determined man to sin.

14. God did not determine man to sin.

15. Therefore, Calvinism is not true.

A careful student will also note this is a reductio ad absurdum where (15) and (1) are logically incoherent when taken together, so that some premise or premises must be rejected in order to avoid the conclusion. Some may ask, “How is this not the fallacious ‘appeal to consequence?’” Good question. In an appeal to consequence, the idea is that one saying, “If X is true, then Y is true, and you don’t want Y to be true. Therefore, X.” Another example is, “if you don’t believe this, then I will never speak to you again!” What makes appeal to consequence fallacious (and vastly different from modus tollens) is that entirely irrational reasons are what gets one to the conclusion. The emotional desire to want something to be true or false does not make it so (nor does it count as evidence generally), and although you may want me to speak to you again, that won’t make it any more appealing logically to believe in whatever it is I am trying to make you believe.

In any case, logical entailments can be powerfully persuasive or they can be used as weapons, attempting to discredit those with whom we disagree (ex. “He believes in pro choice; he believes in killing babies!”). Simply because Y is a logical entailment of X, it does not follow that Joe believes in Y, even if he does in X. We should have a gentle and gracious spirit, but we should stand firm in pointing out these logical entailments. We should further examine our own position, to see if possibly there are some beliefs of ours we need to refine!

EDIT: Thanks to Sean Choi for helping to correct my mistake in listing "strict conditionals" as "material conditionals"!

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                [1] Forget, for the moment, whether or not such a claim is true. The point is to illustrate a common way one sees the logical entailment of a position presented.

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