Carson:

5. Negative inferences

As we have just seen, one form of improper syllogism is the negative inference, but this form is so common that it deserves separate notice and more lavish illustration. It does not necessarily follow that if a proposition is true, a negative inference from that proposition is also true. The negative inference may be true, but this cannot be assumed, and in any case is never true because it is a negative inference. This can easily be presented in syllogistic form.

Consider two examples:

All orthodox Jews believe in Moses.
Mr. Smith is not an orthodox Jew.
Therefore Mr. Smith does not believe in Moses.

This clearly does not hold up, because the conclusion depends on a negative inference from the major premise. Mr. Smith may be an unorthodox Jew who believes in Moses; or he may be a Gentile who believes in Moses.

Try a second example:

All who have faith in Jesus are saved.
Mr. Jones does not have faith in Jesus.
Therefore Mr. Jones is not saved.

From the perspective of New Testament theology, the conclusion is true; but the syllogism is invalid. In other words, this is an improper way of reaching a true conclusion. If the major premise read "Only those who have faith in Jesus are saved" instead of "All who have faith in Jesus are saved," then of course the new syllogism would constitute a valid argument.

In 2 Corinthians 13:5 Paul writes: "Examine yourselves to see whether you are in the faith; test yourselves. Do you not realize that Christ Jesus is in you-unless, of course, you fail the test?”(NIV). Calvin understands Paul to be saying “that all are reprobates, who doubt whether they profess Christ and are a part of His body"²⁶–an interpretation which, as C. K. Barrett observes, "can hardly be said to follow."²⁷ Calvin seems to be arguing as follows:

Those who have confidence Christ is in them are saved.
Some Corinthians and others doubt (i.e., they do not have this confidence).
Therefore those Corinthians and others are reprobates.

Now I do not believe that the major premise rightly interprets the text in any case; but even if we grant that it represents what Paul is saying, the conclusion does not follow because it is a negative inference. It reflects the Reformer’s position that saving faith entails assurance of salvation; but it is not obvious that Paul is trying to make that point.

D.A. Carson, Exegetical Fallacies (Grand Rapids, Michigan: Baker Academic, 1996), 101-102. [Underlining mine, footnote values and content original, and some minor reformatting.]

[Note: the first example is highly illustrative of the problem. Simple propositions such as, “All orthodox Jews believe in Moses” cannot be converted into “Only orthodox Jews believe in Moses.” This is true for all such biblical statements which simply identify specific persons or groups for whom Christ died.]

We could also create an invalid syllogism following the same structure Carson uses.

All sheep are died-for
Smith is not one of the Sheep.
Therefore, Smith is not died-for [namely, not one for whom Christ died].

The conclusion is invalid. However, this is, in essence, the form of the logic used by many proponents of limited satisfaction to argue that Christ died only for the Sheep in John 10. But such logic is invalid, and will never be valid. And recall, it is always wrong to build theological paradigms on invalid syllogisms and logic.]

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²⁶John Calvin, Commentary on the Epistles of Paul the Apostle to the Corinthians, trans. John Pringle, 2 vols . (Grand Rapids: Eerdmans, 1948), 2:397 .

²⁷C. K. Barrett, The Second Epistle to the Corinthians (London : Black, 1973), 338.