In the context of apologetics, the terms "internal critique" and "external critique" are not uncommon. Much of apologetics, from a presuppositional perspective, is performing internal critiques of opposing worldviews. In this post, it is my objective to formally define these notions and give some properties of each.
An internal critique is an attempt to show the incoherence of a system of thought. Put another way, it seeks to find a logical incompatibility between some propositions in a given system. In short, it attempts to find a contradiction within a certain system of thought.
To put this formally, let I be a set of axioms and the propositions that logically follows from them. The goal of an internal critique on the system I is to demonstrate that I ⇒ F, where F refers to falsity or contradiction, and I ⇒ F means that F logically follows from I. This is done by presenting a deduction, or a sequence of inferences, from I, which conclude that there is a contradiction that follows from I. Formally, this is equivalent to showing that I ⊢ F, where '⊢' means that there exists a proof, or derivation, from the left-hand side to the right-hand side. Thus, I ⊢ F means that a contradiction can be formally derived from I. If the inference procedure '⊢' is sound, then if I ⊢ F, then I ⇒ F - that is, that if F can be deduced from I, then F logically follows from I.
A valid internal critique means that I is an incoherent system. Thus, if I ⇒ F, on pain of irrationality, one must seek to make I a coherent system, by eliminating and/or changing some axioms of I. Failing to change one's position, in the face of a valid internal critique, is nothing less than stubborn irrationality. However, this assumes that the critique is valid, and that the deduction is clear and evident. Certainly, if an internal critique is invalid, then one has no reason to change one's position. In fact, to change one's position in such a case would be irrational. Furthermore, even if an internal critique is valid, it may not necessarily be evident, if the deduction used to prove that a contradiction follows from I is not fully demonstrated, is missing steps, or something of the like. This is not an issue with formal proofs, as a formal proof that does not contain all of the necessary inference steps is not a valid proof. Rather, this is an issue in informal arguments, where generalizations and enthymemes are common. In such a case, the burden of proof is upon the critic to set forth an argument that contains the full deduction, with justifications for each inference. Short of this, if how F follows from I is not evident because the deduction I ⊢ F has not been completely set forth, then the person holding to I is not necessarily irrational for not changing his position in this case.
In the ideal world, people would perhaps hold to systems that consist of nothing more than a set of axioms and the propositions that follow from them. Unfortunately, we are fallen, finite beings, and cannot help being inconsistent, at least at some points, no matter how hard we try to be consistent. Thus, a person's system S might in reality consist of I, plus some other propositions P, which do not follow from the axioms of I. In such a case, a valid internal critique could be performed upon S, but this would not necessarily mean that the person holding to S would need to change anything in I - it might simply mean that P needs to be abandoned. In other words, it does not follow that a particular axiomatic system I is incoherent because a particular person happens to hold to I plus some propositions P, which together are incoherent.
Thus, a valid and complete internal critique of a person's position means that person must change something in his position to make it consistent, on pain of irrationality. However, in practice, not all purported internal critiques are actually valid internal critiques. For many common invalid internal critiques, there are two possible responses one can employ to demonstrate their invalidity.
Response #1: Show I ⊢ F to be an invalid or unsound deduction. That is, at some point in the inference procedure, a sound inference rule was incorrectly applied, or an unsound inference rule was applied. Of course, this can only be demonstrated if the critic demonstrates his inference procedure. If a particular part of the inference is unstated, then the respondent is justified in asking the critic to "fill in" the remaining parts of the inference, so that it may be evaluated. Short of this, as previously stated, the respondent is not necessarily irrational in holding to his position, even if the critique happens to be valid.
Response #2: Show that the critic has shown that I' ⊢ F, not I ⊢ F, where I' is a different set of propositions from I, though possibly similar and/or having some overlap. In this case, the critic has committed the straw man fallacy, proving that a position to which the respondent does not hold is incoherent. Unfortunately, such fallacious arguments are commonly touted as being valid internal critiques. In such a case, it suffices for the respondent to show that the critic has used at least one premise that is not in I in order to deduce F. This renders the supposed critique invalid, since the critic must show that I ⊢ F. In this case, the critic has unwittingly performed an external critique.
An external critique is an attempt to show that a particular position is contradicted by another position that it does not subsume - that is, it seeks to show that a particular position is contradicted by something outside of itself. There are generally two motivations for performing an external critique of an opposing position. The first is to show how an opposing viewpoint is ruled out by one's own system. This is useful for dialogue with others who hold to one's own position, in demonstrating how their position rules out another in such and such a way. Thus, this type of external critique is not intended to convince someone of an opposing mindset so much as it is intended to prove something to others of a similar mindset. The second motivation is to show that an opposing position is contradicted by certain propositions that one feels are "self-evident" or that "everyone should accept." Thus, this type of critique attempts to persuade an opponent to a different position, by appealing to some notion of universal rationality.
A valid external critique means that some system I is contradicted by some other system E, where E is "external" to I - that is, that E contains set of propositions that are not in I, and E does not logically follow from I. Formally stated, the goal of an external critique is to show that E ⇒ ~p, where p is a proposition in I, and '~p' means "not p", representing the negation of p. This is done by presenting a deduction E ⊢ ~p, which concludes that ~p follows from E. In the case that '⊢' is sound, then E ⇒ ~p follows from E ⊢ ~p.
Unlike a valid internal critique, a valid external critique does not necessarily say anything about the truth or rationality of holding to a particular position. It could very well be the case that E is incoherent and that I is coherent, in which case an external critique of I from E means nothing to an adherent of I.
There are generally four kinds of responses to external critiques that one can employ.
Response #1: Show E ⊢ ~p to be an invalid or unsound deduction. That is, at some point in the inference procedure, a sound inference rule was incorrectly applied, or an unsound inference rule was applied. This can be useful if the respondent wants to demonstrate, for some reason, that the external critique is invalid.
Response #2: Show that p is not a proposition in I. If this is the case, then the critic has committed a version of the straw man fallacy, by showing that E contradicts a proposition that is not in I to begin with. Thus, the external critique is not valid in this case.
Response #3: Do nothing, or simply point out the fact that it is an external critique. The critic has done nothing to show that the respondent is wrong or irrational to hold to his position, but only that the critic's position contradicts the position of the respondent. Even external critiques that appeal to propositions that are "self-evident" or that "everyone should accept" are of this nature, for those propositions simply belong to the personal system of the critic - that is, of course, unless the critic can prove that those propositions are actually axiomatic for I, in which case his critique would not be external at all, but rather internal.
Response #4: Perform an internal critique upon E, to show that it is an incoherent system. This can be particularly useful if a particular critic is touting his system as "self-evident" or one that "everyone should accept." By performing an internal critique of the critic's position, the respondent can not only silence the critic, but can also show that the critic's objections are objectively baseless, following from an incoherent system.
It is important to keep in mind the difference between internal and external critiques, and what each entails. Hopefully, doing so will eliminate confusion and enable us to be more effective in our apologetics.
Soli Deo Gloria!