Tag Archive | Math

Logic, Math, and Presuppositionalism

Theists who employ what is known as “presuppositional apologetics”, and more specifically, the transcendental argument for God, often make confused claims about logic. John Frame claims that logic is based on the nature of God[1], and Matt Slick says that the laws of logic are absolute and independent of the human mind[2]. There are others who use such argumentation in slightly different ways (such as Eric Hovind and Sye Tenbruggenate), but the general view seems to be that theism, and specifically Christianity, is needed in order to justify one’s use of logic.

But this view of logic is sorely mistaken. The so-called “laws of logic” are merely conventions that exist within a man-made, formal system. To see why, we merely have to look at “logic” for what it really is – an entire field dedicated to the study of inference and relation. In the past century, many methods of doing this have appeared. Graham Priest writes:

Despite this, many of the most interesting developments in logic in the last forty years, especially in philosophy, have occurred in quite different areas: intuitionism, conditional logics, relevant logics, paraconsistent logics, free logics, quantum logics, fuzzy logics, and so on. These are all logics which are intended either to supplement classical logic, or else to replace it where it goes wrong.[3]

Relevant logic, for example, differs from classical logic in that it attempts to solve the paradox of material implication. The paradox of material implication is the observation that “If the moon is made of cheese, then 2+2=4” is true, even though the antecedent is in no way related to the consequent. Material implication just does not capture what we really mean by “if…then” – yet it is classically valid. Relevant logic attempts to solve this by saying that andecedents must be “relevant” to consequents.

This seems to make sense, but it leads to an interesting feature of relevant logic. Relevant logic is not explosive, which means that unlike classical logic, you can sometimes have contradictions which do not entail the trivial truth of every proposition. Thus, “~(A & ~A)”(the so-called “law of non-contradiction”) is not a theorem of relevant logic (while it is a theorem of classical logic).

This is important because it shows that the unchanging, transcendent view of logic that presuppositionalists talk about just isn’t the case. These logical and mathematic systems are just models we invent to examine different ideas. We can and have changed them, or even thrown them out and started over with different “laws”, many times in the past. Relevant logic is just one example. Some other logics even change the definition of truth. Four-valued logics (often used in computing and electronics) don’t have “P v ~P” as a theorem; and fuzzy logics even have an infinite range f truth values. Intuitionist logics aren’t even concerned with truth, but justification. And these all have real-world applications.

Even mathematics does this. In elliptic geometry, for example, the sum of the angles of a triangle is more than 180 degrees; and Euclid’s parallel postulate is false – there are no parallel lines in elliptic geometry. Now, one might object by saying, “well, that’s just a thought experiment, and doesn’t obtain in reality”. The problem with this is that neither does Euclidean  geometry. When’s the last time you saw a triangle? You haven’t. You never have. The only thing that exists in reality is an approximation of a triangle. A “real” triangle would have to have infinitely thin sides in order to have angles that add up to exactly 180 degrees; and would also have to be infinitely flat. We live in a 3-dimensional (at least) universe, and triangles exist in 2 dimensions. A triangle is just a concept, a thought experiment.

All this points to a simple fact: logic and mathematics are made up. We invented them to describe what we see, and they are only approximations.
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[1] http://www.reformed.org/master/index.html?mainframe=/apologetics/martin/frame_contra_martin.html

[2] http://carm.org/transcendental-argument

[3] Priest, Graham. An Introduction to Non-Classical Logic. 2nd ed. New York: Cambridge University Press, 2008. xvii.