If an argument is valid, that doesn’t make it true

What does this mean and why does it matter? I’d like give a brief sketch of an introduction to proofs, formal logic, etc., using a quick follow-up to my review of Dr. Richard Carrier’s Proving History. I need only point to three lines of a “proof” he gives (p. 106) with two premises and his first conclusion:
Premise 1 is “BT [or Bayes’ Theorem] is a logically proven theorem.”
P2 is “No argument is valid that contradicts a logically proven theorem.”
His conclusion? “Therefore, no argument is valid that contradicts BT”.

What does the word “valid” mean to logicians, mathematicians, and all others who deal with formal systems, proofs, etc.? Simplistically, it means that the conclusion “follows from” the premises (or, alternatively, if the premises are true, then the conclusion must be true). Most importantly, a valid proof or argument for logicians, philosophers of logic, mathematicians, etc., can be false. Here’s an example:
P1: The president of the United States is the emperor of the world.
P2: I am the president of the United States.
Conclusion: I am the emperor of the world.

This is a valid argument. If the premises (P1 & P2) are true, then the conclusion is necessarily true. It just so happens, however, that neither premise is true. Formally, this means that the proof/argument is not sound. For an argument/proof to be sound, it must be valid AND the premises must be true. This has two consequences for Dr. Carrier’s first “proved” conclusion. First, it is valid. If the premises were true, then the conclusion would necessarily be true. However, the premises are not true. An argument can be valid and contradict BT, because an argument can be valid and be completely wrong (I am not the president of the US nor am I the emperor of the world, yet the argument I proffered above is valid).

The lesson? Mathematicians & logicians are very careful to distinguish between when they are speaking informally and when they are speaking in some formal language (e.g., the predicate calculus of Russell & Whitehead, complex analysis, linear algebra, etc.). Be wary of anybody who presents something that looks like it is a formal proof or similar use of logic. Sometimes, even a careful thinker can detect when someone is attempting but failing to make a formal argument/proof even if they are not familiar with formal logic. For example, one does not need to know that an argument in logic can be valid and be completely wrong to be suspicious of Carrier’s “proof”. Examine his first premise. What is a “logically proven theorem”? Better yet, what is an illogically proven theorem? A proof is either logical or it isn’t a proof. Also, when someone throws around terms like “theorem”, “proven”, “valid”, etc., without defining them then they are not offering anything a mathematician or logician would. The reason anything can be proven in math or logic is because formal means “rigorous” (there is no ambiguity in what terms or notations means).

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