Are we (not) logicians?

Question

This is an inquiry with several specific questions to answer:

• Can questions about logic be answered wholly with rhetoric?
• Do questions about logic need to be phrased symbolically and formally?
• How/when can answers to questions about logic be recognized as wrong?

Let's start with Is the "omniscient-omnipotent-omnipresent" definition of God consistent? In this question, we are asked whether a collection of first-order axioms is consistent. The accepted answer says that yes, the collection is consistent. There are several deficiencies in the conversation:

• The axioms are not formalized
• There is a broad disagreement about how first-order quantifiers behave
• The use of the word "God" provokes a pile of tangential answers, mostly non-logical
• The accepted answer is wrong

I know that that last point might be contentious. So, in order to isolate the problem, I asked a recent question with a tighter focus, Can omnipotent beings exist?. I also wrote an answer which quotes standard mathematical work by well-established logicians. Because I suspect that there was not much reading of my sources going on, I'll quote explicitly here. First, from Pratt 1999, p5:

A simple case of interference is given by a Chu space having a constant row. If it also contains a constant column, then the two constants must be the same. Thus if A has a row of all 1’s it cannot also have a column of all 0’s. And if it has two or more different constant rows then it can have no constant columns at all.

This phenomenon formalizes a well-known paradox. Viewing points as objects, states as forces, and r(a, x) as 1 just when object a can resist force x, an immovable object is a row of all 1’s while an irresistible force is a column of all 0's.

He gives no proof, although I have requested one on MSE. Second, quoting Tao 2009, the first paragraph:

A fundamental tool in any mathematician’s toolkit is that of reductio ad absurdum: showing that a statement X is false by assuming first that X is true, and showing that this leads to a logical contradiction. A particulary pure example of reductio ad absurdum occurs when establishing the non-existence of a hypothetically overpowered object or structure X, by showing that X‘s powers are ���self-defeating”: the very existence of X and its powers can be used (by some clever trick) to construct a counterexample to that power. Perhaps the most well-known example of a self-defeating object comes from the omnipotence paradox in philosophy (“Can an omnipotent being create a rock so heavy that He cannot lift it?”); more generally, a large number of other paradoxes in logic or philosophy can be reinterpreted as a proof that a certain overpowered object or structure does not exist.

In both cases, the inconsistency of omnipotent beings is taken as so obvious that it is used as a starting point for deeper mathematical inquiry, rather than being worthy of investigation or proof.

My question was closed as a duplicate of the first one. This is incorrect; my question refines the first question, and the accepted answer to the first question is refutably wrong.

So, are we logicians? And if we are, then what does that mean for answers which don't use logic or are wrong?

Answer 1

For myself, I like to see philosophical questions addressed using logic. Many philosophers in the analytical tradition make use of formal logic where appropriate. Though it is fair to say that a lot of philosophical issues cannot simply be resolved by use of formal logic.

The short answers to your questions seem to be:

1. Would it be nice to see more formal logic used where it makes sense to do so? Yes.
2. Should questions about formal logic be expressed using symbols? Yes.
3. Are all questions, even those about logic, amenable to symbolic treatment? No.
4. Can wrong answers about logic be recognized as definitely wrong? Sometimes, and where these are symbolic, it may be possible to prove this.

I don't see how this helps in the example you quote. It is fairly easy to grasp that an immovable object and an irresistible force cannot coexist in actuality without leading to a contradiction. A formal proof of this might be instructive, but it does not change the available responses.

1. We can go along with Descartes and suppose that an omnipotent being can create contradictory states of affairs and that if they do so then it just follows that we mere mortals are unable to understand them. This seems to be the view taken by the accepted answer to the question you reference.

2. We can suppose that omnipotence means that an omnipotent being is able to do anything that it is logically possible to do. A number of theists seem to prefer this option. This would exclude the immovable object and irresistible force from qualifying as a counterexample to omnipotence.

3. We can suppose that omnipotence means something weaker, such as that an omnipotent being has no external limitations on what they are able to do. This seems to be the position preferred by Peter Geach.

4. We can go along with Richard Swinburne and distinguish a first order and second order sense of an omnipotent being's powers and say that an omnipotent being can choose to restrict what they are capable of doing, but these self-imposed restrictions do not entail a lack of omnipotence in the broader sense.

My purposes in answering this question are:

1. I have shown that the question can be answered without recourse to symbolic logic.

2. I have shown that there are a number of possible answers. For myself, when a question has many answers I would rather list them than argue for my own favourite.

3. None of the answers are recognizably wrong and all have been defended by competent logicians. Obviously, any given reader might find some answers to be more plausible than others, but such convictions will inevitably be based on deeper considerations than an appeal to elementary formal logic.

Answer 2

My question was closed as a duplicate of the first one. This is incorrect; my question refines the first question, and the accepted answer to the first question is refutably wrong.

You are right in saying you are right. But, here we find the mechanism to referee disputes over right and wrong democratic. Thus, no matter how irrefutable your logic to a logician, one need only discover that an ignorance of logic among many is often politically more powerful than a crafty argument that obeys the detailed strictures of a well-composed argument. This should be no surprise given the lives of Themistocles, Ignaz Semmelweis, and others.

So, are we logicians? And if we are, then what does that mean for answers which don't use logic or are wrong?

If by we, you mean the average contributor to this site, then I think it's a questionable assertion. I think its fair to say that the utterance 'we are aspiring logicians' is true, but as a non-logician in regards to my formal education, I can say there is no royal road to being a logician, and few have carved out the time and energy to cross the pons asinorum, my self included.