Without computers, how can you conjecture the (in)validity of a long convoluted argument in Predicate Logic?

Question

_{Source: p 287, Sweet Reason: A Field Guide to Modern Logic (2010 2 ed) by Henle, Garfield, Tymoczko.}

Predicate is the reason we started on deductions. In Sentential, remember, we can verify that an argument is valid by using truth tables or the short-cut method. With Predicate, we have no such tool. We can show that an argument is invalid if we exhibit a universe in which the premises are true and the conclusion is false, but (until now) we have had no means of showing that an argument in Predicate is valid.

In the 2 Logic textbooks read (the above and Hurley's), each exercise on deductions (e.g. the following) divulges the (in)validity of the argument, and so enables you to know instantly what to do. But without computers, for arguments with > 10 long, convoluted premises: what if the (in)validity were concealed or tacit?

Per the above, I already know that a Valid Argument can be proven only with Deduction, and an Invalid Argument disproven with a Counterexample. But before 1. writing a Deduction or 2. finding a counterexample, you must have conjectured the argument, respectively, 1. valid and 2. invalid (Otherwise, how would you know to write a Deduction or find a counterexample?!?) So how would you conjecture (in)validity before doing the work?

Answer 1

As the quoted passage says, there is no way of showing that an argument is valid in predicate logic, other than carrying through its deduction. That means that if the argument is not valid, you would never be able to tell that using just deduction.

On the other hand, you can tell that an argument is invalid if you provide a counter model (i.e. an interpretation that makes the premises true and the conclusion false). But if the argument is valid you would never find such a model.

The above make first order logic have the fun property called undecidability. That means that even with computers you cannot tell for an arbitrary first order argument whether it is valid or not.

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A good way to get some intuition as to whether an argument is valid or not is to translate it back to natural language and see if it makes any sense. However, you're talking about long and convoluted arguments, so this method will probably not work in such cases. And as already said, even if you feed such arguments to a computer you cannot hope for it to get you a result in finite time. That's just what predicate logic is like, no way around it.