A God Can Neither Be Proved Nor Disproved

[Calilasseia]

@boomer47

In formal logic, every rule of inference begins IF A------. IE the conclusion is true if and only if the premis is true. There are exceptions, but I honestly forget what they are. I’m sure some one here knows.

Methods of Logic by Willard Van Ormand Quine is your friend here. In particular, Chapter 6, Consistency and Validity. In that chapter, Quine considers simple propositions consisting of atoms (i.e., statements that can be either true or false, represented by simple letters) and combinatorial operators (logical and, logical or, etc). From that chapter:

A truth-functional schema is called consistent if it comes out true under some interpretation of its letters; otherwise inconsistent. A truth-functional schema as called valid if it comes out true under every interpretation of its letters. The schema p(~q), for example, is consistent, for it comes out true when ‘p’ is interpreted as true, and ‘q’ as false; but it is not valid, since there are other interpretations of ‘p’ and ‘q’ which make it come out false.


The way to test a truth-functional schema for validity and consistency is obvious: we carry out a truth-value analysis and see whether we get T in every case, (showing validity) or F in every case (showing inconsistency) or neither. Two examples of validity and inconsistency are respectively “p ⊃ p” and p(~p);


p ⊃ p : p = T, p ⊃ p = T, p = F, p ⊃ p = T


p(~p): p = T, p(~p) = F, p = F, p(~p) = F

The above appears (with somewhat different symbolism) on page 28 of my PDF copy of the book in question.

Quine considers the implications of validity and inconsistency further in the chapter, viz:

Validity is not to be thought of as a term of praise. When a schema is valid, any statement whose form that schema depicts is bound to be, in some sense, trivial. UIt will be trivial in the sense that it conveys no real information regarding the subject matter whereof its component clauses speak.

He then continues with:

Though the statements illustrative of valid schemata are always trivial in the sense noted above, they are not always trivial in the sense of being, like (1), recognisable on sight. Valid schemata may run to any length of complexity; and some even of moderate length cannot be recognised as valid withough substantial computation.

Quine continues to note that certain rules apply when substituting a valid (or inconsistent) schemata in place of one of the predicate letters of another schema, a facility that is of particular utility once one moves on to complex quantificational schemata. Expansion of the relevant concepts begins in Chapter 7, Implication.

Though the text is terse and dense, it is informative if one persists reading it with patience.