This isn’t a logic textbook, yet it isn’t quite a worldview approach to logic. It is something of both, yet completely neither. I still liked it, though.
He begins with a theological “grounding” of logic, which amounts to a summary of his and Frame’s approach to worldview. It’s good, but it lasts about 200 pages before you get into the “nuts and bolts” of logic.
He then gives a primer on deductive syllogisms, propositional logic, quantification, functions, sets, modal logic, and much else. I did enjoy the fact that he pointed out how pure systems like Russell’s and others are so formal as to have little content. This is analogous to the desire for “pure being.”
64: Logic is an aspect of God’s mind. It reveals God’s attributes.
89: Logic is God’s self-consistency
Key argument: Logic is personal, but it doesn’t depend on any one human person, since if all humans perished, logic would still be true. It is transcendent, displays his attributes, and is part of God’s speech (80).
This next part is important, as it provides another foundation for the rest of the book’s argument:
Axioms of Propositional Logic
Principle of Tautology: (p V p) ⊃ p
You might need to learn this one. Poythress’s work is unique in the sense that he puts every single axiom through a truth table.
Principle of Addition
q ⊃ (p V q) “If it is dark, then (either it is raining or it is dark)”
The Principle of Permutation
(p V q) ⊃ (q V p)
If (either it is raining or it is dark), then (either it is dark or it is raining)
The Associative Principle
(p V (q V r)) ⊃ (q V (p V r))
If (either it is raining or (it is dark or it is cold)), then (either it is dark or (it is raining or it is cold))
The Principle of Summation
(q ⊃ r) ⊃ ((p V q) ⊃ (p V r))
If (it is dark implies it is cold), then (the assumption that (it is raining or it is dark) implies the conclusion that (it is raining or it is cold)).
While it might not seem like it, these are powerful tools and the reader is encouraged to work through a few of them in truth tables in the appendices. The book has some severe drawbacks, in that it isn’t a logic textbook, and some important concepts are woefully underdeveloped (like modal logic). But I did enjoy it and parts of it should be read.