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I have an applied math background. What resources are recommend to understand the symbols used here?

The base notation is from formal logic. A (dense) overview of first-order logic can be found on this wikipedia page: https://en.wikipedia.org/wiki/First-order_logic

It's much nicer to get slowly walked through it with a textbook, but I'm blanking on the name of the book I used.

Thanks trobertson. This may be what I need.

So, may I presume that the notation of (Prof) Carl Hewitt’s Direct Logic is consistant with that of First-Order Logic?

Actually, the notation used above is for strongly-typed higher-order theories.

See the following:

Here is a free text book that covers most of the notation, except the quantifiers, around page 30 or so:

For the quantifiers, check out the Wikipedia page here:

For deeper coverage, the section on first order logic here is quite good: