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Thanks, Tim. Maybe it would help to share the problem I'm trying to straighten out in my head.

It regards Hoppe's so-called Apriori of Argumentation. Put simply, one cannot argue that one cannot argue. Is that axiomatic or a priori? If one, why not the other. Thanks.

I'm a relative newbie to philosophical terms like this and already find myself confused by the most basic tenets heheh

terms, not tenets

Axioms are relatively fragile compared to the things which usually get declared as a-priori. You can chop and change them, replace them as you like, and you end up with different formal systems.

Two easy examples of axiomatic systems are propositional logic and predicate logic. Propositional logic consists of an alphabet of propositional symbols standing for sentences, a family of logical connectives, and specified ways of combining them. "A and A"? That's fine. "And and"? Not fine.

Predicate logic is propositional logic augmented with a new structure, quantification, which lets you express things like "All men are mortal" and "Some things are red". Only constants and variables, like the propositional symbols above - names of elements of the alphabet for the language - can be quantified over in the most common ("first order") one.

Why does propositional logic not contain quantifiers? Why can't you quantify over logical relationships of symbols in first order logic? The axioms say so.

Axioms set out rules for manipulating symbols, at a minimum. Mathematical structures can have different axiomatisations which are equivalent (like the usual version of set theory + the axiom of choice and the usual version of set theory + zorn's lemma) in the sense that they prove the same things.

For most of its history and in most of its content (even after Euclid, the father of axiomatic systems), math was done without much reference to axioms. What this reveals is that axioms are something which can be posited or rejected, they are not necessarily 'there', in the sense of 'always applying'.

What is interesting about them, really interesting, is that they don't seem to be arbitrary for the structures we care about. While axioms at a minimum just set out rules for pushing symbols around, you can set up axioms to 'capture' some behaviour., to make the system of rules they engender reflect some useful or elegant things about something else - even if that something else is math itself.

For an example of the latter, setting out axioms to study math itself, there's a notion in logic called compactness. A logic (yes, an entire way of formal reasoning!) is compact whenever a collection of sentences the logic can produce (like "A and A" or "Not (~A or B ) xor (A implies C)", collecting different syllogisms together) can be satisfied so can every finite subcollection of syllogisms. There's a notion in a field of math which used to be distinct, called topology, that calls a space (like 3d space we live in) 'compact' if every way of throwing circles over elements of it (a cover) can be achieved by throwing a finite subcollection of elements over it (a finite subcover). That these two things are so similar, the compactness of a logic and the compactness of a topological space, turned out not to be a coincidence - and there are ways of studying systems of logic using intuitions we developed about notions of space. Setting up axioms like that allows you to penetrate structures which seem to be there, blurring the lines between creation and discovery.

For the a-priori? Things which are true without relying on experience. Things like 'all bachelors are unmarried men' or "red objects are coloured". We can't seem to play about with those things except in fiction or in acts of the imagination, they seem to hold in virtue of the conventions of language, or of logical relationships between facts that can't help but be true, or false in some cases. Necessarily true or necessarily false. This part of of the a-priori, called the analytic a priori, are things that hold by virtue of their meaning alone; they are self evident if understood. One thing is contained in the concept of the other, as it was originally put.

Then there's the 'synthetic' part of it, these are things which are necessarily true not by virtue of one thing simply meaning the other, but of an idea reflecting the essential nature of its topic. For example, "the angles of a triangle sum to the sum of 2 right angles". That's something necessary about triangles which doesn't seem to be inherent in our idea of it, but something we worked out after the fact.

It is popular to conceive of mathematics and logic as being disciplines devoted to the exposition of a priori truths, geometry was even the original paradigmatic example of synthetic a priori in Kant. Whether this is true, whether the distinction between a priori and a posteriori actually holds, whether the distinction between analytic and synthetic actually holds, is all debatable. These are just the basics from a biased commentator.

Thanks, Drake. Combined with some other glances at the difference between apriori and axiom, your post helped this all sink in a little better.

I think it is appropriate to label Hoppe's argumentation as the a priori of argumentation. It's also axiomatic, but I'm thinking that axioms are the starting point of deductive reasoning, no matter how insane the prospects of that deduction.

In other words, all a priori truths are axiomatic, but not all axioms are a priori. That's where my head is at the moment.

Care not to get caught up in concepts of time with respect to a priori. For "before" maybe more accurate to say, "without need of," or logically prior, which logical priority is not at all a temporal priority.

this only just sank in this morning. the implication of 'before' experience implies some kind of universal mind and that is a much bigger argument than I was after. ;)

Yes, they are different. Fdrake said enough to be considered. In, layman's terms, axiom is bound up in logic, while a priori is a monster of its own kind.

I would simply point out their similarities. These, I'll call them devices/conceptual instruments (cartesian doubt, tabulation rasa, a priori, bedrock, axiom, premise, &c.), are attempts to establish an epistemological ground of certainty upon which we can confidently assemble a framework of understanding. None of these devices actually succeed in providing that indubitable self-evident truth, and imo, I don't think anything like that exists. However, they do give us a basis from which we can methodologically conduct philosophical thought experiments (some better than others). So that's something.