# Kant's Notion of Synthetic A Prioiri as Logical Entailment

Is there something wrong about interpreting Kant's notion of synthetic a priori statements to be logical entailments?

I understand, I think, that Kant didn't want to say such statements (e.g math theorems) were a priori because the truth of the predicate was not in the subject. The truth of the propositions follows because of the way that humans think (synthetic), but don't require additional experience (a priori).

Now while Kant didn't mean something like logical entailment, that's how I want to see what Kant was doing. He probably didn't see such a thing because he was still using Aristotelian logic.

The reason I want to say such a thing is because, at least for the cases of math theorems. It doesn't follow from the definition of a triangle that its angles add up to 180 degrees (on a simple definition of triangle). One also requires some Euclidean axioms. So it's those axioms together that entail the conclusion.

In summary, I want to understand a priori propositions as true in virtue of a single definition where synthetic a priori deductions require several propositions.

Is there something wrong with viewing Kant's ideas like this?

• The problem is that such interpretation is neither Kantian nor synthetic. This was more or less Frege's view of mathematics, later elaborated by Russell and others. But they explicitly disagreed with Kant, and called mathematics analytic, not synthetic (although Frege left room for some axioms themselves to be synthetic). It is true that Kant needed synthetic judgments to make up for the poverty of Aristotelian logic, but while what they accomplished can be reconstructed the Frege-Russell way, it is very doubtful that mathematics in practice functions that way. Commented Sep 18, 2019 at 4:57
• "In summary, I want to understand a priori propositions as true in virtue of a single definition where synthetic a priori deductions require several propositions." I'm probably being slow but I can't make sense of this. Deductions in general for Kant require several propositions, so what distinction are you making here? It's actually modern logical entailment that doesn't require multiple propositions, where 'A, therefore A' is valid whereas it is not for Kant. Commented Sep 18, 2019 at 5:39
• So a case of an a priori proposition, I am thinking something like "All bachelors are unmarried men." That bachelors are unmarried men follows from the definition of "bachelor". But Kant won't say that a mathematical theorem like that all triangles have a total of 180 degrees is a prior because that isn't part of the definition of a triangle. For Kant, you need to do some mental work to get to that. What I want to say is that Kant's "mental work" is really just logical entailment (of more than 1 premise). You can take the bachelors are unmarried men to be a 1-premise entailment.
– user32564
Commented Sep 18, 2019 at 5:46
• @transitionsynthesis To say a bit more, Kant will say that the mental work always works as such because it's the way all humans are configured to perceive things, but I want to say he's really just hitting the notion of logical entailment. That triangles have 180 degrees doesn't follow because of the way humans must perceive the world. It's an entailment from several Euclidean propositions.
– user32564
Commented Sep 18, 2019 at 5:55
• 'All bachelors are unmarried men' counts as analytic, I'd say, rather than a priori. Commented Sep 18, 2019 at 8:32

Logical entailments of what, in particular?

Clearly Newtonian physics -- the set of logical entailments of Netwon's principles, is not included. Newtonian definitions clearly involve guessing and observation and are definitely not apriori.

Logic may be how synthetic apriori content has to be related internally, since the illogical is not really "known". But deduction cannot select the body of apriori knowledge out from the remainder for you.

For logic to apply, the most basic synthetic apriori principles still have to come from somewhere. And that does not allow them to be identified as entailments, which would be dependent on more synthetic apriori things. As in that case they aren't the most basic such things. And if those most basic aspects were themselves analytic, then all their entailments would be, too.

Logic is only the combinatorial content of our fund of requirements for thought, other principles, like our internal sense of geometry, or of what it means to be an element of something, need to have those means of combination applied to them. They cannot arise out of the logic. As non-Euclidean geometries and nonstandard models of set theory prove, they are not even literally true or necessary -- they are just necessarily imposed on humans as an initial frame of reference.

• "Newtonian defintions are observed and not a priori." Highly debatable... How you "observe" the Law of inertia ? Commented Sep 18, 2019 at 9:06
• @MauroALLEGRANZA With ramps and balls and hours and hours of bookkeeping, as Galileo and then Newton actually did? Are hou just picking on the choice of vocabulary, or do you actually think some part of that is apriori?
– user9166
Commented Sep 18, 2019 at 10:33