# The Analytic/Synthetic Distinction in false mathematical propositions

Are all mathematical propositions a priori analytic? The most common example used is 2+2=4. The truth lies in the meaning of the parts of the proposition, and is therefore a priori analytic.

But when the proposition is: 2+2=5 is it still an a priori analytic proposition?

• It's pretty easy to turn false statement into a true one, just negate it. If the (true) negation is analytic then presumably so is the original. Whether 2+2=/=5 is analytic depends on one's philosophy of mathematics, Kant thought it was synthetic, Frege thought it was analytic, and Quine thought the distinction itself doesn't make sense because we can't really tell when the "truth lies in the meaning" and when it does not, "meaning" is an obscure concept. – Conifold Oct 21 '16 at 18:50
• it's not clear to me that 2+2=4 is an analytic proposition. it's not like "all bachelors are unmarried men." – user20153 Oct 22 '16 at 21:46
• en.m.wikipedia.org/wiki/Analytic–synthetic_distinction Kant: 7+5=12 is synthetic a priori. not analytic. dunno if that's right, it is wikipedia, but it makes sense: the meaning of 12 is not already included in the other meanings. – user20153 Nov 6 '16 at 21:43
• sorry, SE does not like dashes in URLs. Google "Analytic-synthetic distinction". – user20153 Nov 6 '16 at 21:45
• "The truth lies in the meaning of the parts of the proposition, and is therefore a priori analytic." that's compositionality, not analyticity. the latter means that the meaning of the one is already included in the meaning of the other. not the case for e.g. 2+2=4. – user20153 Nov 6 '16 at 21:48

## 2 Answers

If we stay with the definition of Analytic according to which :

“Analytic” sentences are those whose truth seems to be knowable by knowing the meanings of the constituent words alone, unlike the more usual “synthetic” ones whose truth is knowable by both knowing the meaning of the words and something about the world,

we have that from : 2+2=5 and the other axioms for arithmetic, by purely logical transformations, we can derive : ¬(2+2=4), i.e. the negation of an analytical sentences.

If - according to the above definition - we can know the truth-value of 2+2=4 without recurring to "information" about the world, this holds also for 2+2=5.

• you cannot derive : ¬(2+2=4) from 2+2=5 by purely logical means. such a derivation would depend on substantial exra-logical assumptions, like the meanings of "2", etc. – user20153 Oct 22 '16 at 21:54
• This definition seems out of place here since the sentence is mathematical. If one subscribes to a version of Kantian "synthetic a priori" one can have synthetic without any "truths about the world". The question really is if the axioms of arithmetic are analytic or synthetic, and this definition is not particularly illuminating on that. I think the best we can say is that it is analytic-in-Peano-arithmetic, which simply means deducible in it, but that has little to do with "knowing the meanings". – Conifold Oct 23 '16 at 20:48
• @mobileink: Mauro made it very clear by saying "and the other axioms for arithmetic". There are absolutely no extra-logical assumptions. It is a purely symbolic matter. For a concrete example of such an axiom system, see en.wikipedia.org/wiki/Peano_axioms#Equivalent_axiomatizations, and define "2" as "1+1" and "4" as "1+1+1+1" and "5" as "1+1+1+1+1", and play the PA proof game. – user21820 Oct 31 '16 at 13:29
• @user21820: "other axioms of arithmetic" may be clear to you and me but let's not assume the same is true for everybody. Also, those axioms are already exra-logical. – user20153 Nov 5 '16 at 22:03
• @mobileink: I have no idea what you mean by "extra-logical". Do you know basic first-order logic? Any logical system comes with a set of axioms. Obviously, you need a meta-system to define a logical system, but no logician calls the axioms themselves "extra-logical". Instead, since we can write a proof of "¬ 2+2 = 5" completely within a system axiomatizing arithmetic (an example of which I have already linked for those who don't know any), every logician considers that proof to be not meta-logical. – user21820 Nov 6 '16 at 6:32

If the first proposition (2+2=4) is apriori analytic, then so is the second (2+2=5); the first happens to be true, and the latter false.

This suggests that you're simply asking whether apriori analytic propositions can be false, given that most examples show only true ones. They can be, a proposition has to be judged to be true or false; so there are true propositions, as well as false propositions; a proposition is not one that is neccessarily true, but a question that is proposed, so the first proposition properly posed is:

Is 2 added to 2 equal to four?

to which, the answer is true; and

Is 2 added to 2 equal to five?

to which, the answer is false.