Is it 1+1 or “1+1” that is a formula of addition? To my intuition, it is the former, and the latter seems to be a name of the formula. The reason why I ask this question is that provided my intuition is correct, 1+1 is not equal to 2 by Leibniz’s law, because if we take P to be a predicate for “is a formula of addition,” P(2) is false while P(1+1) is true. I’m not trying to deny that they are somehow in equivalence relation, but questioning if Leibniz’s law is enough to capture the relation. I guess my intuition is wrong in the first place but I don’t get why it is.

  • Welcome to SE Philosophy! Thanks for your contribution. Please take a quick moment to take the tour or find help. You can perform searches here or seek additional clarification at the meta site. Don't forget, when someone has answered your question, you can click on the checkmark to reward the user! – J D Dec 3 '19 at 9:13
  • When this is interpreted in set theory, for example, 1+1 and 2 give two different constructions of the same set. By abuse of notation, 1+1 can name the construction itself ("formula"), or the set constructed. Only with the latter use 1+1 = 2, and neither of these uses mentions 1+1, i.e. uses the name “1+1” itself. – Conifold Dec 3 '19 at 11:16
  • @Conifold Could I ask what you meant by “two different constructions”? How do the construction 1+1 makes and what 2 makes differ? – Tzetachi Dec 3 '19 at 11:32
  • 2 forms the set {Ø, {Ø}} directly, and 1+1 applies the successor operation n ∪ {n} to 1 = {Ø}. – Conifold Dec 3 '19 at 11:48
  • @Conifold Thank you! – Tzetachi Dec 3 '19 at 11:49

1+1=2 is a formula (an expression of mathematical language that express a statement) and "1+1=2" is the way to refer to the expression: correct.

1+1 is a term, i.e. an expression that denotes a number.

Thus, it is not a formula.

The principle of the Indiscernibility of Identicals (the converse of the Identity of Indiscernibles) in its predicate logic formulation states :

x=y → ∀F(Fx ↔ Fy).

If we apply it to "objects", like numbers, we have to use the predicate variable F to express properties of numbers, like e.g. "to be Even".

In this case, what we get is correct :

1+1=2 → (Even(1+1) ↔ Even(2)).

But the same applies if we "move at the meta-", i.e. considering the mention context: "1+1" is a term and "2" is a term, while "1+1=2" is a formula.

  • Thanks for the answer! Now I got it where my thoughts went wrong. I mistook including “+” operator symbol for the property of the object; it’s obviously “1+1” that includes the operator symbol. – Tzetachi Dec 3 '19 at 11:06
  • @Tzetachi - correct; is like "Bob is the Father of John". "Father of" is not a property of John but a function that maps an individual to another individual (its father). – Mauro ALLEGRANZA Dec 3 '19 at 11:08

My understanding of the use-mention distinction is that the former refers to a disposition (behavior) or proposition (meaning bearer) while the latter is merely a reference (syntactical expression such as a string). In this way, dispositions correspond to correspondent truths (combining two individual cookies in results in a state of affairs that a box has a pair of cookies), propositions correspond to coherent truths (1+1=2 -> 2=2 since it is an example of the axiom that any number is equal to itself), and the metalinguistic string "1+1=2" is just a label that works pragmatically and could be "one and one is two" or "eins und eins macht zwei" and so on. As far as formulas are concerned, I believe "1+1=2" would be a fact, not a formula since it is a literal or particular instance of the abstraction "a+a=b" which is universal in nature. Note, I've used the expression "1+1=2" because strictly speaking logical conjunctions aren't predicates, but require a relationship in this case essentially expressed by a copula which is a grammatical form in this expression which expresses identity, equivalence.

Brighter minds might have a better answer.

  • Thank you for the answer! I mistook the relation between “1+1” and 1+1, even though I knew the distinction should have held. – Tzetachi Dec 3 '19 at 11:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.