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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.

Tzetachi
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  • When this is [interpreted in set theory](https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers#Definition_as_von_Neumann_ordinals), 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 03 '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 03 '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 03 '19 at 11:48
  • @Conifold Thank you! – Tzetachi Dec 03 '19 at 11:49

2 Answers2

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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.

Mauro ALLEGRANZA
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  • 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 03 '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 03 '19 at 11:08
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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.

J D
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  • 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 03 '19 at 11:12