One does not have to directly assume that 2 + 2 = 4. Note that all of the following statements correctly describe first-order logic (given the appropriate definitions of 2, 3, and 4):
So ∀x ∀y ((x + y) + 1 = x + (y + 1)) ⊢ 2 + 2 = 4 (in the context of first-order logic). By the soundness theorem for first-order logic, then,
∀x ∀y ((x + y) + 1 = x + (y + 1)) ⊨ 2 + 2 = 4. (i.e. In all models where “∀x ∀y ((x + y) + 1 = x + (y + 1))” is true, “2 + 2 = 4″ is also true.)
So, yes, an assumption is being made, and that assumption is “∀x ∀y ((x + y) + 1 = x + (y + 1))”. To evaluate the plausibility of that assumption, we now must specify a domain of discourse, the meaning of ‘+’, and the meaning of ‘1’. Eliezer means to (I hope!) specify the natural numbers, addition, and one, respectively. Addition over the natural numbers is itself an abstraction of moving different quantities of discrete non-microscopic objects (like apples) next to each other, and observing the quantity after that. Now we can evaluate the plausibility of “∀x ∀y ((x + y) + 1 = x + (y + 1))”, or, to translate into English, “If you take a group of y objects and move it next to a group of x objects, and then move one more object next to that newly formed group of objects, and then observe the quantity, then you will observe the same quantity as if you take a group of y objects and move 1 more object next to it, and then move that newly formed group of objects next to a group of x objects, and observe the quantity, no matter what the original numbers x and y are.” (So symbolic language is useful after all....) We observe evidence for that statement every day! It is a belief that pays rent in terms of anticipated experience. So, yes, we could assume differently, but we would be doing so in the face of mountains of evidence pointing in the other direction, and with something to protect, that won’t do.
Notes:
How do we know the soundness theorem for first-order logic is true? For one thing, we observe inductive evidence for it all the time (the rules of first-order logic are incredibly intuitive, see page 51 (PDF page 57) of A Primer for Logic and Proof for an enumeration), just as we see inductive evidence for “∀x ∀y ((x + y) + 1 = x + (y + 1))” all the time. We can also use mathematical induction (on the length of a deduction) to prove the soundness theorem, but this probably won’t convince someone who is skeptical of even first-order logic.
It might be helpful to think of (X ⊨ Y) as meaning P(Y|X) = 1.
This is a tricky one...here’s my attempt:
One does not have to directly assume that 2 + 2 = 4. Note that all of the following statements correctly describe first-order logic (given the appropriate definitions of 2, 3, and 4):
∀x ∀y ((x + y) + 1 = x + (y + 1)) ⊢ (2 + 1) + 1 = 2 + (1 + 1)
(2 + 1) + 1 = 2 + (1 + 1) ⊢ 3 + 1 = 2 + (1 + 1)
3 + 1 = 2 + (1 + 1) ⊢ 4 = 2 + (1 + 1)
4 = 2 + (1 + 1) ⊢ 4 = 2 + 2
4 = 2 + 2 ⊢ 2 + 2 = 4
So ∀x ∀y ((x + y) + 1 = x + (y + 1)) ⊢ 2 + 2 = 4 (in the context of first-order logic). By the soundness theorem for first-order logic, then, ∀x ∀y ((x + y) + 1 = x + (y + 1)) ⊨ 2 + 2 = 4. (i.e. In all models where “∀x ∀y ((x + y) + 1 = x + (y + 1))” is true, “2 + 2 = 4″ is also true.)
So, yes, an assumption is being made, and that assumption is “∀x ∀y ((x + y) + 1 = x + (y + 1))”. To evaluate the plausibility of that assumption, we now must specify a domain of discourse, the meaning of ‘+’, and the meaning of ‘1’. Eliezer means to (I hope!) specify the natural numbers, addition, and one, respectively. Addition over the natural numbers is itself an abstraction of moving different quantities of discrete non-microscopic objects (like apples) next to each other, and observing the quantity after that. Now we can evaluate the plausibility of “∀x ∀y ((x + y) + 1 = x + (y + 1))”, or, to translate into English, “If you take a group of y objects and move it next to a group of x objects, and then move one more object next to that newly formed group of objects, and then observe the quantity, then you will observe the same quantity as if you take a group of y objects and move 1 more object next to it, and then move that newly formed group of objects next to a group of x objects, and observe the quantity, no matter what the original numbers x and y are.” (So symbolic language is useful after all....) We observe evidence for that statement every day! It is a belief that pays rent in terms of anticipated experience. So, yes, we could assume differently, but we would be doing so in the face of mountains of evidence pointing in the other direction, and with something to protect, that won’t do.
Notes:
How do we know the soundness theorem for first-order logic is true? For one thing, we observe inductive evidence for it all the time (the rules of first-order logic are incredibly intuitive, see page 51 (PDF page 57) of A Primer for Logic and Proof for an enumeration), just as we see inductive evidence for “∀x ∀y ((x + y) + 1 = x + (y + 1))” all the time. We can also use mathematical induction (on the length of a deduction) to prove the soundness theorem, but this probably won’t convince someone who is skeptical of even first-order logic.
It might be helpful to think of (X ⊨ Y) as meaning P(Y|X) = 1.