My suggestion for linking formal logic and natural numbers is to see counting 1 2 3 4 5 6 and so on as a series of propositions each of which contradicts and updates the previous. There's 1 sheep in the field. Oh no there isn't, there are 2. Oh no, I've just spotted another, so 3. Hold on, there aren't 3 but 4. And so on.
In formal logic this could be expressed as P, Q ⊃ not-P, R, R ⊃ not-Q, . . .. The material implication symbol "⊃" can be interpreted as the updating function. The conventional two-valued truth table for this function whereby 1 1 = 1, 0 1 = 1, 1 0 = 0, 0 0 = 1, means that 1 0 is a failure to update.
My suggestion for linking formal logic and natural numbers is to see counting 1 2 3 4 5 6 and so on as a series of propositions each of which contradicts and updates the previous. There's 1 sheep in the field. Oh no there isn't, there are 2. Oh no, I've just spotted another, so 3. Hold on, there aren't 3 but 4. And so on.
In formal logic this could be expressed as P, Q ⊃ not-P, R, R ⊃ not-Q, . . .. The material implication symbol "⊃" can be interpreted as the updating function. The conventional two-valued truth table for this function whereby 1 1 = 1, 0 1 = 1, 1 0 = 0, 0 0 = 1, means that 1 0 is a failure to update.