Sorites paradox natural deduction problem

Question

I'm new to natural deduction and am attempting a question which sounds like the Sorites paradox: 1 million grains of sand is a heap. If 1 million grains of sand is a heap, then 999,999 grains is a heap. If 999,999 grains is a heap then 999,998 grains is a heap...2 grains is a heap, therefore 1 grain of sand is a heap. (paraphrased) There is an implicit premise that "a heap of sand -1 grain is still a heap" which is premise A.

I have formalised the premises as: $A$, $(A∧B)→C$, $(A∧C)→D$, $(A∧E)→F$ therefore $F$

Is it possible to show this is valid through Fitch-style natural deduction?

Answer 1

1. $\forall x [\text {Own}(x , \text { 1,000,000 cents}) \to \text {Rich}(x)]$

2. $\forall x \forall n [(\text {Own}(x , n+1 \text { cents}) \to \text {Rich}(x)) \to (\text {Own}(x , n \text { cents}) \to \text {Rich}(x))]$

3. $\forall x [(\text {Own}(x , 1 \text { cent}) \to \text {Rich}(x))]$

Answer 2

Natural deduction based on classical logic can only be applied to logical propositions that are unambiguously either true or false. Being a heap of sand is an ambiguous proposition. As far as I can tell, there is no workable definition in terms of numbers of grains of sand and their spatial distribution. It is also not at all self-evident that a single grain of sand is a heap.