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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?

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  • $\begingroup$ As it stands your formalization is not valid. I think the original heap argument is valid, for a similar reason that mathematical induction is valid. I would formalize it this way: $A_{1,000,000},\;(A_{n}\land n\ge 2)\to A_{n-1},\;$ therefore $A_1.$ I think you might be able to prove that in Fitch-style natural deduction. $\endgroup$ – Adrian Keister Jan 17 at 14:46
  • $\begingroup$ NO; it is not valid. $\endgroup$ – Mauro ALLEGRANZA Jan 17 at 14:46
  • $\begingroup$ @AdrianKeister thank you for your reply. I'm new to natural deduction and so found this difficult - is there any way to formalise it without the use of $n>2$ and subscripts or is this impossible? $\endgroup$ – ashalrik Jan 17 at 14:54
  • $\begingroup$ thanks @MauroALLEGRANZA is there any way you'd suggest to formalise it using more implicit premises or is this not possible? thanks $\endgroup$ – ashalrik Jan 17 at 14:55
  • $\begingroup$ @ashalrik: Well, sure. The subscripts are a convenience, not a necessity. You're going to run out of single-letter Latin characters pretty quick. The condition on $n\ge 2$ might not be necessary; it's there to prevent a reductio ad absurdam counter-argument, where you could have negative grains of sand. Also, strictly speaking, the "premise" of $A_n\to A_{n-1}$ is actually more of a premise schema than a usual premise. It comes down to this: you don't have time to prove this argument exhaustively, so you're going to have to come up with some way to "skip steps". How you do that is up to you. $\endgroup$ – Adrian Keister Jan 17 at 14:59
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  1. $\forall x [\text {Own}(x , \text { 1,000,000 cents}) \to \text {Rich}(x)]$

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

  1. $\forall x [(\text {Own}(x , 1 \text { cent}) \to \text {Rich}(x))]$
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  • $\begingroup$ thanks so much! $\endgroup$ – ashalrik Jan 17 at 15:40
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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.

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