Doubt implies thought.
I'm doubting.
So I'm thinking.

[(d→t) ∧ d]→t

Thought implies existence.
I'm thinking.
So I exist.

[(t→e) ∧ t]→e

d→t; ¬t→¬d; [¬t ∧ (d→t)]→¬d
t→e; ¬e→¬t; [¬e ∧ (t→e)]→¬t

[(d→t) ∧ (t→e)]→(d→e)
[(¬e→¬t) ∧ (¬t→¬d)]→(¬e→¬d)

d→t, d, ∴ d ∧ t; t→e, t, ∴ t ∧ e
d ∧ t, ∴ t ∧ d; t ∧ e, ∴ e ∧ t

My question: How much further can we go with this and what meaning can we extract from it?

  • Is there a (pointed) question? Or just opinion exchange? Jul 19 '19 at 13:39
  • @Rusi In general, I dislike opinions. I'm hoping for objective answers and no "point" intended.
    – QWERTY_dw
    Jul 19 '19 at 14:02
  • Here, I'm interested in: first principles, logic, and semantics. (In that order)
    – QWERTY_dw
    Jul 19 '19 at 15:00
  • You formalise the first premise with d denoting "doubt" while in the right-hand conjunct you are using d to denote the second premise, "I'm doubting". The point of cognito is to derive "I" but Descartes has assumed it.
    – NWR
    Jul 19 '19 at 20:50
  • 1
    This will not go very far. "Doubt implies thought", and "thought implies existence" are presumably analytic inferences based on the definitions of words "doubt", "thought", and "existence". One can keep up the wordplay by bringing in more words, e.g. "thought implies action", I think, therefore I act, etc. But to get anything non-trivial, more substantive (and controversial) premises are needed, as Descartes discovered with his "clear and distinct ideas" and "God is not a deceiver", locked into the Cartesian Circle.
    – Conifold
    Jul 20 '19 at 6:17

I will use the following symbolization to fit the constraints of the proof checker associated with forallx:

  • D: I'm doubting. (The OP uses d.)
  • T: I'm thinking. (The OP uses t.)
  • F: I exist. (The OP uses e)

Consider the following as premises: D, D → T, and T → F we can conclude F.

enter image description here

Line 4 is the result of using modus ponens or conditional elimination (→E) on lines 1 and 2. Line 5 is the result of using modus ponens or conditional elimination on lines 3 and 4.

The conclusion F on line 5 has been validly derived.

Here is the question:

How much further can we go with this and what meaning can we extract from it?

Given the three assumptions: (1) I am doubting, (2) doubting implies thinking, and (3) thinking implies existing, we can conclude that I am thinking (line 4) and that I am existing (line 5).

If I use conjunction introduction on lines 4 and 5 to obtain T ∧ F, this is the same result that the OP obtained: t ∧ e. I am thinking and I am existing.

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Fall 2019. http://forallx.openlogicproject.org/forallxyyc.pdf

  • I don't know how those proof programs work, but can it go "further"? [D→(T ∧ F)]→?
    – QWERTY_dw
    Jul 19 '19 at 16:26
  • 1
    @user40358 This is a proof checker. Put in the premises and a desired conclusion and then you can try to derive the conclusion. As an example, you should be able to enter what I illustrated above to get a feel for how it works. It doesn't reach the conclusion for you like a proof assistant or tree proof generator, so you will have to add each rule yourself and know when to use it. That is what the forallx textbook, linked in the answer, should help you learn. To go further set a different goal, but make the goal something you would like to reach. Jul 19 '19 at 17:16

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