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.
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.
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