See: Logic
Truth functions
Inputs | Outputs.
No inputs yield {$2^{(2^0)}$} constant functions as follows:
- | T Always true
- | F Always false
This grounds truth and falsehood.
One input A with value T or F yields {$2^{(2^1)}$} truth functions with outputs as follows:
- TT | T
- FF | F
- TF | A
- FT | Not-A
This stage grounds affirmation and negation.
A, B, each with value T or F, generate {$2^2$} input pairs TT-TF-FT-FF and yield {$2^{(2^2)}$} truth functions with outputs as follows:
- TTTT | T always true
- FFFF | F always true
- TTTF | A-or-B = Not(Not-A-and-Not-B) = if-not-A-then-B = if-not-B-then-A
- TTFT | A-or-Not-B = Not(Not-A-and-B) = if-B-then-A = if-not-A-then-not-B
- TFTT | Not-A-or-B = Not(A-and-Not-B) = if-A-then-B = if-not-B-then-not-A
- FTTT | Not-A-or-Not-B = Not(A-and-B) = if-A-then-not-B = if-B-then-not-A = Nand(A,B)
- FFFT | Not-A-and-Not-B = Not(A-or-B) = not(if-not-A-then-B) = not(if-not-B-then-A) = Nor(A,B)
- FFTF | Not-A-and-B = Not(A-or-Not-B) = not(if-B-then-A) = not(if-not-A-then-not-B)
- FTFF | A-and-Not-B = Not(Not-A-or-B) = not(if-A-then-B) = not(if-not-B-then-not-A)
- TFFF | A-and-B = Not(Not-A-or-Not-B) = not(if-A-then-not-B) = not(if-B-then-not-A)
Grounds if...then..., and, or, nand, nor
- TTFF | A = (A or B) and (A or not-B)
- TFTF | B
- TFFT | A-iff-B = Not-A-iff-Not-B = (if-A-then-B) and (if-B-then-A) = (A and B) or (not-A and not-B) = not(A and not B) and not(not A and B)
- FTTF | A-iff-Not-B = Not-A-iff-B = (if-A-then-not-B) and (if-B-then-not-A) = (A and not-B) or (not-A and B)
- FTFT | Not-B
- FFTT | Not-A
Grounds A, not-A, iff
Nand and Nor = supernegation = do nothing action
- Nand outputs signal if any input is silent, and outputs silence if all inputs have signals.
- Nor outputs silence if any input has signal, and outputs signal if all inputs have silence.
So apparently can express the halting problem: Nor expresses that the problem has halted.
- Not-A = A nand A.
- A and B = Not (A nand B) = (A nand B) nand (A nand B)
- A or B = (A nand A) nand (B nand B)
A, B, C, each with value T or F, generate {$2^3$} input pairs TTT-TTF-TFT-TFF-FTT-FTF-FFT-FFF and yield {$2^{(2^3)}$} truth functions with outputs as follows:
- TTTTTTTT Always true
- FFFFFFFF Always false
- TTTTTTTF A or B or C
- TFFFFFFF A and B and C
- FTTTTTTT Nand(A,B,C)
- FFFFFFFT Nor(A,B,C)
- TTTTTTFF A or B
- TTFFFFFF A and B
- TTTTTFFF
- TTTTFFFF A