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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