MyWikiBiz, Author Your Legacy — Thursday November 28, 2024
Jump to navigationJump to search
4,405 bytes removed
, 16:54, 17 August 2009
Line 995: |
Line 995: |
| With that preamble behind us, let us turn to consider the case of semiosis, or sign transformation process, that is generated by our first proof of the propositional equation <math>E_1.\!</math> | | With that preamble behind us, let us turn to consider the case of semiosis, or sign transformation process, that is generated by our first proof of the propositional equation <math>E_1.\!</math> |
| | | |
− | {| align="center" cellpadding="10" style="text-align:center; width:90%" | + | {| align="center" cellpadding="10" |
− | | | + | | [[Image:Logical Graph (P (Q)) (P (R)) = (P (Q R)) Proof 1.jpg|500px]] |
− | <pre>
| |
− | o-----------------------------------------------------------o
| |
− | | Equation E_1. Proof 1. |
| |
− | o-----------------------------------------------------------o
| |
− | | |
| |
− | | |
| |
− | | q o o r |
| |
− | | | | |
| |
− | | p o o p |
| |
− | | \ / |
| |
− | | @ |
| |
− | | |
| |
− | | (p (q)) (p (r)) |
| |
− | | |
| |
− | o=============================< Double Negation >===========o
| |
− | | |
| |
− | | q o o r |
| |
− | | | | |
| |
− | | p o o p |
| |
− | | \ / |
| |
− | | o |
| |
− | | | |
| |
− | | o |
| |
− | | | |
| |
− | | @ |
| |
− | | |
| |
− | | (( (p (q)) (p (r)) )) |
| |
− | | |
| |
− | o=============================< Collection >================o
| |
− | | |
| |
− | | q o o r |
| |
− | | | | |
| |
− | | o o |
| |
− | | \ / |
| |
− | | o |
| |
− | | | |
| |
− | | p o |
| |
− | | | |
| |
− | | @ |
| |
− | | |
| |
− | | (p ( ((q)) ((r)) )) |
| |
− | | |
| |
− | o=============================< Double Negation >===========o
| |
− | | |
| |
− | | q r |
| |
− | | o |
| |
− | | | |
| |
− | | p o |
| |
− | | | |
| |
− | | @ |
| |
− | | |
| |
− | | (p (q r)) |
| |
− | | |
| |
− | o=============================< QED >=======================o
| |
− | </pre>
| |
| | (30) | | | (30) |
| |} | | |} |
Line 1,058: |
Line 1,003: |
| | | |
| We are in the process of examining various proofs of the propositional equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))},</math> and viewing these proofs in the light of their character as semiotic processes, in essence, as sign-theoretic transformations. | | We are in the process of examining various proofs of the propositional equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))},</math> and viewing these proofs in the light of their character as semiotic processes, in essence, as sign-theoretic transformations. |
− |
| |
− | Here is a reminder of the equation in question:
| |
− |
| |
− | {| align="center" cellpadding="10" style="text-align:center; width:90%"
| |
− | |
| |
− | <pre>
| |
− | o-----------------------------------------------------------o
| |
− | | Equation E_1 |
| |
− | o-----------------------------------------------------------o
| |
− | | |
| |
− | | q r |
| |
− | | q o o r o |
| |
− | | | | | |
| |
− | | p o o p p o |
| |
− | | \ / | |
| |
− | | @ = @ |
| |
− | | |
| |
− | | (p (q)) (p (r)) = (p (q r)) |
| |
− | | |
| |
− | | [p=>q] & [p=>r] = [p=>[q&r]] |
| |
− | | |
| |
− | o-----------------------------------------------------------o
| |
− | </pre>
| |
− | | (31)
| |
− | |}
| |
| | | |
| The second way of establishing the truth of this equation is one that I see, rather loosely, as ''model-theoretic'', for no better reason than the sense of its ending with a pattern of expression, a variant of the ''disjunctive normal form'' (DNF), that is commonly recognized to be the form that one extracts from a truth table by pulling out the rows of the table that evaluate to true and constructing the disjunctive expression that sums up the senses of the corresponding interpretations. | | The second way of establishing the truth of this equation is one that I see, rather loosely, as ''model-theoretic'', for no better reason than the sense of its ending with a pattern of expression, a variant of the ''disjunctive normal form'' (DNF), that is commonly recognized to be the form that one extracts from a truth table by pulling out the rows of the table that evaluate to true and constructing the disjunctive expression that sums up the senses of the corresponding interpretations. |