Content deleted Content added
Axiomtutor (talk | contribs) added the formal definition by recursion. |
Randy Kryn (talk | contribs) →Computer science: uppercase per direct link (Apollo Guidance Computer) |
||
| (3 intermediate revisions by 2 users not shown) | |||
Line 293:
Thus if ''S'' is a sentence that is a string of symbols consisting of logical symbols ''v''<sub>1</sub>...''v''<sub>''n''</sub> representing logical connectives, and non-logical symbols ''c''<sub>1</sub>...''c''<sub>''n''</sub>, then if and only if {{math|size=100%|''I''(''v''<sub>1</sub>)...''I''(''v''<sub>''n''</sub>)}} have been provided interpreting ''v''<sub>1</sub> to ''v''<sub>''n''</sub> by means of ''f''<sub>nand</sub> (or any other set of functional complete truth-functions) then the truth-value of {{tmath|I(s)}} is determined entirely by the truth-values of ''c''<sub>1</sub>...''c''<sub>''n''</sub>, i.e. of {{math|size=100%|''I''(''c''<sub>1</sub>)...''I''(''c''<sub>''n''</sub>)}}. In other words, as expected and required, ''S'' is true or false only under an interpretation of all its non-logical symbols.
==
Using the functions defined above, we can give a formal definition of a proposition's truth function.<ref>{{Cite web |title=An Introduction to Mathematical Logic |url=https://store.doverpublications.com/products/9780486497853?srsltid=AfmBOoo9mFyD06QpIypwYtJnNn2CYOf-Ps2CCwMYl_IgAfLRwgeh7v1s |access-date=2025-02-20 |website=Dover Publications |language=en}}</ref>
Let ''PROP'' be the set of all propositional variables,
Line 315:
Finally, now that we have defined the extended truth assignment, we can use this to define the truth-function of a proposition. For a proposition, ''A'', its '''truth function''', <math>f_A</math>, has ___domain equal to the set of all truth assignments, and range equal to <math>\{T,F\}</math>.
It is defined, for each truth assignment <math>\phi</math>, by <math>
== Computer science ==
Line 322:
The "logical equivalence" of "NAND alone", "NOR alone", and "NOT and AND" is similar to [[Turing equivalence (theory of computation)|Turing equivalence]].
The fact that all truth functions can be expressed with NOR alone is demonstrated by the [[Apollo
== See also ==
| |||