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Discussion: LogicalImplication
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%3 N_20250218182802d_ColinWright    By ColinWright 2025/02/18 @ 18:28:02d -------------------------------- If (A&B) is true, then certainly A is true.        (select only this node)     N_20250218183047a_ColinWright    By ColinWright 2025/02/18 @ 18:30:47a -------------------------------- Re-write this as: (A&B) => A        (select only this node)     N_20250218182802d_ColinWright->N_20250218183047a_ColinWright N_20250217221641a_ColinWright    By ColinWright 2025/02/17 @ 22:16:41a -------------------------------- Here we investigate the Truth Table for P=>Q in the Propositional Calculus.        (select only this node)     N_20250217221641b_ColinWright    By ColinWright 2025/02/17 @ 22:16:41b -------------------------------- In the Propositional Calculus we have variables or unknowns which can take the value True (T) or False (F).        (select only this node)     N_20250217221641a_ColinWright->N_20250217221641b_ColinWright N_20250217223718a_ColinWright    By ColinWright 2025/02/17 @ 22:37:18a -------------------------------- This discussion is targetted at someone who largely knows what's going on. It is gently paced to try to build the final conclusion in a careful, logical fashion.        (select only this node)     N_20250217221641a_ColinWright->N_20250217223718a_ColinWright N_20250218201930a_ColinWright    By ColinWright 2025/02/18 @ 20:19:30a -------------------------------- You are reading in "Neighbourhood mode" ... see if you can unset that to be able to see the entire discussion, all at once.        (select only this node)     N_20250217221641b_ColinWright->N_20250218201930a_ColinWright N_20250217221641c_ColinWright    By ColinWright 2025/02/17 @ 22:16:41c -------------------------------- We form expressions by using AND, OR, sometimes XOR, NOT, and "=>" which we read as "Logical Implication".        (select only this node)     N_20250217221641b_ColinWright->N_20250217221641c_ColinWright N_20250217221641c_ColinWright->N_20250218201930a_ColinWright N_20250221114706a_ColinWright    By ColinWright 2025/02/21 @ 11:47:06a -------------------------------- The idea in Propositional Calculus is to regard these as operators that take inputs and produce an output.        (select only this node)     N_20250217221641c_ColinWright->N_20250221114706a_ColinWright N_20250217221641d_ColinWright    By ColinWright 2025/02/17 @ 22:16:41d -------------------------------- "Logical Implication" is sometime read as "IF...THEN..."        (select only this node)     N_20250217221641c_ColinWright->N_20250217221641d_ColinWright N_20250217221842a_ColinWright    By ColinWright 2025/02/17 @ 22:18:42a -------------------------------- When we have an operator OP we can then construct a Truth Table showing the value it takes for each possible assignment of values to its variables.        (select only this node)     N_20250218175445a_ColinWright    By ColinWright 2025/02/18 @ 17:54:45a -------------------------------- Using the symbol "#" for "XOR"        (select only this node)     N_20250217221842a_ColinWright->N_20250218175445a_ColinWright N_20250217222724a_ColinWright    By ColinWright 2025/02/17 @ 22:27:24a -------------------------------- Using the symbol "|" for "OR"        (select only this node)     N_20250217221842a_ColinWright->N_20250217222724a_ColinWright N_20250217222552a_ColinWright    By ColinWright 2025/02/17 @ 22:25:52a -------------------------------- Using the symbol "&" for "AND"        (select only this node)     N_20250217221842a_ColinWright->N_20250217222552a_ColinWright N_20250217222642a_ColinWright    By ColinWright 2025/02/17 @ 22:26:42a -------------------------------- Using the symbol "~" for "NOT"        (select only this node)     N_20250217221842a_ColinWright->N_20250217222642a_ColinWright N_20250217224745a_ColinWright    By ColinWright 2025/02/17 @ 22:47:45a -------------------------------- Using the symbol "=" for EQUALS        (select only this node)     N_20250217221842a_ColinWright->N_20250217224745a_ColinWright N_20250218182802e_ColinWright    By ColinWright 2025/02/18 @ 18:28:02e -------------------------------- If A is true, then certainly (A|B) is true.        (select only this node)     N_20250218183030a_ColinWright    By ColinWright 2025/02/18 @ 18:30:30a -------------------------------- Re-write this as: A => (A|B)        (select only this node)     N_20250218182802e_ColinWright->N_20250218183030a_ColinWright N_20250217221842c_ColinWright    By ColinWright 2025/02/17 @ 22:18:42c -------------------------------- This is reflecting the idea that "A AND B" is only true if A is true and B is also true.        (select only this node)     N_20250217223928a_ColinWright    By ColinWright 2025/02/17 @ 22:39:28a -------------------------------- Given these rules, an expression can be evaluated and/or simplified.        (select only this node)     N_20250217221842c_ColinWright->N_20250217223928a_ColinWright N_20250218180530a_ColinWright    By ColinWright 2025/02/18 @ 18:05:30a -------------------------------- These are De Morgan's Laws.        (select only this node)     N_20250218181425a_ColinWright    By ColinWright 2025/02/18 @ 18:14:25a -------------------------------- So this is all intended to capture the "Natural Language" meanings of EQ, AND, OR, NOT, and XOR.        (select only this node)     N_20250218180530a_ColinWright->N_20250218181425a_ColinWright N_20250218181425b_ColinWright    By ColinWright 2025/02/18 @ 18:14:25b -------------------------------- It was Gottfried Leibniz who envisaged the end of philosophic disputes, replacing argument with calculation.        (select only this node)     N_20250218181425c_ColinWright    By ColinWright 2025/02/18 @ 18:14:25c -------------------------------- "if controversies were to arise, there would be no more need of disputation between two philosophers than between two calculators. For it would suffice for them to take their pencils in their hands and to sit down at the abacus, and say to each other (and if they so wish also to a friend called to help): Let us calculate."        (select only this node)     N_20250218181425b_ColinWright->N_20250218181425c_ColinWright N_20250218181542c_ColinWright    By ColinWright 2025/02/18 @ 18:15:42c -------------------------------- We refer to this as Logical Implication.        (select only this node)     N_20250218181542d_ColinWright    By ColinWright 2025/02/18 @ 18:15:42d -------------------------------- A => B        (select only this node)     N_20250218181542c_ColinWright->N_20250218181542d_ColinWright N_20250218182802c_ColinWright    By ColinWright 2025/02/18 @ 18:28:02c -------------------------------- Consider the following statements:        (select only this node)     N_20250218182802c_ColinWright->N_20250218182802e_ColinWright N_20250218182802c_ColinWright->N_20250218182802d_ColinWright N_20250217222134a_ColinWright    By ColinWright 2025/02/17 @ 22:21:34a -------------------------------- A B : | --------- F F : F F T : T T F : T T T : T        (select only this node)     N_20250217222134b_ColinWright    By ColinWright 2025/02/17 @ 22:21:34b -------------------------------- This is reflecting the idea that "A OR B" is true if either A is true or B is true, or both.        (select only this node)     N_20250217222134a_ColinWright->N_20250217222134b_ColinWright N_20250217222134b_ColinWright->N_20250217223928a_ColinWright N_20250217222329a_ColinWright    By ColinWright 2025/02/17 @ 22:23:29a -------------------------------- A : ~ --------- F : T T : F        (select only this node)     N_20250217222329b_ColinWright    By ColinWright 2025/02/17 @ 22:23:29b -------------------------------- This is reflecting the idea that "NOT(A)" is true when A is false, and vice versa.        (select only this node)     N_20250217222329a_ColinWright->N_20250217222329b_ColinWright N_20250217222329b_ColinWright->N_20250217223928a_ColinWright N_20250218182222a_ColinWright    By ColinWright 2025/02/18 @ 18:22:22a -------------------------------- A B : => ----------- F F : ?? F T : ?? T F : ?? T T : ??        (select only this node)     N_20250218182523a_ColinWright    By ColinWright 2025/02/18 @ 18:25:23a -------------------------------- Let's be guided by our common-sense understanding of the natural language concept of "A implies B"        (select only this node)     N_20250218182222a_ColinWright->N_20250218182523a_ColinWright N_20250218184852a_ColinWright    By ColinWright 2025/02/18 @ 18:48:52a -------------------------------- So that means that F=>A *must* have the value T.        (select only this node)     N_20250218184852b_ColinWright    By ColinWright 2025/02/18 @ 18:48:52b -------------------------------- And this is the case, regardless of the value of A.        (select only this node)     N_20250218184852a_ColinWright->N_20250218184852b_ColinWright N_20250218192927a_ColinWright    By ColinWright 2025/02/18 @ 19:29:27a -------------------------------- Setting A to be F we see that F=>F has value T        (select only this node)     N_20250218184852a_ColinWright->N_20250218192927a_ColinWright N_20250218193023a_ColinWright    By ColinWright 2025/02/18 @ 19:30:23a -------------------------------- So F=>T has value T        (select only this node)     N_20250218184852a_ColinWright->N_20250218193023a_ColinWright N_20250217221842b_ColinWright    By ColinWright 2025/02/17 @ 22:18:42b -------------------------------- A B : & --------- F F : F F T : F T F : F T T : T        (select only this node)     N_20250217222552a_ColinWright->N_20250217221842b_ColinWright N_20250217222642a_ColinWright->N_20250217222329a_ColinWright N_20250217222724a_ColinWright->N_20250217222134a_ColinWright N_20250218195957a_ColinWright    By ColinWright 2025/02/18 @ 19:59:57a -------------------------------- Click on a node to expand it and its immediate neighbours. For example, click HERE !!!        (select only this node)     N_20250217223718a_ColinWright->N_20250218195957a_ColinWright N_20250217223928b_ColinWright    By ColinWright 2025/02/17 @ 22:39:28b -------------------------------- If the values of all the variables are known, the expression can be evaluated ... we can determine if the expression as a whole has the value T(rue) or F(alse)        (select only this node)     N_20250217223928a_ColinWright->N_20250217223928b_ColinWright N_20250217224311a_ColinWright    By ColinWright 2025/02/17 @ 22:43:11a -------------------------------- For some expressions the value is always True, no matter what the values are of the constituent variables.        (select only this node)     N_20250217223928b_ColinWright->N_20250217224311a_ColinWright N_20250217224311f_ColinWright    By ColinWright 2025/02/17 @ 22:43:11f -------------------------------- Evaluate this when A is False and it will be True        (select only this node)     N_20250217224502a_ColinWright    By ColinWright 2025/02/17 @ 22:45:02a -------------------------------- So the expression "(A|(~A))" is a tautology.        (select only this node)     N_20250217224311f_ColinWright->N_20250217224502a_ColinWright N_20250217224311b_ColinWright    By ColinWright 2025/02/17 @ 22:43:11b -------------------------------- Such expressions are called "Tautologies".        (select only this node)     N_20250217224311c_ColinWright    By ColinWright 2025/02/17 @ 22:43:11c -------------------------------- For example: These are all tautologies        (select only this node)     N_20250217224311b_ColinWright->N_20250217224311c_ColinWright N_20250218175922a_ColinWright    By ColinWright 2025/02/18 @ 17:59:22a -------------------------------- [ A & (B|C) ] = [ (A&B) | (A&C) ]        (select only this node)     N_20250217224311c_ColinWright->N_20250218175922a_ColinWright N_20250218175922c_ColinWright    By ColinWright 2025/02/18 @ 17:59:22c -------------------------------- [ A | (B&C) ] = [ (A|B) & (A|C) ]        (select only this node)     N_20250217224311c_ColinWright->N_20250218175922c_ColinWright N_20250217225319c_ColinWright    By ColinWright 2025/02/17 @ 22:53:19c -------------------------------- [~(A&B)]=[(~A)|(~B)]        (select only this node)     N_20250217224311c_ColinWright->N_20250217225319c_ColinWright N_20250217224311d_ColinWright    By ColinWright 2025/02/17 @ 22:43:11d -------------------------------- A | (~A)        (select only this node)     N_20250217224311c_ColinWright->N_20250217224311d_ColinWright N_20250221132850a_ColinWright    By ColinWright 2025/02/21 @ 13:28:50a -------------------------------- (A=B) = ~(A#B)        (select only this node)     N_20250217224311c_ColinWright->N_20250221132850a_ColinWright N_20250217225319a_ColinWright    By ColinWright 2025/02/17 @ 22:53:19a -------------------------------- [~(A|B)]=[(~A)&(~B)]        (select only this node)     N_20250217224311c_ColinWright->N_20250217225319a_ColinWright N_20250218193506a_ColinWright    By ColinWright 2025/02/18 @ 19:35:06a -------------------------------- Pulling it all together:        (select only this node)     N_20250218193023a_ColinWright->N_20250218193506a_ColinWright N_20250217224311e_ColinWright    By ColinWright 2025/02/17 @ 22:43:11e -------------------------------- Evaluate this when A is True and it will be True.        (select only this node)     N_20250217224311e_ColinWright->N_20250217224502a_ColinWright N_20250218195957b_ColinWright    By ColinWright 2025/02/18 @ 19:59:57b -------------------------------- Click on the grey bar to select only that single node. Obviously.        (select only this node)     N_20250218200141a_ColinWright    By ColinWright 2025/02/18 @ 20:01:41a -------------------------------- Now click on the grey bar on the very top node to start reading the actual content...        (select only this node)     N_20250218195957b_ColinWright->N_20250218200141a_ColinWright N_20250218182523b_ColinWright    By ColinWright 2025/02/18 @ 18:25:23b -------------------------------- Consider the statement: "A implies B"        (select only this node)     N_20250218182523a_ColinWright->N_20250218182523b_ColinWright N_20250218184425b_ColinWright    By ColinWright 2025/02/18 @ 18:44:25b -------------------------------- Then the Right-Hand-Side is (A|T), which is T, regardless of the value of A.        (select only this node)     N_20250218184716a_ColinWright    By ColinWright 2025/02/18 @ 18:47:16a -------------------------------- So that means that A=>T *must* have the value T.        (select only this node)     N_20250218184425b_ColinWright->N_20250218184716a_ColinWright N_20250217224311d_ColinWright->N_20250217224311e_ColinWright N_20250217224311d_ColinWright->N_20250217224311f_ColinWright N_20250217224502a_ColinWright->N_20250218181425a_ColinWright N_20250217224745b_ColinWright    By ColinWright 2025/02/17 @ 22:47:45b -------------------------------- A B : = --------- F F : T F T : F T F : F T T : T        (select only this node)     N_20250217224745a_ColinWright->N_20250217224745b_ColinWright N_20250218192954a_ColinWright    By ColinWright 2025/02/18 @ 19:29:54a -------------------------------- Setting A to be T we see that T=>T has value T        (select only this node)     N_20250218184852b_ColinWright->N_20250218192954a_ColinWright N_20250218184852b_ColinWright->N_20250218193023a_ColinWright N_20250218184852b_ColinWright->N_20250218192927a_ColinWright N_20250217224745c_ColinWright    By ColinWright 2025/02/17 @ 22:47:45c -------------------------------- This is reflecting the idea that "A=B" is true when A and B take the same value.        (select only this node)     N_20250217224745c_ColinWright->N_20250217223928a_ColinWright N_20250218181542e_ColinWright    By ColinWright 2025/02/18 @ 18:15:42e -------------------------------- So what should the truth table be for "A=>B" ??        (select only this node)     N_20250218181542d_ColinWright->N_20250218181542e_ColinWright N_20250218182523c_ColinWright    By ColinWright 2025/02/18 @ 18:25:23c -------------------------------- If A is true but B is false, then that statement is false.        (select only this node)     N_20250218182523b_ColinWright->N_20250218182523c_ColinWright N_20250218192954a_ColinWright->N_20250218193506a_ColinWright N_20250218175922b_ColinWright    By ColinWright 2025/02/18 @ 17:59:22b -------------------------------- So AND distributes over OR        (select only this node)     N_20250218175922a_ColinWright->N_20250218175922b_ColinWright N_20250218224321a_ColinWright    By ColinWright 2025/02/18 @ 22:43:21a -------------------------------- Conclusions and wrapping up still to come        (select only this node)     N_20250218180700a_ColinWright    By ColinWright 2025/02/18 @ 18:07:00a -------------------------------- There are similar to how multiplication distributes over addition, but that only works "one way", because addition does not distribute over multiplication.        (select only this node)     N_20250218180700a_ColinWright->N_20250218181425a_ColinWright N_20250218181542a_ColinWright    By ColinWright 2025/02/18 @ 18:15:42a -------------------------------- So now we can ask about the common natural language construction:        (select only this node)     N_20250218181542b_ColinWright    By ColinWright 2025/02/18 @ 18:15:42b -------------------------------- "If ... then ..."        (select only this node)     N_20250218181542a_ColinWright->N_20250218181542b_ColinWright N_20250218181542e_ColinWright->N_20250218182222a_ColinWright N_20250218182802a_ColinWright    By ColinWright 2025/02/18 @ 18:28:02a -------------------------------- What about the other entries?        (select only this node)     N_20250218182802b_ColinWright    By ColinWright 2025/02/18 @ 18:28:02b -------------------------------- We can again start from some common sense considerations.        (select only this node)     N_20250218182802a_ColinWright->N_20250218182802b_ColinWright N_20250218193506b_ColinWright    By ColinWright 2025/02/18 @ 19:35:06b -------------------------------- A B : => ----------- F F : T F T : T T F : F T T : T        (select only this node)     N_20250218193506a_ColinWright->N_20250218193506b_ColinWright N_20250217221842b_ColinWright->N_20250217221842c_ColinWright N_20250218182523d_ColinWright    By ColinWright 2025/02/18 @ 18:25:23d -------------------------------- A B : => ----------- F F : ? F T : ? T F : F T T : ?        (select only this node)     N_20250218182523c_ColinWright->N_20250218182523d_ColinWright N_20250217224311a_ColinWright->N_20250217224311b_ColinWright N_20250217224745b_ColinWright->N_20250217224745c_ColinWright N_20250217225319b_ColinWright    By ColinWright 2025/02/17 @ 22:53:19b -------------------------------- For each, evaluate each side of the "=" for each possible combination of assignments of T(rue) and F(alse) to the variables A and B and you'll see that each side comes out with the same value.        (select only this node)     N_20250217225319a_ColinWright->N_20250217225319b_ColinWright N_20250218193506b_ColinWright->N_20250218224321a_ColinWright N_20250217225319c_ColinWright->N_20250217225319b_ColinWright N_20250218182523d_ColinWright->N_20250218182802a_ColinWright N_20250218175445b_ColinWright    By ColinWright 2025/02/18 @ 17:54:45b -------------------------------- A B : # --------- F F : F F T : T T F : T T T : F        (select only this node)     N_20250218175445a_ColinWright->N_20250218175445b_ColinWright N_20250218175445c_ColinWright    By ColinWright 2025/02/18 @ 17:54:45c -------------------------------- This is reflecting the idea that "A XOR B" is only true if A is true or B is true, but not both.        (select only this node)     N_20250218175445c_ColinWright->N_20250217223928a_ColinWright N_20250218175445d_ColinWright    By ColinWright 2025/02/18 @ 17:54:45d -------------------------------- It's the same as "NOT EQUAL"        (select only this node)     N_20250218175445c_ColinWright->N_20250218175445d_ColinWright N_20250218182802b_ColinWright->N_20250218182802c_ColinWright N_20250218195957a_ColinWright->N_20250218195957b_ColinWright N_20250218175445b_ColinWright->N_20250218175445c_ColinWright N_20250218175922b_ColinWright->N_20250218180700a_ColinWright N_20250218175922d_ColinWright    By ColinWright 2025/02/18 @ 17:59:22d -------------------------------- So OR distributes over AND        (select only this node)     N_20250218175922d_ColinWright->N_20250218180700a_ColinWright N_20250218175445d_ColinWright->N_20250221132850a_ColinWright N_20250221114706b_ColinWright    By ColinWright 2025/02/21 @ 11:47:06b -------------------------------- So an expression has a value determined by, and only determined by, the values of the things being combined.        (select only this node)     N_20250221114706a_ColinWright->N_20250221114706b_ColinWright N_20250218175922c_ColinWright->N_20250218175922d_ColinWright N_20250218181425a_ColinWright->N_20250218181542a_ColinWright N_20250218181425a_ColinWright->N_20250218181425b_ColinWright N_20250218181542b_ColinWright->N_20250218181542c_ColinWright N_20250218183253a_ColinWright    By ColinWright 2025/02/18 @ 18:32:53a -------------------------------- So these should be tautologies.        (select only this node)     N_20250218183030a_ColinWright->N_20250218183253a_ColinWright N_20250218184425a_ColinWright    By ColinWright 2025/02/18 @ 18:44:25a -------------------------------- Let's consider what happens when we assign B to be T(rue).        (select only this node)     N_20250218183030a_ColinWright->N_20250218184425a_ColinWright N_20250218183030a_ColinWright->N_20250218184716a_ColinWright N_20250218183047a_ColinWright->N_20250218183253a_ColinWright N_20250218184507a_ColinWright    By ColinWright 2025/02/18 @ 18:45:07a -------------------------------- Let's consider what happens when we assign B to be F(alse).        (select only this node)     N_20250218183047a_ColinWright->N_20250218184507a_ColinWright N_20250218183047a_ColinWright->N_20250218184852a_ColinWright N_20250218183530a_ColinWright    By ColinWright 2025/02/18 @ 18:35:30a -------------------------------- As they are tautologies, no matter what values we assign to A and B, these statements should take the value T(rue).        (select only this node)     N_20250218183253a_ColinWright->N_20250218183530a_ColinWright N_20250218184507b_ColinWright    By ColinWright 2025/02/18 @ 18:45:07b -------------------------------- Then the Left-Hand-Side is (A&F), which is F, regardless of the value of A.        (select only this node)     N_20250218184507b_ColinWright->N_20250218184852a_ColinWright N_20250218192927a_ColinWright->N_20250218193506a_ColinWright N_20250217225319b_ColinWright->N_20250218180530a_ColinWright N_20250218183530a_ColinWright->N_20250218184507a_ColinWright N_20250218183530a_ColinWright->N_20250218184852a_ColinWright N_20250218183530a_ColinWright->N_20250218184425a_ColinWright N_20250218183530a_ColinWright->N_20250218184716a_ColinWright N_20250218184425a_ColinWright->N_20250218184425b_ColinWright N_20250218184507a_ColinWright->N_20250218184507b_ColinWright N_20250218184716a_ColinWright->N_20250218184852b_ColinWright N_20250218184716a_ColinWright->N_20250218192954a_ColinWright N_20250218184716a_ColinWright->N_20250218193023a_ColinWright N_20250221114706b_ColinWright->N_20250217221842a_ColinWright