{"id":180,"date":"2023-08-01T16:27:52","date_gmt":"2023-08-01T14:27:52","guid":{"rendered":"https:\/\/actilud.com\/info\/en\/?p=180"},"modified":"2025-03-29T16:38:59","modified_gmt":"2025-03-29T15:38:59","slug":"implication","status":"publish","type":"post","link":"https:\/\/actilud.com\/info\/en\/implication\/","title":{"rendered":"Implication"},"content":{"rendered":"

Implication is one of the advanced techniques. It works with Booleans and requires the establishment of hypotheses. It produces a simplification of the number of Booleans. It is an extremely powerful instruction and is widely used in puzzles marked “often difficult.”<\/span><\/p>\n

The implication is not easy to understand. To help, we can translate it with the formula “if… then.”<\/p>\n

If it rains, then there are clouds.<\/h5>\n

This sentence is perfect for understanding \u2013 keeping it in mind helps avoid many mistakes.<\/p>\n

Actilud deals with the following case:<\/p>\n

If a \u21d2 b and b \u21d2 a, then a\u21d4b and we can replace one with the other.<\/p><\/blockquote>\n

To perform an implication, we choose a boolean a<\/span><\/em> and observe what happens when we set it to true.\u00a0<\/span><\/em> If we find a boolean\u00a0 b\u00a0<\/span><\/em> that becomes\u00a0 true\u00a0<\/span><\/em> as a result of\u00a0 a,<\/span><\/em> we reset a<\/span><\/em> to its initial state and observe what happens when we set b<\/span><\/em> to true<\/span><\/em> . If a<\/span><\/em> becomes true in turn, we have demonstrated the equivalence and we can replace one with the other. On the site we can memorize a configuration, this allows us to make hypotheses and go back. It is advisable to start with the most frequent booleans.<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

But be careful!<\/p>\n

A common mistake is to believe that if a \u21d2 b and \u00aca \u21d2 \u00acb, then a \u21d4 b. This is false! As false as saying “if it is not raining, then there are no clouds.” Applied to Booleans, this translates as follows: if a true\u00a0<\/em> =>\u00a0 b true\u00a0<\/em> and if a false\u00a0<\/em> =>\u00a0 b false, we are not allowed to conclude that a \u21d4 b and we should not replace one with the other,<\/em> even if we really want to, because the behaviors of the two Booleans seem to agree.<\/p>\n

Note: the symbol \u00ac is the negation: b \u21d2 \u00aca means b<\/em> implies ”\u00a0 not a”<\/em> ; if b<\/em> is true, then “\u00aca “<\/em> \u00a0is true, therefore a<\/em> is false.<\/p>\n

The truth table of implication is as follows; it is not intuitive. The implication a \u21d2 b is false in only one case, when a is true and b is false. This means that a true premise must not imply a false conclusion; but, if the premise is false, it does not matter what the conclusion is: the implication is always true.<\/p>\n\n\n\n\n\n\n\n
a<\/th>\nb<\/th>\na \u21d2 b<\/th>\n<\/tr>\n
TRUE<\/td>\nTRUE<\/td>\nTRUE<\/td>\n<\/tr>\n
TRUE<\/td>\nFALSE<\/td>\nFALSE<\/td>\n<\/tr>\n
FALSE<\/td>\nTRUE<\/td>\nTRUE<\/td>\n<\/tr>\n
FALSE<\/td>\nFALSE<\/td>\nTRUE<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Let’s analyze the common error. When we set a\u00a0<\/em> to true, we select the first two rows of our table. Then, we prove that a \u21d2 b is true (by setting a<\/em> to true,<\/em> we see that b<\/em> becomes true<\/em> in turn); we see in the table that we must retain the first row: a true,<\/em> a \u21d2 b true,\u00a0<\/em> therefore b is true.
\nLet’s see what happens with \u00aca \u21d2 \u00acb. When we set a\u00a0<\/em> to false, we select the last two rows of our table. Then, when we prove that \u00aca \u21d2 \u00acb (by setting a<\/em> to false we see that b<\/em> becomes false<\/em> in turn), the implication is true, so we are always on the last two rows. And we see in this case, that the implication is true regardless of the state of b,<\/em> true or false. So we have not proven anything.<\/p>\n

Inconsistencies<\/h2>\n

This is a common case that is often appreciated by players, because it allows them to move forward very quickly in the resolution! Sometimes an implication produces an inconsistency. There are several types:<\/p>\n