True/Liar Fathers and Sons

We don't know who made what statement between the grandfather, father and son, but we are tasked to identify which was it that was spoken by the father (the middle male). Thus, our first clue is that his must be distinctive from the other statements in its truth value.

By this, we know that statement 1 is a lie AND that statement 2 is the truth. From these findings, we know that statement 3 is also the truth.

Since statement 1 is a unique lie among the 3 statements, then it must have been told by Tom, the father.

We can also do it by making assumptions, and this will be clearly presented on a table.

Only one of the possibilities check out, the first of the third one, with Pop-Pop & Sonny both told the truth of either Statement 2 & Statement 3 each and Tom lied by way of Statement 1.

If you guess and check, there are only $8$ different possibilities.

Name the class of the persons who always speak the truth True and the class of those who always speak a lie False.

By naming this two classes observe that , for the scope intended , the first two statements can be interpreted as meaning that "the fathers belong to the same class" and "the sons belong to different classes".

If the first two statements are both true or both false then either the fathers belong to the same class and one son belongs to a different class or the sons belong to the same class and the fathers belong to different classes meaning that there will always be (for any of these 2 cases) 2 statements which are told by persons which belong in the same class and have the same truth value and 1 statement which has a different truth value. Observe further that because the first statements are both true or both false the only way to obtain a configuration of 2 statements of one value and 1 of a different value is for 3 to have a different value from them which doesn't happen as if statements 1 and 2 are true 3 is also true and if 1 and 2 are false 3 is also false leading to a contradiction.

Therefore because the first 2 statements don't have the same value they should have different truth values. Observe that if they have different truth values then for 1 true and 2 false both fathers and both sons belong to the same class meaning that they are all belonging to the same class and therefore would contradict that the statements have different truth values leading to an impossible case. If statement 1 is false and statement 2 is true then the fathers belong to different classes and the sons also to different classes. Since statement 3 must be true anyway for the case of 1 false and 2 true and in for such a configuration to be possible the person who belongs this anyway being Tom to both classes must belong to a different class than the other 2 Tom must have said statement 1 and anyway he is a liar while the other two are truth tellers both based on the information received can't be deduced.

It's interesting to note that the problem is solved by observing it's logical structure.

By this , it's interesting to note that what determines the consistency of the statements are statements 1 and 2 which leads not completely uni-dimensional to determining the value of 3 on one hand and also to information about the fathers and sons , the two needing to be consistent and therefore implying in some cases this type of case checking anyway.