Vacuously True
On any given day, Sam either always lies or he always tells the truth. Today, he says:
1) I like math.
2) If I like math, then I also like physics.
Does Sam like math?
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2 solutions
Moderator note:
Correct! And if you want to be a tad more formal about it, say "this is a contradiction" instead of "this is impossible."
Lets call "I like math" A and "I like physic" B
We have statement 1 is A and statement 2 is A-->B.
Now if we consider "A-->B is false" means A-/->B then both statements could be false, therefore we could not determine the truth?
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" A ⇒ B is false" means A is true and B is false. This is contradiction with the first statement A that is also false.
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What I actually find by looking at it is that the form of the implication which assigns for A the value false isn't rather possible and that therefore is nonsensical to speak of an implication like A -> B where A is false in the first place and by being nonsensical it can't be said of it as either being true or false. When I consider by grasping an implication , that is , when i am thinking at an implication the way I do this is by considering that if something , in this case A happens then something else B would also happen the implication being a relation which I actually have in mind as the object of which i speak when it is said that the statement about the implication is true or false. Since this is about a relation I can verify if the relation itself as the object which I have in mind is actually the case or isn't and this verifying is easy to do for the case in which A has the value true since by being the case it can be seen if B happens or not and say about the value of the statement. But I find that for the case where A is false I can't speak about the validity of the implication because I can't verify it and this on the idea or rather principle of thinking that in order to affirm something about a thing in this case about what A implies I would firstly need to have in mind that thing (A) itself. Here when i speak about verifying i should emphasize that I am speaking of verifying based on the criteria of the wrongness or rightness of reasoning or inference that is that by assigning truth values (true or false) for A and B and processing them by reasoning provided that the reasoning is correct i can achieve the truth values of the statement "A->B" and that therefore when I say that for A false the implication can't be verified I mean that for taking A false I can't conceive any implication which means that it is by not conceiving such an implication nonsensical of speaking of an implication which starts a priori whit a false premise or "hypothesis". This can be stated simpler and by some improvised terms by the fact that the form of an implication always implies A being true and that it is impossible for the form "A->B" to be even conceived where A is false and therefore that thinking of an implication where the hypothesis is false is nonsensical because whenever I conceive an implication i will affirm by the very form of it the hypothesis and then verify it's wrongness or rightness based on the other second term but i'd love to understand why what I say here is wrong.
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@A A – The "if A then B " here is material implication, not causation; it's equivalent to " ¬ A ∨ B ". When the hypothesis is false, the whole statement is true, even if it doesn't make sense in "normal world".
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@Ivan Koswara – Yes , i looked at wikis and stuff like that on the net but I still don't get it or find it is wrong and in my comment i already told that when i speak of verify i don't refer at any kind of verifying based on empirical criteria yet if it is right what you say then i should make a distinction between the logical form of an implication and a causation. What I stated then would be so to say true for causation since it is impossible to conceive a causation that is , it is impossible by it's very form , by it's structure , or by the fact that the mind in the first moment of conceiving a causation must affirm something that is it is impossible "a priori" where A is false I think but i still fail to understand why the form is valid and by what direct thinking (direct grasping of the concept) if this thinking is not the thinking under the terms of causes and effect the form of "material implication" is different. I don't state that you are wrong , but that I don't find it acutely enough to be right i thinking of it. If what you say here is a proof by making -A V B equivalent with the implication A -> B this should be seen directly and confirmed by thinking at an implication where A is false or rather show how A -> B is different from a causation sometimes ; I will moreover emphasize again that the statement is not about a "normal world" as you say but rather about the way of conceiving and thinking itself rather so to say about the phenomenology of thinking when we take into account the reasoning of an implication and therefore the statement wouldn't make sense (has no meaning and can be neither true or false) because it couldn't be thought of , it would be a contradiction in the form of causal thinking therefore being necessary to see emphasized what is the difference between "material implication" and causation.
Edited: I think I see now what you mean so i'll try to explain it clearly.
I will firstly say that in general , apart from any special cases , and at it's most formal an implication is a relation between some object and another so that for the case in which if the first is , the second must also be and that if there are particular cases for such a relation then there are some "species" of the general implication which necessarily will have this thing in common.
What seems to be stated is that an implication is not necessary a causation therefore meaning that there is at least some other way of thinking an implication , of conceiving (the thing which we do when we process in thinking) an implication which doesn't have the particular traits of a causation while still remaining an implication and this is the "material implication". Causation and "material implication" would be therefore two different ways of conceiving which have their own direct conceptual reality in thinking therefore meaning that in order to see what are the differences and how they are still an implication we should understand better enough what is one what is the other. Since causation is the most familiar and simplest I'll start with a description of what that is in order to understand how i think , what is the particular act of thinking of an implication for causation and then move to the way it can be done differently. When I think of "A -> B" in conceiving a causation what I do is that I see that A causes B and in this , conceiving a cause , I see in the intimacy of thinking that B derives from A the derivation being innate , by itself or in other words in the thinking of causation always being continuously implied , when I think of B , A as the one from which B derives. Since this way of seeing how A causes B has such an intimacy it can be said that it is substantial. In this case therefore when I say that A -> B I read (and think) A causes B it means A being the one from which B derives is necessary and is in relation to B the origin of this other second term of the logical expression. But , when I say A -> B in the case of material implication , if the thinking of the implication is different from the thinking of causation this means that for the relation between A and B would still be true that for the case in which A is B would also be but this relation not on the ground of causation in which the first is the origin of the second. Taking this case it can be said that A -> B but not as a cause of B but as a formal conceiving of the relation -A or B (reasoning this equivalence on the grounds of the rule of inference called modus tollens I think) the relation between the two terms being on the grounds of an exterior and formal rule which encompasses both and where both A and B are therefore just externally related and therefore also externally implied as a result of some already given rule which is used. The difference would be then that the relation between A and B in a causation is intimate while in a material implication I conceive such a relation but just externally as a result of their adhering to some rule or criteria being therefore a difference in conceiving an implication internally and externally but however if this is what is meant by "material implication" then a proper naming would be "formal implication" even if indeed any implication in formal logic is considered formal. Is that what you meant by "material implication" ?
Shouldn't a counter statement be given? For example, counter statement of the second statement can be:
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If I like math, then I don't like physics.
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If I don't like math, then I like physics.
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If I don't like math, then I also don't like physics.
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It's not necessary... ( A ⇒ B is false), if and only if, (A is true and B is false). It's a rule from Logic of Calculus of Propositions or Logic of first order, which are different to "Logic of folk"
I really liked this problem. It's a variation of another one I posted. It ilustrates the conditional function in a very interesting way. Thumbs up!
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I saw your problem and I resolved it of this form, I think you still have to learn much about logic, althought your solution was good
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why do you say I have a lot to learn about logic? I mean, I agree, but do you have any reasons to say that?
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@Elena Gomes – Yes, do you know just the truth tables or what is a tautology? or do you know the Logic of Calculus of propositions, or something of Logic of first order?
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@Guillermo Templado – you're asking me if I know? That means you have actually no reason to state what you did... Interesting.
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@Elena Gomes – I said I think... furthemore, I have also to learn a lot
I dont get your solutions can you please make it more simple? I dont get how second one will be true
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I'd like too a better understanding of this so called rule of logic. I want to see if my reasoning for why an implication is true when A is false is the same or not so to say.
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Yup, please can you see the implication table A ⇒ B (if A then B) here? truth tables
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@Guillermo Templado – Yes , exactly this is why I said that I wanted to see the explanation not just the truth tables but the understanding of them as seeing and understanding why 2+2 = 4 and not just by what is given in a table of addition ; in fact logic is about understanding and seeing why the reasoning is so as for example in causation it wouldn't make sense to say there is nothing to cause.
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@A A – Sorry, I can't answer you then, because I don't have any explication why this is like this?
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@Guillermo Templado – Some stuff are self evident in logic (as the principle of identity) indeed but not everything , where some concepts are rather composite.
That is they are made out from this simple and self evident things but imply a more elaborated reasoning about them and in that case it is necessary to understand better what is the reasoning behind it cause it may be assumed that such a reasoning is indeed since there is an understanding of it therefore a way of thinking you have of it.
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@A A – maybe, the reason why this is like this,it's because when in maths we want to prove a proposition of the kind A ⇒ B , (if A then B ) we are supposing that if A is, in fact, true... the only possibility what the proposition makes sense for, is that B is true too... I say maybe... I'm not completely sure
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@Guillermo Templado – I see things like this. Implication is meant to be distinguished because it is thought differently in two types of implication , one being causation (A causes B) the other material implication.
If it is differently thought then implication which is found in both causation and material implication has some difference in causation and material implication which needs to be pointed out clearly. Implication considered naturally means causation , that means that thinking that A -> B you think naturally that because A then B and therefore that there is a relation between A and B such that you can say that from A derives naturally B or that when I am thinking of A I see that from it should come B in an organic way. That means that for A -> B in a causation to have any truth value it must be supposed that A happens because otherwise it would not be what to causes the second term B and also it means that in a causation there must be some sort of relation between A and B to be a causation. Implication as material implication is different from causation and nonetheless is still considered implication. In material implication there is no need for a term to derive from the other like in causation which seems artificial as long as it is thought in terms of causation (and maybe actually it really is artificial and has no meaning at all) and therefore you can assume that A is false. If A is false then B is true/false would be true because B can happen with or without A , therefore it is meant to be read something like A doesn't happen yet because the A doesn't cause B but actually implies B there is no relation between the existence of A such that the existence of B to be derived from it and therefore B can or not happen. Yet to me such stuff seems a little artificial because it is not clear how can A imply B if there is no relation between A and B in the first place , and is artificial exactly to point what you said about the fact that it is only one way the proposition by assigning truth values is contradictory anyways. In short I really have doubts regarding what "material implication" is meant to mean and what it's thought sense is anyways.
Yup, please can you see the implication table A ⇒ B (if A then B) here? truth tables
We are looking for solution(s) where both "P" from the first statement and "if P then Q" from the second one to have the same truth value. It has one solution of which the two to be both true, and therefore, it must be on Sam's truthful day.
TRUTH TABLE : IF P THEN Q
| P = like Maths (Statement 1) | Q = like Physics | IF-THEN (Statement 2) |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
We are comparing the first and third columns in each of the 4 last rows for the same alphabets, either 2 Ts or 2 Fs.
If Sam was lying, the statement 1 would be false but then the stament 2 would be true because its hypothesis (If Sam likes maths) is false, so this is contradictory and therefore, Sam is telling the truth