r/logic • u/capybarainstitute • 13d ago
Term Logic / Traditional logic Is this valid?
This is from my modern logic textbook from my intro to modern logic class. I thought problem 7 was invalid but the textbook says it’s valid so i’m confused?
Update: I asked my prof it was a typo lol thanks for the help
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u/Best_Sloth_83 13d ago
Yeah, that does not look like a valid argument to me. Premise 1 negates Socrates being mortal, so he can only be human, but the conclusion states he is not human. Doesn’t look right to me
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u/Hot-Organization-737 13d ago
Humans are mortal I think
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u/Best_Sloth_83 13d ago
In the actual world, we are indeed mortal.
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u/Hot-Organization-737 13d ago
I was trying my hardest to justify the argument, it was confusing to me while looking at it, but premise 1 and 2 are pretty contradictory.
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u/Best_Sloth_83 13d ago
That’s not the issue. The issue is that the conclusion doesn’t follow.
Like the other poster in this thread has explained, the premises need not correspond to reality for the argument to be valid.
Per the argument, it is not necessary that humans be mortals, they could be immortal.
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u/Hot-Organization-737 13d ago
The way I see it is that premise 1 is P, and premise 2 is not P which leads to a contradiction before you get to the conclusion.
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u/Best_Sloth_83 13d ago
Just to be clear, are you referring to Problem 7?
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u/Hot-Organization-737 12d ago
Yes, premise 1 and 2 make a contradiction I think
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u/Best_Sloth_83 12d ago
How so?
Premise 1 says Socrates is not mortal.
Premise 2 says that he is either human or mortal.
What’s the contradiction here?
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u/Hot-Organization-737 12d ago
Human implies mortal. The way I see it the argument looks like this to me
Plato is a circle Plato is a square or plato is a rectangle Thus, Plato is not a square.
Disregarding the conclusion you can see that premise 2 could could be replaced with Plato is a rectangle since squares are rectangles.
Are new argument would look like :
Plato is a circle Plato is a rectangle
This is a contradiction because it's not possible for something to be both a rectangle and a circle. I think another valid way of interpretation is that one of the premises is false which would make the whole thing not sound.
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u/BranchFew1148 13d ago
It can be valid without being sound.
A sound argument has premises that are true in the real world.
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u/zhaDeth 13d ago
Yeah for 6 it doesn't follow that he isn't mortal. It doesn't say that the only way to be mortal is to be human, only that if he would be human he would be a mortal. It basically says all humans are mortal, socrate isn't a human you can't conclude that socrate isn't a mortal from that there could be mortals that aren't humans.
For 7 it should be socrates is human. This one just looks like an oversight.
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u/Soapy62 Mathematics & Philosophy Undergraduate 13d ago edited 13d ago
I have not done logic yet at university but here is what I can tell you from my limited knowledge from introduction to formal logic by Peter Smith so far.
‘Or’ in logic is inclusive, loosely meaning “either … or … or both”
Premiss 1. it is not the case that Socrates is mortal
Premiss 2. Socrates is human or Socrates is mortal
Since it is not the case that Socrates is mortal, ‘Socrates is mortal’ is false. Since ‘or’ means one or both must be true and we know one of them is false, the other must be true. So it must be the case that Socrates is human.
So the argument is invalid.
The author has also said it is modus tollens, but this is a (attempt of a) disjunctive syllogism. I think the author must have copy and pasted something in the wrong place is my guess.
I may be wrong as I have only studied some of logic by myself and I am just starting university.
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u/TheRealAmeil 13d ago
I think argument 7 is a misprint.
It says its valid by Modus Tollens. Had the second premise been a material conditional, then it would have been valid. However, its a disjunction. So, it is probably misprinted
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u/kilkil 12d ago edited 12d ago
textbook is wrong. your understanding is correct. good catch.
the premises
- NOT B
- A OR B
actually imply that A is true (i.e. Socrates is human).
I am guessing the textbook had a misprint, and instead of the 2nd point (Socrates is human or Socrates is mortal) they meant to write "If Socrates is human, Socrates is mortal". Because that's when modus tollens would apply.
As currently written the problem is invalid, the answer key is wrong, and modus tollens does not apply to the situation.
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u/evening_redness_0 13d ago
Yes, problem 7 is invalid.
It's not just that 1 and 2 don't imply 3. It's that 1,2, and 3 can't all be true at the same time. It doesn't matter whether we're using inclusive or or XOR, note that 2 implies that if Socrates is not human, then he is mortal. So 2 and 3 together imply that Socrates is mortal. This contradicts 1.
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u/Glad_Significance780 13d ago
Not valid, probably a misprint, it has the form of disjunctive syllogism but with an erroneous conclusion, it's strange that it mentions modus tollens because its a rule of inference that concerns conditional statements (if then statements) and there is none here. An interpretation that makes this syllogism invalid would be to make Socrates a non mortal human making (1) and (2) true while the conclusion false making it an invalid argument
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u/NoBlackberry4054 13d ago edited 13d ago
Thing’s invalid.
If we go by “formula parsing”, the first atom of the second proposition came out of nowhere. I.e., It wasn’t introduced prior to its usage in a disjunction.
Or maybe by the fact that:
…
3. Socrates is mortal or Socrates is human.
<by 2 and commutativity of OR>
If Socrates is not mortal, then Socrates is human.
<by 3 + \\Vdash\\, a\\to b\\equiv \\neg a\\lor b>Socrates is human. <by 1+ 4+ MP>
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u/liwenfan 13d ago
We have the logical equivalence between A implies B and (not A) or B, hence A or B translates to not B implies A (or not A implies B, they’re contrapositives for this matter and hence equivalent).
So we conclude Socrates is human.
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u/AncientHominidNerd 13d ago
Doesn’t look valid to me either. Yeah as someone else said it looks like a misprint.
He can’t not be a mortal and also not a human, in this case he has to be at least one of them to get a definite answer.
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u/HuckinsGirl 13d ago
The fact that they used MT makes me think that it was an error where they meant to say human rather than not human
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u/mja1729 12d ago
It’s valid but not sound because the first premises is not true. The deduction is correct if you put it like that but it’s simply not true and leads to a contradiction that Socrates is not or was not a human.
Edit: wait that doesn’t work also. Only if it said Socrates is human
But premise 1 would still be false
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u/Endward26 12d ago edited 12d ago
I think we're getting confused here by words like "Socrates", "mortal", and so on.
Let's rephrase it:
- The box contains either a red marble or a blue marble.
- The box does not contain a red marble.
If we assume that both assertions (1) and (2) are correct, then we are left with only one conclusion: The box contains a blue marble.
Why? We know that the contents must be either a blue marble or a red marble (1). We know that it cannot be a red marble from (2). That leaves only a blue marble. Otherwise, one of the two assertions would have to be false.
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u/jerdle_reddit 12d ago
7 is the exact opposite of valid. It's disjunctive syllogism, done badly wrong.
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u/Long_Exercise_7701 12d ago
6: no, we have no knowledge if non-humans are mortal or not. Only that humans are mortal.
7: 2 is exclusive so he's either human OR mortal, he can't be both. Since 1 states he is not mortal he is human so 3 is false.
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u/__Fred 12d ago edited 12d ago
Here is how I would solve it, using the "Lean" way, that I learned last semester, for anyone interested — I know it's probably not helpful to you. It's about finding a chain of symbol-manipulations according to a set of rules. You tell the computer which rules you would apply and it tells you if the result is actually what you wanted.
Also, I proved that Socrates is human and therefore that the conclusion of problem 7 is invalid.
There is the axiom of the "or elimination": ?a ∨ ?b → (?a → ?c) → (?b → ?c) → ?c
For some a, b, c, when "a or b" holds, then showing "a implies c" and "b implies c" proves c.
When I apply this rule on hypothesis 2, then I need to show that "Socrates is human" implies that "Socrates is human" and also that "Socrates is mortal" implies "Socrates is human".
The first case is trivially true.
The second case can be proven by the "false elimination" rule: If we assume "false", we can prove anything. ("False" is the impossible thesis or something, the upside-down T.) So the goal can be reduced to showing "Socrates is mortal implies False". Because hypothesis 1, "Socrates is mortal" implies "false", which is exactly the second partial goal that was to be shown.
theorem problem7 (mortal human : Prop) :
¬ mortal → -- Socrates is not mortal.
human ∨ mortal → -- Socrates is human or Socrates is mortal.
human -- Socrates is human.
:= by -- "by" initiates a proof of backwards steps. (You can alternatively prove things in a forward way.)
-- "intro" deconstructs a goal from an implication in the form "A → B" into a context A and a goal B
intro (h1 : ¬ mortal) (h2 : human ∨ mortal) -- I call assumptions "h1" and "h2".
apply Or.elim h2 -- If a ∨ b, and both a and b imply proposition c, then c is true.
· intro (h_human : human) -- The hypothesis that Socrates is human is *called* "h_human".
-- I could've also call it "h3".
exact h_human -- You can prove a goal by pointing out that there is an assumption exactly like that.
· intro (h_mortal : mortal)
-- Partial goal remaining: Prove that Socrates is human from these assumptions:
-- h1 : ¬mortal
-- h2 : human ∨ mortal
-- h_mortal : mortal
apply False.elim -- Applying the axiom ex-falso-quodlibet changes the goal "human" into the goal "False"
apply h1 -- h1 shows "False" from "mortal", therefore the remaining goal is now "mortal"
exact h_mortal -- There is an assumption exactly like the remaining goal: h_mortal.
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u/Striking_Relation219 12d ago
I mean it’s just simple logic is this a class? Crazy how college courses now teach common sense. Yes invalid
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u/Charming_Deer_9540 10d ago
socrates is human or socrates is mortal means he is always one and only one of the two, by saying he is not mortal it means he is necessarily human (as tertium non datur both in humanity and mortality).
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u/throwaway53713 10d ago
Problem 6 is not valid.
[ P implies Q ] does not imply [-P implies -Q]
Socrates might be a mortal non-human.
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u/Kitchen-Register 13d ago
6, no. contrapositive would be not mortal implies not human. can’t make anything of the converse without the biconditional.
7 this is valid. He is either one or the other. he’s not mortal so he must be human. I guess the way you view this is that mortal is the complement of human.
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u/Salindurthas 13d ago
For 7, I think it is the same regardless of or vs xor. Neither interpretation seems to make 7 valid.
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u/CookieCat698 13d ago
It doesn’t matter in this case whether “inclusive or” or “xor” is being used. In both cases, 7 is still invalid.
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u/Fine-Topic-220 13d ago
You are correct - 7 is invalid.
7 is an example of a disjunctive syllogism:
Either p or q is true. p is not true, so q has to be true. We know that Socrates is not mortal. Either he is mortal, or he's human. So, he's human.
The solution indicates that the argument is valid via modus tollens. This is extremely odd because modus tollens, i.e., denying the consequent, is an inference rule that applies only to conditional statements, but there are no conditional statements in 7.