Vacuously True Statements
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- Опубликовано: 19 окт 2024
- Learning Objectives: Determine when a conditional statement is vacuously true
A conditional statement of the form "if P then Q" is called vacuously true whenever P is false. This might seem counterintuitive, but consider the reverse. The conditional "if P then Q" is false whenever the assumption P is true and the conclusion Q is false. So when the assumption is false it doesn't matter what the conclusion is, and we call the conditional true, but in a vacuous sense.
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This video was created by Dr. Trefor Bazett
I love how the deeper we go, the more wild his hair gets. I can't wait to see how it looks by the end of the course
You are doing the Lord's work. Please do more videos on informal logic, more specifically truth tables and Venn diagrams.
amen
thank you for explaining this. it was overlooked in class!
Your videos are really awesome, its a shame they don’t have much popularity. Maybe you could do popular videos of math, like popular simple math exercises, so people start getting interested.
its amazing how you write backwards on the glass so well. also thank you this is helping me study for my test :)
I assume they just mirror the video
@@christhomas4739 yes, you can tell it is mirrored because the breast pocket is on the wrong side.
his breast is on the wrong side. @@adrycough
thank you. that cleared up a lot of confusion for me.
Just realized he is either writing backwards or the video is flipped when they do the overlay. Cool
came for discrete math, stayed for daddy
straight to the point!
I have a question. given a statment "for a whole number n if n > 2 then n^2 > 4", is it trivially true, vacuosly true, both or neither?
you have to prove the disjuctive form is equivalent to the conditional form, but you need to define vacuous truth to prove that. This video essentially is circular logic isn't it
Perhaps you mean that the definition of vacuous truth relies on conditional statements. Lovely.
In a sense, math is indeed circular, that is, certain definitions are made in order to cook up certain results down the road (or to avoid certain consequences). This is precisely the case for the necessity of vacuously true implication. You can check out my video “why is P implies Q true when P is false”. Cheers
Ok so let me get this straight:
Within a conditional statement, if P AND Q are true, then the statement itself results in a "T" (truth) symbol on the Logic Truth Table.
However, if P is false, and Q is true, or both P and Q are false, the statement in general would be a "F" (false) on the Logic Truth Table? Even though technically the first two are vacuously true as you put it?
No, the statement would be considered true in the truth table whenever P is false.
@@orianasfeir9477
Consider the statement:
If I flip over a card, it will be a 7 of spades.
If I never flip over the card, it becomes a 7 of spades?
@@mygills3050If you didn't flip over a card, then the card you flipped over is a 7 of spades.
Imagine I have 1 card placed face down. Let’s say I want to test out:
“If *I flip it over,* I will *see a seven of spades.*”
When P is *true* and Q is *true*, I find P > Q is *true* because P occurred and Q followed. I flipped over the card, and it was a seven of spades.
When P is *true* and Q is *false*, I find P > Q is *false* because P occurred and Q did not. I flipped over the card, and it was something other than a seven of spades.
When P is *false,* well, I didn’t flip over the card. I can’t see whether P > Q is true or not because P hasn’t happened.
The example you give might be better understood with temporal logic, but ultimately if you don’t flip it over, then if you flip it over, it will be a seven of spades but it also won’t be a seven of spades. The point is, contradictions imply anything in Classical Logic, and in pretty much any logic with rules for negation, contradictions imply other contradictions. Further, statements of the form P->(Q->R) are logically equivalent to statements of the form (P&Q)->R in almost all relevant truth-functional logics, if not all of them. So, let’s say “I flip the card” is P. In the cases where P is false, you’re essentially saying ~P->(P->Q), which is obvious if you think contradictions imply anything (a property known as the principle of explosion). The principle of explosion follows from allowing proof by contradiction and assuming that ~~P is logically equivalent to P.
For an intuition, consider your friend says something absurd like “Mars is further from the Sun than Jupiter”. Well, logic says your response: “if that’s true, then the Moon must be made of cheese” is a correct response to such absurd claims.
to refine your sentence, When P is false, means I didn't flip over the card, so I vacuously think that P > Q will be true, means that if I flip I might get seven of spades but not 100% sure.
But then what is the value of P > Q in the case that P is false? *Maybe* true?
Hy, can you also make videos on 6th edition of discrete maths plz
Thank you.
Insane to me that I pay my university $1,400 / class, yet I have to go on RUclips to search for lectures because my professor can barely speak English...
this is me rn
If Trevor IDENTIFY as a unicorn, he IS a unicorn, therefore it is a hard fact, and the contradiction is vacuous, thus P becomes a tautology. Who says Maths is not logical?
love from nepal
Thank you!
so there's no causal relation? cuz everyone could still get an A, yea?
Why will a False proposition imply a True consequence?
If (p->q) (~p ν q), then 'either I will not study hard or I will pass the exams (by other means😂) .
"I've never met a mountain lion that eats meat" is a misleadingly true statement if you've never met a mountain lion. Would that be considered a vacuously true statement?
We don't call this vacuously true, but it certainly is its own thing where if a more general statement could be said it is better to say that then a more specific statement as otherwise it implies the general statement couldn't be said!
But is Trevor really not a Unicorn 🤔?
🔥🔥🔥
You spell Trevor with an 'f'???
You spell Stephen with an 'f'??? lol
You really shouldn't be talking.....
English is a funny language .
@@wachowski9525It is possible that he identified the similarity in the spelling of their names as a relatable feature they share and merely highlighted this observation. Absent explicit clarification from him, his intention remains indeterminate. Consequently, your conjecture can only be regarded as speculative. Each approval (like) you receive likely represents individuals, akin to yourself, who have not engaged in thorough contemplation of the matter.