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Posted byu/[deleted]2 years ago
Archived

## Logic: If 1 equals 2 then 4 equals 5. Is this a true statement?

[deleted]

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43% Upvoted
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level 1
Theory of Computing14 points · 2 years ago

The statement "if p then q" is true if p is false.

This is known as a vacuous truth.

level 1
24 points · 2 years ago

If 1 equals 2 then 4 equals 5

This is vacuously true.

level 2
3 points · 2 years ago

Ah. Trivial.

Thanks.

level 1
5 points · 2 years ago · edited 2 years ago

Logic: If 1 equals 2 then 4 equals 5. Is this a true statement?

Yes. This is how material implication works---if the antecedent is false, then the conditional is true, regardless of the truth of the consequent.

Since our example has no relations between different elements we assumed it must be transitive because the condition doesn't need to be satisfied (or is satisfied already?)

Your R is transitive, but your reasoning is not quite correct. It can be that a R b and b R c. But if this happens then a = b = c so a = c so a R c. We don't require that a, b, and c be distinct just because we name them with different variables.

level 1
Undergraduate1 point · 2 years ago

Yep! That's true! It's called "vacuous truth" when the premise of an implication is false.

level 1
1 point · 2 years ago

As others have mentioned, your statement is vacuously true, but there's also a direct, arithmetical proof:

We have 1 = 2 and 3 = 3. Adding the equations, we get 3 + 1 = 3 + 2, that is, 4 = 5.

The fact that you can do this turns out to be important in the foundations of mathematics.

level 2
1 point · 2 years ago

Isn't it also true that if 1=2, then 4=27 ?

level 3
2 points · 2 years ago

It's not hard to prove that if 0=1 in any ring, then all elements of that ring are equal to 0 (0 being the general additive identity, and 1 the multiplicative). And since if we assume that 1=2 (in the integers), we must also have 0=1, so all the integers are 0. Most mathematicians I know think this is rather boring, and doesn't result in a particularly useful ring.

level 4
2 points · 2 years ago

The trivial ring is not particularly useful, but the fact that you can prove any equality starting from 0=1, without invoking the concept of vacuous truth, is actually useful in foundations.

For example, you can base arithmetic on intuitionistic logic instead of classical logic: the resulting theory, Heyting arithmetic, is strictly weaker than usual arithmetic.

This yields an obvious question: what do we get if we base arithmetic on minimal logic instead? Minimal logic does not have the principle of explosion, so arguments by "vacuous truth" are no longer available. But the argument above demonstrates that we can prove any equality starting from 0=1, without invoking explosion! Therefore, the resulting theory is just as strong as Heyting arithmetic.

level 3
1 point · 2 years ago

Of course it is. It's not this particular equality that turns out to be important, but the general principle used in the proof above: you can prove any equality starting from 0=1, without invoking the concept of vacuous truth.

level 1
1 point · 2 years ago

It's vacuously true, as would be something like "If whales are space shuttles, then I have seventeen eyes and three noses" When you have a statement "if A then B" and A is false, then the conditional as a whole is vacuously true, so B can be whatever you want it to be.

level 1
0 points · 2 years ago · edited 2 years ago

If you assume a false statement to be true, you can conclude anything you want. This is the principle of explosion.

It also calls to mind the logic of "if then".

"If the sky is green, pigs can fly" is a true statement because the sky isn't green, so it doesn't matter what the predicate is. Likewise, "if the sky is green, I will eat a slice of pie" is also true, whether or not I decide to eat pie regardless as long as the first portion is false.

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Welcome to r/math!

This subreddit is for discussion of mathematical links and questions. Please read the FAQ and the rules below before posting.

If you're asking for help understanding something mathematical, post in the Simple Question thread or /r/learnmath. This includes reference requests - also see our lists of recommended books and free online resources. Here is a more recent thread with book recommendations.

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Basic Math Symbols

≠ ± ∓ ÷ × ∙ – √ ‰ ⊗ ⊕ ⊖ ⊘ ⊙ ≤ ≥ ≦ ≧ ≨ ≩ ≺ ≻ ≼ ≽ ⊏ ⊐ ⊑ ⊒ ² ³ °

Geometry Symbols

∠ ∟ ° ≅ ~ ‖ ⟂ ⫛

Algebra Symbols

≡ ≜ ≈ ∝ ∞ ≪ ≫ ⌊⌋ ⌈⌉ ∘∏ ∐ ∑ ⋀ ⋁ ⋂ ⋃ ⨀ ⨁ ⨂ 𝖕 𝖖 𝖗

Set Theory Symbols

∅ ∖ ∁ ↦ ↣ ∩ ∪ ⊆ ⊂ ⊄ ⊊ ⊇ ⊃ ⊅ ⊋ ⊖ ∈ ∉ ∋ ∌ ℕ ℤ ℚ ℝ ℂ ℵ ℶ ℷ ℸ 𝓟

Logic Symbols

¬ ∨ ∧ ⊕ → ← ⇒ ⇐ ↔ ⇔ ∀ ∃ ∄ ∴ ∵ ⊤ ⊥ ⊢ ⊨ ⫤ ⊣

Calculus and Analysis Symbols

∫ ∬ ∭ ∮ ∯ ∰ ∇ ∆ δ ∂ ℱ ℒ ℓ

Greek Letters

𝛢𝛼 𝛣𝛽 𝛤𝛾 𝛥𝛿 𝛦𝜀𝜖 𝛧𝜁 𝛨𝜂 𝛩𝜃𝜗 𝛪𝜄 𝛫𝜅 𝛬𝜆 𝛭𝜇 𝛮𝜈 𝛯𝜉 𝛰𝜊 𝛱𝜋 𝛲𝜌 𝛴𝜎 𝛵𝜏 𝛶𝜐 𝛷𝜙𝜑 𝛸𝜒 𝛹𝜓 𝛺𝜔

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© 2018 Reddit, Inc. All rights reserved
The text passage contains a mathematical question and multiple answers provided by different users. The question is a logic problem about the truth value of a conditional statement.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Press J to jump to the feed. Press question mark to learn the rest of the keyboard shortcuts
0
Posted byu/[deleted]2 years ago
Archived

## Logic: If 1 equals 2 then 4 equals 5. Is this a true statement?

[deleted]

12 comments
43% Upvoted
This thread is archived
New comments cannot be posted and votes cannot be cast
Sort by
level 1
Theory of Computing14 points · 2 years ago

The statement "if p then q" is true if p is false.

This is known as a vacuous truth.

level 1
24 points · 2 years ago

If 1 equals 2 then 4 equals 5

This is vacuously true.

level 2
3 points · 2 years ago

Ah. Trivial.

Thanks.

level 1
5 points · 2 years ago · edited 2 years ago

Logic: If 1 equals 2 then 4 equals 5. Is this a true statement?

Yes. This is how material implication works---if the antecedent is false, then the conditional is true, regardless of the truth of the consequent.

Since our example has no relations between different elements we assumed it must be transitive because the condition doesn't need to be satisfied (or is satisfied already?)

Your R is transitive, but your reasoning is not quite correct. It can be that a R b and b R c. But if this happens then a = b = c so a = c so a R c. We don't require that a, b, and c be distinct just because we name them with different variables.

level 1
Undergraduate1 point · 2 years ago

Yep! That's true! It's called "vacuous truth" when the premise of an implication is false.

level 1
1 point · 2 years ago

As others have mentioned, your statement is vacuously true, but there's also a direct, arithmetical proof:

We have 1 = 2 and 3 = 3. Adding the equations, we get 3 + 1 = 3 + 2, that is, 4 = 5.

The fact that you can do this turns out to be important in the foundations of mathematics.

level 2
1 point · 2 years ago

Isn't it also true that if 1=2, then 4=27 ?

level 3
2 points · 2 years ago

It's not hard to prove that if 0=1 in any ring, then all elements of that ring are equal to 0 (0 being the general additive identity, and 1 the multiplicative). And since if we assume that 1=2 (in the integers), we must also have 0=1, so all the integers are 0. Most mathematicians I know think this is rather boring, and doesn't result in a particularly useful ring.

level 4
2 points · 2 years ago

The trivial ring is not particularly useful, but the fact that you can prove any equality starting from 0=1, without invoking the concept of vacuous truth, is actually useful in foundations.

For example, you can base arithmetic on intuitionistic logic instead of classical logic: the resulting theory, Heyting arithmetic, is strictly weaker than usual arithmetic.

This yields an obvious question: what do we get if we base arithmetic on minimal logic instead? Minimal logic does not have the principle of explosion, so arguments by "vacuous truth" are no longer available. But the argument above demonstrates that we can prove any equality starting from 0=1, without invoking explosion! Therefore, the resulting theory is just as strong as Heyting arithmetic.

level 3
1 point · 2 years ago

Of course it is. It's not this particular equality that turns out to be important, but the general principle used in the proof above: you can prove any equality starting from 0=1, without invoking the concept of vacuous truth.

level 1
1 point · 2 years ago

It's vacuously true, as would be something like "If whales are space shuttles, then I have seventeen eyes and three noses" When you have a statement "if A then B" and A is false, then the conditional as a whole is vacuously true, so B can be whatever you want it to be.

level 1
0 points · 2 years ago · edited 2 years ago

If you assume a false statement to be true, you can conclude anything you want. This is the principle of explosion.

It also calls to mind the logic of "if then".

"If the sky is green, pigs can fly" is a true statement because the sky isn't green, so it doesn't matter what the predicate is. Likewise, "if the sky is green, I will eat a slice of pie" is also true, whether or not I decide to eat pie regardless as long as the first portion is false.

Community Details

612k

Subscribers

516

Online

Welcome to r/math!

This subreddit is for discussion of mathematical links and questions. Please read the FAQ and the rules below before posting.

If you're asking for help understanding something mathematical, post in the Simple Question thread or /r/learnmath. This includes reference requests - also see our lists of recommended books and free online resources. Here is a more recent thread with book recommendations.

If you are asking for a calculation to be made, please post to /r/askmath or /r/learnmath.

If you are asking for advice on choosing classes or career prospects, please post in the stickied Career & Education Questions thread.

Please be polite and civil when commenting, and always follow reddiquette.

r/math Rules
1.
No homework problems
2.
Stay on-topic
3.
Be excellent to each other
4.
No low-effort image posts
5.
No career or education related questions
Recurring Threads & Resources

Everything about X - every Wednesday

What Are You Working On? - posted Mondays

Career and Education Q&A - Every other Thursday

Simple Questions - Posted Fridays

Click here to chat with us on IRC!

Using LaTeX

To view LaTeX on reddit, install one of the following:

MathJax userscript (install Greasemonkey or Tampermonkey first)

[; e^{\pi i} + 1 = 0 ;]

Post the equation above like this:

[; e^{\pi i}+1=0 ;]

Useful Symbols

Basic Math Symbols

≠ ± ∓ ÷ × ∙ – √ ‰ ⊗ ⊕ ⊖ ⊘ ⊙ ≤ ≥ ≦ ≧ ≨ ≩ ≺ ≻ ≼ ≽ ⊏ ⊐ ⊑ ⊒ ² ³ °

Geometry Symbols

∠ ∟ ° ≅ ~ ‖ ⟂ ⫛

Algebra Symbols

≡ ≜ ≈ ∝ ∞ ≪ ≫ ⌊⌋ ⌈⌉ ∘∏ ∐ ∑ ⋀ ⋁ ⋂ ⋃ ⨀ ⨁ ⨂ 𝖕 𝖖 𝖗

Set Theory Symbols

∅ ∖ ∁ ↦ ↣ ∩ ∪ ⊆ ⊂ ⊄ ⊊ ⊇ ⊃ ⊅ ⊋ ⊖ ∈ ∉ ∋ ∌ ℕ ℤ ℚ ℝ ℂ ℵ ℶ ℷ ℸ 𝓟

Logic Symbols

¬ ∨ ∧ ⊕ → ← ⇒ ⇐ ↔ ⇔ ∀ ∃ ∄ ∴ ∵ ⊤ ⊥ ⊢ ⊨ ⫤ ⊣

Calculus and Analysis Symbols

∫ ∬ ∭ ∮ ∯ ∰ ∇ ∆ δ ∂ ℱ ℒ ℓ

Greek Letters

𝛢𝛼 𝛣𝛽 𝛤𝛾 𝛥𝛿 𝛦𝜀𝜖 𝛧𝜁 𝛨𝜂 𝛩𝜃𝜗 𝛪𝜄 𝛫𝜅 𝛬𝜆 𝛭𝜇 𝛮𝜈 𝛯𝜉 𝛰𝜊 𝛱𝜋 𝛲𝜌 𝛴𝜎 𝛵𝜏 𝛶𝜐 𝛷𝜙𝜑 𝛸𝜒 𝛹𝜓 𝛺𝜔

Other math subreddits
r/learnmath

83,812 subscribers

r/mathbooks

8,948 subscribers

r/cheatatmathhomework

27,990 subscribers

r/matheducation

12,969 subscribers

r/CasualMath

8,018 subscribers

r/puremathematics

9,587 subscribers

r/mathpics

16,287 subscribers

r/mathriddles

7,694 subscribers

Related subreddits
r/Mathematica

4,151 subscribers

r/matlab

16,428 subscribers

r/sagemath

625 subscribers

r/actuary

11,465 subscribers

r/algorithms

40,662 subscribers

r/compsci

384,228 subscribers

r/interdisciplinary

1,598 subscribers

r/statistics

61,659 subscribers

© 2018 Reddit, Inc. All rights reserved
user
The text passage contains a mathematical question and multiple answers provided by different users. The question is a logic problem about the truth value of a conditional statement.

TYPE: SINGLE_QUESTION_WITH_ANSWER