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Sankar Mukhopadhyay, Penn Economics, 2003 writes an example of why $p \implies q$ evaluates to true when both p and q are false. He proposes $3+1=7 \implies 6-1=2$. Then he writes: $3+1-3=7-3=4$, so 1=4. Then, $6-1=6-4=2$. Stating that this proves the implication.

From the standpoint of numbers, both statements are false. But, if 1 is symbol for 4, then both statements are true. He used the numeral 1 first as a number and then as a symbol. Were the statements false until he wrote $1=4$?

In thinking about this and trying to make sense of it, I tried producing other implications. The one that seemed to help the most is $3+1=7 \implies BananasareYellow$. In this case, p is false, q is true (during some time period) and the two statements are unrelated. The statement p can be numbers or symbols and made to be true or false without affecting q. The statement q, however, may or may not be true depending on time. Bananas are first green, then yellow, then black. These aspects of the two statements seem to be no more or less significant than statements that are false then true.

I can not see how to make sense of any of this unless I assume that Mukhopadhyay's statements and mine need more statements that restrict how and when they are true or false.

I want to just cry "Help" and hope that I can be helped.

  • Also, would someone please direct me to instructions regarding correctly attributing work that I reference. –  Mar 10 '20 at 22:45
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    I think the attempt that you cite to explain why $\mathsf{false} \implies \mathsf{false}$ is conventionally taken to be $\mathsf{true}$ is amusing but unhelpful ($ 0 = 1$ implies that the moon is made of cheese, but there is no sensible way to justify that by equational reasoning). See https://math.stackexchange.com/questions/70736/in-classical-logic-why-is-p-rightarrow-q-true-if-p-is-false-and-q-is-tr and many other MSE questions for more about why this is the convention we adopt. – Rob Arthan Mar 10 '20 at 22:59
  • Every time $3+1 = 7$ bananas are yellow. I've never come across a case where $3+1 = 7$ and bananas weren't yellow. If you can document a case where $3+1=7$ but bananas aren't yellow, but right now, I'm standing by my convictions that $3+1=7\implies$ Bananas are yellow. – fleablood Mar 10 '20 at 23:12
  • "0=1 implies that the moon is made of cheese, but there is no sensible way to justify that by equational reasoning" But every single instance in the history of the universe, every single time that $0=1$ it has been true that the moon is made of cheese. The only way you can prove $0=1\implies$ the moon is made of cheese is false, would be to find a case where $0=1$ but the moon isn't made of cheese. But you can never find such a case because $0\ne 1$. So $0=1$ and the moon not being made of cheese is impossible. – fleablood Mar 10 '20 at 23:15
  • @RobArthan There may be parts of the moon that contain $0\times 100%$ cheese, but should it be true that $0=1$, then those parts will contain $1\times100%$ cheese; therefore the moon will be made of cheese if $0=1$. – Graham Kemp Mar 10 '20 at 23:35
  • @GrahamKemp: I don't think that is a sensible way of justifying that particular instance of ex falso quodlibet, by equational reasoning. I appreciate it that it's fun to come up with these little witticisms, but they don't help beginners in logic like the OP. – Rob Arthan Mar 10 '20 at 23:43

2 Answers2

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You may need learn first order logic. In propositional logic, there is no such thing as 'quantifier', for example the time you mentioned. If $BananasareYellow$ is a statement in propositional logic, then it has only one truth value, independent to any quantifiers, such as time or species of bananas. In propositional logic, banana stays forever yellow. (If the statement is true) However in first order logic, you may write your statement $$\forall t(BananasareYellow(t)),\forall t(t<10secondsfrompresent\rightarrow BananasareYellow(t))$$ or$$\forall t(t<10years\rightarrow BananasareYellow(t)).$$ Where $BananasareYellow(t)$ indicates a formula which tells banana is yellow or not at time $t$. $The first statement is (probably) wrong since banana will turn black eventually. The second statement is (probably) true since banana will stay Yellow within 10 seconds. The last statement will be wrong, justyfiy yourself.

And for your notations to describe the statements, to have more concrete understanding, you need to know the difference of Syntax and Semantics. If semantics of notations (syntax) $1,2,3,\cdots$ are usual natural numbers $1,2,3,\cdots$, then the statement $3+1=7$ will be false. If the semantics of notations $1,2,3,4,5,\cdots$ are to be usual natural numbers $1,2,3,7,5\cdots$, then the statement $3+1=7$ will be true. So if not obvious, you need to explain your notation's definition (semantics) before writing your sentences.

Jingeon An-Lacroix
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  • The 3+1=7 was not mine. I was just trying to find a way to understand it. Regardless, I see that I need to start with your "first order logic". I am trying to self study, so I guess some silly questions will come out. –  Mar 11 '20 at 00:40
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    The first order logic will be contained any kind of mathematical logic book, good luck! – Jingeon An-Lacroix Mar 11 '20 at 00:46
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$3+1−3=7−3=4$, so $1=4$

This is begging the question. That is, it is only a true claim if "false implies false".

There are at least 2 reasons false implications (false to true, false to false) aren't left undefined. The first is that there is no harm in defining them, because they are vacuous. For example, defining $1=2 \text{ implies } 3=4$ is harmless because no one will ever be able to prove $1=2$.

The second reason is that it is painful to work in a partial logic. If you left falsely predicated implication undefined, you'd have to have conversations like this:

  • Would you like to buy some jewelry?
  • If the jewelry costs less than $100 then I would like to buy it, otherwise I don't want to buy it

Everything after the comma would be necessary to turn the above statement into a total claim if the implication was undefined for false conditionals.

Then the question of why it is defined the way it is, roughly because it is the strongest possible vacuous claim. If you are going to define something vacuously, you may as well get as much mileage out of it as possible.

DanielV
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  • Oh how I want to exclaim, "It was not me!" The author wrote the implication using the two false statements then performed the math that showed 1=4. It may be that there is logic in his method. At least it has helped me along the road to self study. –  Mar 11 '20 at 13:04