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It's well known that vacuous truths are a concept, i.e. an implication being true even if the premise is false.

What would be the problem with simply redefining this to be evaluated to false? Would we still be able to make systems work with this definition or would it lead to a problem somewhere? Why must it be the case that false -> false is true and false -> true is true?

Bram28
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user539262
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    Then implication just becomes and. This is easily seen by writing out a truth table. – Riley Mar 12 '18 at 02:50
  • Unless we made false->false true – user539262 Mar 12 '18 at 03:07
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    But then it would become a biconditional. – Bram28 Mar 12 '18 at 03:13
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    It is not a matter of "an implication being true even if the premise is false". The implication is assumed to be true. However, if the antecedent is false, then the truth value of the consequent is not constrained. – Rodrigo de Azevedo Mar 12 '18 at 12:16
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    Vacuous truth normally means ∀x ⋲ {} x ⊨ Q; that is all statements about all members of the empty set are true. – Joshua Mar 12 '18 at 15:50
  • See also https://math.stackexchange.com/q/48161/442 ... is this a duplicate? – GEdgar Mar 13 '18 at 19:03
  • I accidentally commented on one of the answers, when I meant to put it here: I don't have much background in logic, but my understanding is that paraconsistent logics - in particular, relevance logic (en.wikipedia.org/wiki/Relevance_logic) - does not suffer from vacuous truths, and indeed appears to be exactly what this question is asking for. Could someone who does have a logic background confirm this? – Robin Saunders Mar 15 '18 at 07:11
  • @Riley I forgot to ask, what is the intuition behind why a and b is wrong? If we say a implies b is "whenever a happens, b happens", the notion of "a and b" does strangely seem to fit, no? – user539262 Mar 16 '18 at 03:10
  • No. Imagine the statement "If it's raining, it's cloudy." Does this mean that someone can come along and say "Ha, you're wrong! It's not raining!", thereby disproving your statement? Clearly and intuitively, this does not disprove your statement. In fact, the only way to disprove your statement would be to find an example where it's raining but it's not cloudy. – Riley Mar 16 '18 at 14:16
  • No need to. For any logical true-or-false propositions A and B, we can prove $\neg A \implies [A\implies B]$. The only assumptions we need to make about implication are: modus ponens, deduction and reductio ad absurdum (all self-evident). See my recent blog posting on material implication at http://www.dcproof.com/IfPigsCanFly.html – Dan Christensen Apr 06 '18 at 14:49

6 Answers6

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Notice that 3=5 is false. but if 3=5 we can prove 8=8 which is true.

$$ 3=5$$

therefore $$ 5=3$$

Add both sides, $$8=8$$

We can also prove that $$ 8=10$$ which is false.

$$ 3=5$$

Add $5$ to both sides, we get $$8=10$$

The point is that if we assume a false assumption, then we can claim whatever we like.

That means " False $\implies$ False " is true.

And " False $\implies$ True " is true.

Mohammad Riazi-Kermani
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    This really made it click for me. From a false premise we can arrive at both true and false statements alike. But from a true premise we can't arrive at false conclusions, only true ones. Is this right? – user539262 Mar 12 '18 at 03:23
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    Right., I am glad that a simple example made the point through. – Mohammad Riazi-Kermani Mar 12 '18 at 03:28
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    But how would you go from $3=5$ to 'snow is white' or to 'snow is purple'? You have shown a specific false claim and a specific true claim that we can infer from a false claim, but I don't see how that proves the general: 'if we make a false assumption, then we can claim anything we like'. Or, as the OP put it: if we use different variables for $P$ and $Q$, then maybe we cannot go from $P$ to $Q$, even if $P$ is false. – Bram28 Mar 12 '18 at 03:44
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    @Bram28 Once, Berttand Russell said in a lecture that, if 1+1 = 1, then everything can be proven true. Someone challenged him, “Then prove you’re the Pope.” Russell said, “One and one is one. I am one. The Pope is one. Therefore, I and the Pope am one.” – Davislor Mar 12 '18 at 05:46
  • @Bram28 We can prove anything we like if we assume a contradiction (this fact is called explosion), which is a special case of a false statement; but if we can prove a statement false then assuming that false statement allows us to prove a contradiction. – J.G. Mar 12 '18 at 08:53
  • @Bram28 Truth in propositional logic depends only on the true/false value of the individual propositions, not on their interpretation as facts about the real world. Mohammad has given examples that show that it is necessary for some false propositions to imply some true propositions and some false propositions. Since the definition of "implies" can only depend on the truth values of the propositions on the left and right, the only possible interpretation is that false always implies false and false always implies true. – David Richerby Mar 12 '18 at 10:21
  • @Davislor Thanks for the Russell quote! :) – Bram28 Mar 12 '18 at 11:43
  • @J.G. right: from a contradiction we can infer anything we want, but the OP was merely talking about a false statement, not necessarily a contradiction, and likewise this answer is starting with a mere false statement, not a contradiction. – Bram28 Mar 12 '18 at 12:56
  • @Bram28 Yeah; it would have helped if they'd clarified anything can be derived from a suitably chosen false statement. – J.G. Mar 12 '18 at 12:57
  • @DavidRicherby OK, but what if there are also cases where a false proposition does not imply some other false and some other true statements? I still don't see how a false statement (not a contradiction, but a mere false statement) would logically imply everything. – Bram28 Mar 12 '18 at 12:58
  • @Bram28 If you want "or" to work the way it should, then you have to accept the following sort of argument: $0=1$ implies ($1=2$ or "snow is purple"). Since $1 \neq 2$, then we must have "snow is purple". – Steven Gubkin Mar 12 '18 at 13:27
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    @Davislor: I'm sorry, I don't know what the word "purple" means, but fortunately I understand mathematics, physics and engineering and can build a machine to detect "purpleness" if you clearly and unambiguously say what you mean by "purple". Care to take a shot at it? Once we have a clear and unambiguous definition of what this word means then we can much more easily show how a false proposition implies that anything is "purple". – Eric Lippert Mar 12 '18 at 14:06
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    Comment #2 above is incorrect. @user539262 asks "But from a true premise we can't arrive at false conclusions, only true ones. Is this right?" Correct answer is: we cannot know for sure, because of issues related to Gödel's theorems, undecidability/non-halting, axiomatic systems, and logical system consistency. TL;DR version: a logical system cannot prove its own consistency, nor can you get round that by simply using a more powerful logical system. Which directly means/implies that you just cannot know, if you only start with true statements, that you only ever get true statements out. – Stilez Mar 12 '18 at 14:38
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    @Stilez This isn't what Gödel's theorem says. Starting from true premises always leads to true conclusions. Gödel says that formal systems powerful enough to express arithmetic cannot prove all true statements. In particular, they cannot prove their own consistency. You do, I guess, have to allow for the possibility (however faint) that Peano arithmetic is inconsistent. But then, we are still not deriving false from truth: we have been all the time been deriving false (and truth) from falsity. – Steven Gubkin Mar 12 '18 at 14:41
  • @StevenGubkin agreed ... but now you throw in an extra piece of information, namely that $1 \not = 2$ ... and the question is whether a false statement by itself implies any other statement. In fact, I think that likewise Mohammed is making additional assumptions regarding basic arithmetic when going from $3=5$ and $5=3$ to $8=8$, and going from $3=5$ to $8=10$. I think his examples would have been better going from $3=5$ and $5=3$ to $3=3$, and going from $3=5$ to $5=3$; at least those are purely * logical * implications. – Bram28 Mar 12 '18 at 14:44
  • @Bram28 The definition of $1=2$ being false is that $1\neq 2$ is true. So I would not view this as "pulling in an extra piece of information". – Steven Gubkin Mar 12 '18 at 14:50
  • @Steven Gubkjn - it's somewhere in that bundle.of things I mentioned. I probably should have mentioned Peano as well. But when the comment effectively asks "is it certain (I.e. provable) that "true statements" + a suitable and "adequately powerful" logical system (whatever that means) always result in true statements, the answer has to be a firm "not known/not decideable". The rest is a grab-bag of thing that will help explain why not. – Stilez Mar 12 '18 at 15:02
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    @Stilez You are still not quite correct. It is known that implications from true premises are true: that is the definition of implication. What Godel showed is we cannot get ALL the true statements from any one finite set of axioms. – Steven Gubkin Mar 12 '18 at 15:15
  • Correct me if I am wrong but I believe the whole point behind proofs is that truth leads to more truth by definition (e.g. the Peano axioms which tell us what operations and logical constructs are valid). If it were possible to arrive at an "ostensibly" false result then we'd be adding another axiom somewhere to work around it (for example making the successor function S an injective function to prevent something like 8 coming right after 2 as well as right after 5). But from a false premise, the resulting nonsense will be wrong much of the time and right sometimes by chance alone. – user539262 Mar 12 '18 at 16:22
  • @user539262 There are systems in which true premises imply logical contradictions. We call those inconsistent. These aren’t normally useful, though, and we want our theories to be sound. This is different from how no proof system complex enough to be a model of arithmetic over the natural numbers is complete, that is, powerful enough to prove every true statement about itself (a fact which Alan Turing showed has profound implications for computing). – Davislor Mar 12 '18 at 17:10
  • @EricLippert I don’t think your comment was meant for me? Although the Bertrand Russell (sorry I typoed his name the first time) anecdote is a funny example of finding a clever way to prove something seemingly unrelated to arithmetic. It does seem likely to me that your objective physical definition of purple would break down if we could prove the theorem that all quantities, including the wavelength of all photons, were equal. – Davislor Mar 12 '18 at 17:22
  • @Davislor: Whoops, you're right, I intended to address that to Bram28. And yes, that was my point: once we have a crisp enough technical definition of snow and purple, you can then attack it at any point. – Eric Lippert Mar 12 '18 at 17:25
  • @J.G., ah, but the principle of explosion when applied to physics and truth in the real universe doesn't take into account propagation delay, which is probably the only reason the universe still exists. ;) (Had breakfast at Milliways recently?) – Wildcard Mar 12 '18 at 19:08
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    @Bram28 If you digitise color, you camera-computer would return something like: white has a brightness of 5, purple has a brightness of 3. Once you accept this reasonable premise(e.g. quantifying color by the sum of its R,G,B channels), proving 'white is purple' is just a step away. – zyc Mar 12 '18 at 22:30
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    @Bram28 1+1=1 is true. 1+1=1 is false. Therefore true is false. Snow is blue is a false statement. Therefore snow is purple is a true statement. (of course, this seems to only work informally) – user253751 Mar 13 '18 at 02:20
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Clearly we want $P\rightarrow P$ to be true, wouldn't you agree?

I mean, if i say:

If Pat is a bachelor, then Pat is a bachelor

do you really dispute the truth of that claim, or claim that it depends on whether or not Pat really is a bachelor? The whole point of conditionals is that we can say 'if', and thereby imagine a situation where something would be the case, whether it is actually the case or not. And guess what: if Pat would be a bachelor, then Pat would be a bachelor, even if Pat is not actually a bachelor.

So, if $P$ is false, it better be the case that $false \rightarrow false = true$, for otherwise $P \rightarrow P$ would be false, which is just weird.

Of course, we also want $true \rightarrow true = true$ by this same argument, for otherwise again we would have $P \rightarrow P$ being false.

As far as $false \rightarrow true$ is concerned: given that we have that $true \rightarrow true =true$, $false \rightarrow false$, and ( I think you would certainly agree) $true \rightarrow false = false$, we better set $false \rightarrow true =true$, because otherwise the $\rightarrow$ would become commutative, i.e. We would have that $P \rightarrow Q$ is equivalent to $Q \rightarrow P$ ... which is highly undesired, since conditionals have a 'direction' to them that cannot be reversed automatically. Indeed, while I think you would agree with the truth of:

'if Pat is a bachelor, then Pat is male'

I doubt you would agree with:

'if Pat is male, then Pat is a bachelor'

EDIT

Re-reading your question, and considering some of the ensuing discussions and comments, I wonder if the following might help:

Suppose that we know some statement $P$ is false, i.e. We know that:

$1. \neg P \quad Given$

Then we can show that $P$ implies any $Q$, given the standard definition of logical implication:

$2. P \quad Assumption$

$3. P \lor Q \quad \lor \ Intro \ 2$

$4. Q \quad Disjunctive \ Syllogism \ 1,3$

And, using our typical rule for $\rightarrow \ Intro$, we can then also get:

$5. P \rightarrow Q \quad \rightarrow \ Intro \ 2-5$

And this of course works whether $Q$ is true or false.

Bram28
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  • "Clearly we want $P\rightarrow P$ to be true" I'm not sure of this? Why would we necessarily want it to be true? Why couldn't $P\rightarrow P$ be false when $P$ is false? – user539262 Mar 12 '18 at 03:02
  • "We would have that P→Q is equivalent to Q→P ... which is highly undesired." Why undesired? – user539262 Mar 12 '18 at 03:05
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    @user539262 The $\rightarrow$ is supposed to capture conditionals, i.e. 'If ... then ...' statements. Frankly, I can't think of a more clear example of an 'if ... then ..' statement being true than if the 'if' and 'then' parts are the very same. – Bram28 Mar 12 '18 at 03:07
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    @user539262 Also, conditionals have an inherent 'direction' about them. It makes sense that 'if Pat is a bachelor, then Pat is male', but not that 'If Pat is male, then Pat is a bachelor' – Bram28 Mar 12 '18 at 03:09
  • I just don't see why the false -> false = true thing must be the case. It's like saying "P and P is true because we have the same P twice" but if P is false (false and false) is still false. Why must "false -> false = true" hold? false -> false = false seems a defensible choice especially if we use different variables P and Q. – user539262 Mar 12 '18 at 03:17
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    @user539262 True. In fact, here is an example: 'If Pat lives in Paris, then Pat lives in Germany'. Assuming Pat lives in London, we have a false anteceddent, a false consequent, and (I agree) a false conditional. hey, no one said that the $\rightarrow$ perfectly captures the English conditional, and in fact examples like this show that the eNglish conditional simply isn't truth-functional. But, for formal logic purposes, we want our operator to be truth-functional. And, out of all the options, the one we chose works the best, for reasons I indicated, and further reasons yet. – Bram28 Mar 12 '18 at 03:33
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I don't have a lot to say on this, but I used to be very annoyed by the concept of vacuous truth and only these two observartions soothed my ailment.

1.) One clearly wants $A\land B\implies A$, and this wouldn't be true without $false \implies true$ being true.

2.) $false \implies true $ is exactly the same statement as "the empty set is contained in every other set" which to me is intuitive.

JKEG
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  • Why would $A \land B \implies A$ be something one clearly wants? – user539262 Mar 12 '18 at 03:04
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    @user539262 Well, if A is true, and B is also true, then A is true. – JKEG Mar 12 '18 at 03:08
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    I don't understand how the two statements are exactly the same in 2. Could you please explain? – Eric Duminil Mar 12 '18 at 08:17
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    @Eric I think the argument goes something like this: let $X$ and $Y$ be sets, and $z$ be an instance of the type of thing contained in these sets. Then $X\subset Y$ is defined as ($\forall z: z\in X\implies z\in Y$). If $X\subset Y$ is to be true when $X = \emptyset$, then because $z\in\emptyset$ is always false, we need $\text{false}\implies z\in Y$ to be true for all $z$'s, both those in $Y$ and those not in $Y$, which in turn requires both $\text{false}\implies\text{true}$ and $\text{false}\implies\text{false}$ to be true. – David Z Mar 13 '18 at 01:45
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Right now, we like making statements like

If $a$ and $b$ are both even, then $a+b$ is even.

Or, we could write this mathematically as $$ a \equiv b \equiv 0 \pmod{2} \implies a+b \equiv 0 \pmod{2}. $$ If we were to redefine $\implies$ to disallow vacuously true implications, this would no longer be a true statement, because of cases such as "$1$ and $3$ are both odd, but $1+3$ is even". But we still want to talk about such statements, so we'd probably just end up saying longer sentences such as

The statement "$a$ and $b$ are both even" implies in the sense that allows vacuously true implications the statement "$a+b$ is even".

This is a very useful relationship to talk about, so you'd just condemn us to longer phrasing for no good reason. Meanwhile, we already have conjunctions such as "and" and "iff" to describe cases where both statements must be true, or where both statements must have the same truth value.

Mathematical terminology is driven by utility. If it were useful to have "if" mean the thing you want it to mean, we'd do it. But it's useful to have if-then statements be true in vacuous cases, just like it's useful to (to give some other examples) have $0$ be an even number and have $1$ be neither prime nor composite. So we go with the meaning that makes our lives easier.

Misha Lavrov
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The assumption here is that we are working with a bivalent logic in which all statements are true or false. They are never both true and false, nor are they neither true or false.

Now if this is your universe, let's see...

  1. Everything is either true or false.

  2. Now assume false (is true).

  3. We already know that true is true, so if false is ALSO true, ...

  4. Then everything is true!

This answer doesn't say how you can derive any true statement from a false statement. It just says that if all false things were true, then all things are true (assuming you want everything to be false or true).

Ray Toal
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It's well known that vacuous truths are a concept, i.e. an implication being true even if the premise is false.

What would be the problem with simply redefining this to be evaluated to false?

One answer to the question is that we would no longer be able to know that P → Q from the fact that we know ¬P ∧ Q. We would also not be able to know that P → Q from ¬(P ∧ ¬Q). And we would no longer know from ¬Q → ¬P that P → Q.

So, for example if I knew that either a given number was not divisible by 3, or that it was even, I would not be able to deduce that if the number was divisible by 3 it would be even.

Similarly, if I knew that a given number was not both divisible by 3 and not even, I would similarly not be able to infer that if the number was divisible by 3 it would be even.

Lastly, if I knew that if a given number was not even it was also not divisible by 3, then I would not be able to know that if it was divisible by three, it was even.

In each of the cases above, if If P, Q was false when P was false, the conclusion could be false because the number in question might be divisible by 3.

Note

If we made P → Q true when P was false then → would just be ∧. We can't replace material implication (MI) with ∧ because, MI is what it is by definition. I therefore take this question to mean what would happen if we took 'If P, Q' to be false when P is false.

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