Understanding "If... then..." Logic: Examples and Explanations

[sponsoredwalk]

"If... then..."

p: I eat breakfast. q: I do not eat lunch. T: true, F: false.

If I eat breakfast then I do not eat lunch. p → q is T
If I eat breakfast then I eat lunch. p → q is F
If I do not eat breakfast then I do not eat lunch. p → q is T
If I do not eat breakfast then I eat lunch. p → q is T

The third one, just because I do not eat breakfast it does not mean I do not eat lunch,
I may be eating lunch! Me not eating lunch does not necessitate the fact that I do not
eat lunch.

The fourth one, if I do not eat breakfast, it does not mean I'm eating lunch! I may
not be eating lunch!

How do I get my head around this?

Another example:

x: It will rain. y: The grass will grow.

If it rains then the grass will grow. x → y is T
If it rains then the grass will not grow. x → y is F.
If it does not rain then the grass will grow. x → y is T
If it does not rain then the grass will not grow. x → y is F.

I made this example up, it may be flawed. The third example to me seems right because
the grass will grow whether or not the rain comes. This also explains why I made the
fourth one false, the grass will grow wither or not the rain comes. After an extended
period of time this may be false though, eventually rain will be required. I don't know...

A final example:

δ: The sun is shining. ε: Pigs eat turnip.

If the sun is shining then Pigs eat turnip. δ → ε is T

Before I even go on, this may or may not be true, there is no rule set in stone on this
matter, does this even apply to here? If not, can it be made to fit the mold?

If the sun is shining then Pigs will not eat turnip. δ → ε is T?

Again it's futile, I think. Any ideas?

 

[D H]

If I do not eat breakfast then I do not eat lunch. p → q is T
If I do not eat breakfast then I eat lunch. p → q is T

The third one, just because I do not eat breakfast it does not mean I do not eat lunch,
I may be eating lunch! Me not eating lunch does not necessitate the fact that I do not
eat lunch.

The fourth one, if I do not eat breakfast, it does not mean I'm eating lunch! I may
not be eating lunch!

How do I get my head around this?

One way to get your head around that p→q is always true when p is false by definition. Don't try to wrap your head around it. Logicians define material implication this way for one simple reason: it works. This definition turns out to be very useful.

 

[micromass]

You can see this in the following way:

The implication "If 4 divides x" then "2 divides x" is undoubtely true.

But we can choose x such that the first part of the implication is false and the second is true. Take for example x=2, then 4 does not divide 2, but 2 divides 2. Thus we have a situation F-->T. Yet the implication is still true. The only way to make an implication false, is to make the first part true and the second part false. There is no other way...

 

[Hurkyl]

How do I get my head around this?

I'm confused by your calculations. Your first example seems to assume as a hypothesis that P and Q are both true, although you didn't state it explicitly. So considering a hypothetical where P is false is completely vacuous.

 

[sponsoredwalk]

I'm confused by your calculations. Your first example seems to assume as a hypothesis that P and Q are both true, although you didn't state it explicitly. So considering a hypothetical where P is false is completely vacuous.

I'm just going off the example on page 6 of Discrete Mathematics for New Technology by
Garnier/Taylor for the first example. It's the only example they give & then they show a
truth table.

http://books.google.ie/books?id=WnkZSSc4IkoC&printsec=frontcover#v=onepage&q&f=false

Page 6 of this different book by them, but same content , gives the example.

 

[Hurkyl]

Suppose I told you that P is true, and P->Q is true. Using the truth table, what information can you gain about Q?

Suppose I told you that P is false, and P->Q is true. Using the truth table, what information can you gain about Q?

 

[sponsoredwalk]

Suppose I told you that P is true, and P->Q is true. Using the truth table, what information can you gain about Q?

Suppose I told you that P is false, and P->Q is true. Using the truth table, what information can you gain about Q?

If P is true in this example and P → Q is true then Q is true.

If P is true in this example and P → Q is true but Q is false
then P → Q is not true, i.e. we have a contradiction,
that or else P was not true to begin with.

If P is false and P → Q is true then Q is true, I wrote this as:
If I do not eat breakfast then I do not eat lunch. p → q is T.
It is true because I don't have to eat lunch when I don't eat
breakfast. But if I do eat lunch when I do not eat breakfast
then this is false. Jeesh idk, nobody eats lunch all day but it
could conceivably happen!

You see for a conjunction, a disjunction & an exclusive disjunction I can reconstruct
the truth table simply and logically and judge when a sentence fits a particular answer,
but here I can't even form the truth table, there's just so much room for freedom.
The rain example is a good one of what I mean. I thought the breakfast example I gave
was fine, but you seem to have a problem with it. Either P is true or P is false, same with Q,
I thought the 4 parts of the first example I gave covered all of these contingencies?
When I wrote I eat breakfast in words that means P is true, when I wrote I do not eat
breakfast that represents P as being false, similarly with Q I just negated the language
to represent Q as being false in the second and fourth examples.

 

[D H]

If P is true in this example and P → Q is true then Q is true.

Correct.

If P is false and P → Q is true then Q is true

Incorrect. Knowing that P is false and that P → Q is true tells you nothing about Q. Similar circumstances arise with other By way of analogy, suppose you know P is true and P ∨ Q is true. What does this tell you about Q? (Answer: Nothing.)

Let's flip material implication around. Suppose you know P → Q is true and you know Q is false. What does this tell you about P?

 

[sponsoredwalk]

Let's flip material implication around. Suppose you know P → Q is true and you know Q is false. What does this tell you about P?

P: I eat breakfast. Q: I do not eat lunch.

Earlier I wrote

If I do not eat breakfast then I eat lunch. P → Q is T

as representing this situation, and I do understand this
as being the case for these two statements. It makes
sense, if Q is false, i.e. I do eat lunch, then I cannot
be eating breakfast as well so P would have to be
false. So, if P → Q is T it would mean that P
would also be false as if it were true Q could not
be false. But I am mentally relying on the situation
to decipher this situation, the start of the book says
we're trying to formulate a set of rules independent of
the content of the statements. If you look at the other
examples I gave it is the content of those statements
that has me so confused.

If I totally abstract and use arbitrary statements P & Q.

If P then Q,

If P is true and Q is true then P → Q is true.
If P is true and Q is false then P → Q is false,
here P was supposed to imply Q was true but because
Q was not true the implication is a false claim.
If P is false and Q is true then P → Q is true,
My thinking is that the statement P → Q relies on
P being true for P → Q to mean anything and we
can't prove anything about it, it seems a safe bet
but I could be wrong.
If P is false and Q is false then P → Q is true,
Same reasoning, it doesn't seem too iron clad,
let me know what you think.

Edit: Just looking at the last example here,
my reasoning may be wrong. If P is false
and Q is false then P → Q is true because
P being false implied Q was false so P → Q
is true. Is that how you think of it, seems
plausible but probably wrong?

I actually didn't think so abstractly because the
book was relying on specific examples and
specific examples were helping with conjunctions
and disjunctions. My abstracted view does
not answer my example:

If it rains then the grass will grow. x → y is T
If it rains then the grass will not grow. x → y is F.
If it does not rain then the grass will grow. x → y is T
If it does not rain then the grass will not grow. x → y is F

as the content of the statements is complex,
in the last example the grass will grow without
rain, also in the pigs and turnip example,
I'm pretty sure pigs will not stop eating turnip
just because it is not a sunny day

 

[D H]

You are hung up in trying to define implication in a way that makes sense to you. Don't do that. The given definition is just that: A definition, and it is very useful one. Your definition is much less useful.

You did not answer my question, at the end of post #8. Try doing that. You know P implies Q and you know that Q is false. What does this tell you about P? Don't reason it out. Just look at the truth table.

 

[sponsoredwalk]

You did not answer my question, at the end of post #8. Try doing that. You know P implies Q and you know that Q is false. What does this tell you about P? Don't reason it out. Just look at the truth table.

I did answer you, it implies P is true. I have actually given three separate reasons in that
post why I think this is the case, one in the specific example at the start and then two
conflicting reasons in the abstract case in which I'm trying to determine which one is
applicable.

You are hung up in trying to define implication in a way that makes sense to you. Don't do that. The given definition is just that: A definition, and it is very useful one. Your definition is much less useful.

No offense but you're literally telling me to accept on blind faith something just because
it works & that there is no reason why. This isn't quantum mechanics, and even still
QM has a good enough explanation in linear algebra. The logic here isn't inaccesible,
I mean I'm almost there & you've helped me by looking at it from different angles but
there are a few ways to think about these things. Seriously, another reason I'm not
memorizing this is because I'll just forget it anyway. All I'm trying to do is to find the
thought process that works all the time in this situation. As you can clearly see I am
confused by different examples - this way or that way. I doubt it'll take a read of
Bertrand Russell to figure out this particular example but I'm sure he explains it
in there
My last post has my worries in it, if you think you understand where I am confused
please let me know, it'd be really really helpful .

 

[D H]

I did answer you, it implies P is true.

Try again! Look at the truth table.

$$\begin{array}{ccc} P & Q & P\to Q \\ T & T & T \\ T & F & F \\ F & T & T \\ F & F & T \end{array}$$

 

[sponsoredwalk]

Try again! Look at the truth table.

$$\begin{array}{ccc} P & Q & P\to Q \\ T & T & T \\ T & F & F \\ F & T & T \\ F & F & T \end{array}$$

Lol, no my mistake! I meant that P is false, sorry! I wrote that in haste but if you read my
last big response I wrote that it implies P is false, sorry! If you reread it you'll see
Read my explanation of the abstract P & Q situation & you'll see I gave two reasons
as to why this is the case, which one is correct? I was right though!

 

[D H]

No offense but you're literally telling me to accept on blind faith something just because it works & that there is no reason why.

No, I'm telling you that that is the definition. Don't argue with it. It is axiomatic. The only reason to argue with axioms is when they lead to inconsistencies. There are no inconsistencies here. It is just a truth table.

I take it you have no problem with the first part of the truth table. The issue is how to complete it.

$$\begin{array}{ccc} P & Q & P\to Q \\ T & T & T \\ T & F & F \\ F & T & ? \\ F & F & ? \end{array}$$

There are four possibilities to fill in those missing values: FF, FT, TF, and TT. Let's try each one.


$$\begin{array}{ccc} P & Q & P\to Q \\ T & T & T \\ T & F & F \\ F & T & F \\ F & F & F \end{array}$$

This is exactly the same as the truth table for conjunction. Certainly we want if P then Q to mean something different from P or Q.


$$\begin{array}{ccc} P & Q & P\to Q \\ T & T & T \\ T & F & F \\ F & T & F \\ F & F & T \end{array}$$

This appears to be how you want implication to be defined. Suppose it doesn't rain and the grass grows. This definition means "If it rains the grass will grow" is false. What if I water the lawn? Note also that this is the same truth table as that for not equals or exclusive or. Once again there is no reason to give a new name to something that already has a perfectly good name. (Doubly so in this case, since this truth table already has two names!)


$$\begin{array}{ccc} P & Q & P\to Q \\ T & T & T \\ T & F & F \\ F & T & T \\ F & F & F \end{array}$$

This truth table also already has a name: It is Q regardless of P, or for short, just Q.


$$\begin{array}{ccc} P & Q & P\to Q \\ T & T & T \\ T & F & F \\ F & T & T \\ F & F & T \end{array}$$

This is the only one left.

 

[sponsoredwalk]

Yeah I see what you mean by just giving a new name to the only possible table left
but every situation of every truth table has a logical explanation, the only
situation in which I can't be sure I can logically determine the truth value is in the
two situations you've given question marks to. I do have reasons here as to why I've
gotten the correct answer, but those reasons aren't working in the other two examples
I gave in my OP.

My reasoning is here:

If I totally abstract and use arbitrary statements P & Q.

If P then Q,

If P is true and Q is true then P → Q is true.
If P is true and Q is false then P → Q is false,
here P was supposed to imply Q was true but because
Q was not true the implication is a false claim.
If P is false and Q is true then P → Q is true,
My thinking is that the statement P → Q relies on
P being true for P → Q to mean anything and we
can't prove anything about it, it seems a safe bet
but I could be wrong.
If P is false and Q is false then P → Q is true,
Same reasoning, it doesn't seem too iron clad,
let me know what you think.

Edit: Just looking at the last example here,
my reasoning may be wrong. If P is false
and Q is false then P → Q is true because
P being false implied Q was false so P → Q
is true. Is that how you think of it, seems
plausible but probably wrong?

-----------------------------------------------

See, there are two ways I'm thinking about it, here is more justification for my reasoning,
I did a search of old PF posts and got an interesting response from HallsOfIvy in an old
locked post:

Think of it as "innocent until proven guilty". The statement says "If Maria learns discrete mathematics then she will get a good paying job" (obviously this is a discrete mathematics course!). It says nothing about what will happen if she does not learn discrete mathematics and so the situation in which Maria does not learn discrete mathematics tells us nothing about the truth or falseness of the statement. Since it hasn't be "proven" false, call it true.
https://www.physicsforums.com/showthread.php?t=175051

This accords with my explanation of why the third situation has the value it has, but
you see that when I try to understand the fourth situation I have two answers.
Is it because P is false that we can't say anything at all about Q, ergo staying on
the safe side by just setting it true because we can't say anything about it,
as Halls says, or is the fact that "If P is false then Q is false" the reason why it
has it's value?

Still though, all of this does not help me with my second and third example in my OP
& that's worrying me. Also, the example in the post I quoted brings up questions but
I'll hold off on them. Who would have thought this was so hard to wrap your head around!

 

[D H]

All I can say is you need to loosen up. It is something you need to do if you want to study advanced mathematics. Just go with the flow. Definitions are what they are, definitions. Nothing else. If the definition is useful it sticks around.

So what good is this definition? Google the terms "modus ponens" and "modus tolens" as a start.