Hi guys, if there's a definition of specific statement that declares for satisfying that "specific statement"
if A->B
(A,B is an events.)

my question if A doesn't occur, that means there's a possibility that A satisfy that "specific statement", also there's a possibility A will not satisfy that "specific statement", so we don't have definitely data what's going if A doesn't occur, so we can't claim that if A doesn't occur, then definitely the specific statement satisfied !

 

In propositional logic, the conditional statement "if A then B" is vacuously true when A is false. In other words, for any proposition B whatsoever, regardless of whether it is true or false, it is always the case that A ⇒ B is true when A is false.

For example, "if 5 > 10, then 5 > 100" is a true statement of logic. But so is: "if 5 > 10, then Chuck Norris was the first man on the Moon".

The reason we say that these statements are (vacuously) true is because there is no way to find a counterexample. To prove that a statement is false, all we need is a single counterexample. In the case of A ⇒ B, this means finding a true A where B is false. But if A is always false, then it is impossible to find a true A where B is false. Since no counterexample exists, we call the whole thing (vacuously) true. The "vacuous" qualifier indicates that the truth of the statement carries no information, i.e., it's "true" only in form, not in meaning.

Incidentally, this is why we need to be careful about our logical arguments. You've surely seen "proofs" that 1 = 2 and such. A proof is essentially a sequence of "if, then" statements. Invariably, one of the steps in the given proof involves division by zero or some other proposition that can never be true. So, the "truth" of the 1 = 2 conclusion is vacuous, and the "proof" is meaningless.

 

In propositional logic, the conditional statement "if A then B" is vacuously true when A is false. In other words, for any proposition B whatsoever, regardless of whether it is true or false, it is always the case that A ⇒ B is true when A is false.

For example, "if 5 > 10, then 5 > 100" is a true statement of logic. But so is: "if 5 > 10, then Chuck Norris was the first man on the Moon".

The reason we say that these statements are (vacuously) true is because there is no way to find a counterexample. To prove that a statement is false, all we need is a single counterexample. In the case of A ⇒ B, this means finding a true A where B is false. But if A is always false, then it is impossible to find a true A where B is false. Since no counterexample exists, we call the whole thing (vacuously) true. The "vacuous" qualifier indicates that the truth of the statement carries no information, i.e., it's "true" only in form, not in meaning.

Incidentally, this is why we need to be careful about our logical arguments. You've surely seen "proofs" that 1 = 2 and such. A proof is essentially a sequence of "if, then" statements. Invariably, one of the steps in the given proof involves division by zero or some other proposition that can never be true. So, the "truth" of the 1 = 2 conclusion is vacuous, and the "proof" is meaningless.

Sir .. Sorry but didnt get ur point .. How if 5>10 then 5>100? Dont make sense
Secondly how seriously if A->B then its the se to say if A false then the statement satisfied..?

Thanks

 

In propositional logic, the conditional statement "if A then B" is vacuously true when A is false. In other words, for any proposition B whatsoever, regardless of whether it is true or false, it is always the case that A ⇒ B is true when A is false.

For example, "if 5 > 10, then 5 > 100" is a true statement of logic. But so is: "if 5 > 10, then Chuck Norris was the first man on the Moon".

The reason we say that these statements are (vacuously) true is because there is no way to find a counterexample. To prove that a statement is false, all we need is a single counterexample. In the case of A ⇒ B, this means finding a true A where B is false. But if A is always false, then it is impossible to find a true A where B is false. Since no counterexample exists, we call the whole thing (vacuously) true. The "vacuous" qualifier indicates that the truth of the statement carries no information, i.e., it's "true" only in form, not in meaning.

Incidentally, this is why we need to be careful about our logical arguments. You've surely seen "proofs" that 1 = 2 and such. A proof is essentially a sequence of "if, then" statements. Invariably, one of the steps in the given proof involves division by zero or some other proposition that can never be true. So, the "truth" of the 1 = 2 conclusion is vacuous, and the "proof" is meaningless.

How actually if A false then the statement A=>B is always true? Incredible

 

Sir .. Sorry but didnt get ur point .. How if 5>10 then 5>100? Dont make sense
Secondly how seriously if A->B then its the se to say if A false then the statement satisfied..?

What do we mean by A ⇒ B? Intuitively, we're saying that whenever A is true, it must be the case the B is true: that is, A implies B.

Now, to prove that A ⇒ B is false, it is necessary and sufficient to provide a single counterexample. For example, suppose A stand for the proposition "x > 10", and B for the proposition "x > 100" over the domain of integers. Then, A ⇒ B is the statement

if x > 10, then x > 100​

To show that this is false, we need a single counterexample. Let x = 11 and we're done, as A is true (11 > 10) yet B is false (11 > 100). We've found a counterexample and proved that the statement A ⇒ B is false.

Now, let's assume that the statement "if 5 > 10, then 5 > 100" is false. Since it is a false statement, there must exist a counterexample showing that A is true while B is false. However, since A = "5 > 10" is never true, there is no way to find an example of A being true and B being false. And since it is impossible to find a counterexample to show that A ⇒ B is false, it must be true by definition -- in propositional logic, a statement is either true or it is false, never both or neither.

Notice that our proof that the statement A ⇒ B is true relied only on the fact that A (5 > 10) is always false; the proof was independent of the truth of B. This tells us that, as long as A is always false, it doesn't matter whether B is true or false. Hence, both of these statements are true:

if 5 > 10, then 5 > 100
if 5 > 10, then 5 < 100​

Note that this logic is not abstract or purely mathematical. Your computer uses precisely the same logic, and so can be used to easily demonstrate that A ⇒ B is true when A is false. First, note that

A ⇒ B ≡ ~A ∨ B​

In English, the statement "if A then B" is logically equivalent to (has the same truth table as) the statement "not-A or B". Running the following code:

int A = 0; // A is false
int B = 1; // B is true

if ( !A || B )
  puts("true statement!");
else
  puts("false statement!");

B = 0; // now B is false

if ( !A || B )
  puts("true statement!");
else
  puts("false statement!");

You will see "true statement!" printed twice, showing that -- regardless of the value of B -- the statement A ⇒ B is always true whenever A is false.

 

What do we mean by A ⇒ B? Intuitively, we're saying that whenever A is true, it must be the case the B is true: that is, A implies B.

Now, to prove that A ⇒ B is false, it is necessary and sufficient to provide a single counterexample. For example, suppose A stand for the proposition "x > 10", and B for the proposition "x > 100" over the domain of integers. Then, A ⇒ B is the statement

if x > 10, then x > 100​

To show that this is false, we need a single counterexample. Let x = 11 and we're done, as A is true (11 > 10) yet B is false (11 > 100). We've found a counterexample and proved that the statement A ⇒ B is false.

Now, let's assume that the statement "if 5 > 10, then 5 > 100" is false. Since it is a false statement, there must exist a counterexample showing that A is true while B is false. However, since A = "5 > 10" is never true, there is no way to find an example of A being true and B being false. And since it is impossible to find a counterexample to show that A ⇒ B is false, it must be true by definition -- in propositional logic, a statement is either true or it is false, never both or neither.

Notice that our proof that the statement A ⇒ B is true relied only on the fact that A (5 > 10) is always false; the proof was independent of the truth of B. This tells us that, as long as A is always false, it doesn't matter whether B is true or false. Hence, both of these statements are true:

if 5 > 10, then 5 > 100
if 5 > 10, then 5 < 100​

Note that this logic is not abstract or purely mathematical. Your computer uses precisely the same logic, and so can be used to easily demonstrate that A ⇒ B is true when A is false. First, note that

A ⇒ B ≡ ~A ∨ B​

In English, the statement "if A then B" is logically equivalent to (has the same truth table as) the statement "not-A or B". Running the following code:

int A = 0; // A is false
B = 1; // B is true

if ( !A || B )
  puts("true statement!");
else
  puts("false statement!");

B = 0; // now B is false

if ( !A || B )
  puts("true statement!");
else
  puts("false statement!");

You will see "true statement!" printed twice, showing that -- regardless of the value of B -- the statement A ⇒ B is always true whenever A is false.

I appreciate your help very much!!
So in abbreviation you sa whenever A is false we say it's true in form but in meaning..aka there's like a realization in math whenever something is logically vacuous then we pick up the true condition..

 

I appreciate your help very much!!
So in abbreviation you sa whenever A is false we say it's true in form but in meaning..aka there's like a realization in math whenever something is logically vacuous then we pick up the true condition..

Right, it's true in form only (because it can never be false). A related concept is tautology, statements that are always true by virtue of their form. For example, A = A is always true but meaningless -- it conveys no information. Likewise, A or not-A is always true by form. On the other side are contradictions, statements that are always false by form: A and not-A.

Generally speaking, we only care about propositional statements that are semantically true or false, i.e., the type of statements that can be true for some interpretations and false for others, as these convey meaning about the particular model we've chosen. In contrast, tautologies, contradictions, and vacuously true statements are always true or always false, regardless of the chosen semantics/model/interpretation. That makes them rather useless for conveying information.

 

Right, it's true in form only (because it can never be false). A related concept is tautology, statements that are always true by virtue of their form. For example, A = A is always true but meaningless -- it conveys no information. Likewise, A or not-A is always true by form. On the other side are contradictions, statements that are always false by form: A and not-A.

Generally speaking, we only care about propositional statements that are semantically true or false, i.e., the type of statements that can be true for some interpretations and false for others, as these convey meaning about the particular model we've chosen. In contrast, tautologies, contradictions, and vacuously true statements are always true or always false, regardless of the chosen semantics/model/interpretation. That makes them rather useless for conveying information.

So to sum up!
given A--->B , if A occurred then B will be occurred. if A not occurred then it also satisfying A will lead to B because "false" is satisfying everything ....

 

So to sum up!
given A--->B , if A occurred then B will be occurred. if A not occurred then it also satisfying A will lead to B because "false" is satisfying everything ....

Another way of looking at the truth table for the implication operator is that if we have a (valid) statement like:

IF (the animal is a bird) THEN (it reproduces by laying eggs)

We know that if the animal is a bird, then we also know that it reproduces by laying eggs. If you can find an animal that is a bird and it reproduces by some means other than laying eggs, then this statement is NOT a valid implication statement.

But if the animal is NOT a bird, then we know nothing about whether it reproduces by laying eggs or not and either case is completely consistent with the implication statement because the implication isn't making any claims about animals that are not birds.

 

Another way of looking at the truth table for the implication operator is that if we have a (valid) statement like:

IF (the animal is a bird) THEN (it reproduces by laying eggs)

We know that if the animal is a bird, then we also know that it reproduces by laying eggs. If you can find an animal that is a bird and it reproduces by some means other than laying eggs, then this statement is NOT a valid implication statement.

But if the animal is NOT a bird, then we know nothing about whether it reproduces by laying eggs or not and either case is completely consistent with the implication statement because the implication isn't making any claims about animals that are not birds.

Yeah ..but what it was confusing me in our world if I told you this : all the red horses have red hat
what you conclude from this ? "assume I'm talking to you face to face and told you this"
then you say the following things:
red horse has red hat
there's a horse has red hat
there's a horse hasn't red hat
if haven't red hat as there's no red horse.
those all the consecutive conclusion of our life logic and totally I accept them, but in math the option that we concluded "there's a horse hasn't red hat" isn't into consideration .. that's why I'm confused about and I think the answer because the world of math is not always like our world..

 

Yeah ..but what it was confusing me in our world if I told you this : all the red horses have red hat
what you conclude from this ? "assume I'm talking to you face to face and told you this"
then you say the following things:
red horse has red hat
there's a horse has red hat
there's a horse hasn't red hat
if haven't red hat as there's no red horse.
those all the consecutive conclusion of our life logic and totally I accept them, but in math the option that we concluded "there's a horse hasn't red hat" isn't into consideration .. that's why I'm confused about and I think the answer because the world of math is not always like our world..

Let P(x) be the predicate function "is a red horse", Q(x) "has a red hat", and let x range over the set of horses. Then, the English statement "all red horses have red hats" is equivalent to the logical statement (∀x)[P(x) → Q(x)]. In other words, if x has property P ("is a red horse"), then x has property Q ("has a red hat").

Now, let's suppose that ∃xP(x) is true, i.e., "there is a red horse". Then, by the implication above, ∃xQ(x) must be true -- "there is a horse with a red hat" -- and its negation ~∃xQ(x) ≡ ∀x~Q(x) -- "all horses do not have red hats" must be false. These make intuitive sense. Your confusion, I think, lies with the case when ∃xP(x) is not true. In that case, P(x) → Q(x) is vacuously true, and we can have both ∃xQ(x) and ~∃xQ(x). You seem to think that this only a math thing, not applicable to the real world. But consider what these statements mean, given that P(x) → Q(x) and ~P(x) are true:

∃xQ(x): "there is a horse with a red hat"
~∃xQ(x): "all horses do not have a red hat"

Clearly, both of these statements cannot be true at the same time, but that doesn't matter. What the vacuous truth is telling us is that, from the information given (P → Q with ~P), we can't say whether there is or is not a horse with a red hat. Since those are the only two options, and both are possible, then the implication P → Q must be true. If it were instead false, we would be saying that one or both of the options are impossible, but clearly both are possible!

To make this more concrete, suppose that the original had statement had been a biconditional statement: "a horse is red if and only if it has a red hat", i.e., ∀x[P(x) Q(x)]. In this case, P implies Q and Q implies P. Now, if we assert ~P (there is no red horse), then the only way that P Q can be true is if Q is also false (there is no horse with a red hat). That Q implies P gives us enough information to know that if Q is true, then P must also be true; and since P is false, it follows that Q must also be false. We don't have that extra information with P → Q, and so we say that it is vacuously true when P is false.

By the way, your final statement "if it does not have a red hat, then it is not a red horse" translates to ~Q(x) → ~P(x), which is the contrapositive of P(x) → Q(x), and precisely equivalent to it. It adds no new information.

 

When will all of this stuff get moved to "Homework" based on the predicate so pellucidly described by WBahn here: https://forum.allaboutcircuits.com/threads/arbitrary-sequence-counter-up-down.155428/#post-1339991 .

 

When will all of this stuff get moved to "Homework" based on the predicate so pellucidly described by WBahn here: https://forum.allaboutcircuits.com/threads/arbitrary-sequence-counter-up-down.155428/#post-1339991 .

Unlike the other active thread (current dependent resistor challenge) in the Math subforum, this thread is actually about mathematics. I don't see why it should be moved to Homework.

 

Yeah ..but what it was confusing me in our world if I told you this : all the red horses have red hat
what you conclude from this ? "assume I'm talking to you face to face and told you this"
then you say the following things:
red horse has red hat
there's a horse has red hat
there's a horse hasn't red hat
if haven't red hat as there's no red horse.
those all the consecutive conclusion of our life logic and totally I accept them, but in math the option that we concluded "there's a horse hasn't red hat" isn't into consideration .. that's why I'm confused about and I think the answer because the world of math is not always like our world..

Many of your conclusions are unreasonable and don't follow from your claim.

The only thing is that being claimed is that each and every red horse has a red hat. Nothing more. It says nothing about what kind of hat horses that aren't red might or might not have and it says nothing about whether there are any red horses at all.

Now, one of your claims IS valid, namely that if a particular horse does not have a red horse then that horse is not red.

This is known as the contrapositive.

IF (A) IMPLIES (B) is true, then IF (NOT B) IMPLIES (NOT A) is also true.

Just as if I said, "All purple horses that are telepathic can fly." I'm not claiming that there are actually any horses than can fly. I'm also not claiming that any horse that can fly is a telepathic purple horse. I'm merely asserting that if you can find a telepathic purple horse, that that horse can fly.

EDIT: Fix typos.

 

Many of your conclusions are unreasonable and don't follow from your claim.

The only thing is that being claimed is that each and every red horse as a red hat. Nothing more. It says nothing about what kind of had horses that aren't red might or might not have and it says nothing about whether there are any red horses at all.

Now, one of your claims IS valid, namely that if a particular horse does not have a red horse then that horse is not red.

This is known as the contrapositive.

IF (A) IMPLIES (B) is true, then IF (NOT B) IMPLIES (NOT A) is also true.

Just as if I said, "All purple horses that are telepathic can fly." I'm not claiming that there are actually any horses than can fly. I'm also not claiming that any horse that can fly is a telepathic purple horse. I'm merely asserting that if you can find a telepathic purple horse, that that horse can fly.

My probelm was that i convert everything to real life analogous which that's not good approach.


Thank u all for ur help.

 

My probelm was that i convert everything to real life analogous which that's not good approach.


Thank u all for ur help.

It's not that it's not a useful approach, but you definitely have to be careful when doing so because humans tend to infer information that is not present and make assumptions based on what seems reasonable.

Thus, if someone tells me that all brown cars in the parking lot have their lights on, I tend to assume that the person wouldn't be saying this unless there were at least two brown cars in the parking lot that have their lights on and also that only brown cars have their lights on. But this is based on what I infer from the mere fact that a claim is being made in the first place and not on what the claim actually is. I unconsciously assume that the person intends to tell me enough information so that I can determine the entire set of cars in the parking lot that have their lights on and that there's no point in doing so unless there are, indeed, cars in the parking lot that have their lights on.

 

In propositional logic, the conditional statement "if A then B" is vacuously true when A is false. In other words, for any proposition B whatsoever, regardless of whether it is true or false, it is always the case that A ⇒ B is true when A is false.

For example, "if 5 > 10, then 5 > 100" is a true statement of logic. But so is: "if 5 > 10, then Chuck Norris was the first man on the Moon".

The reason we say that these statements are (vacuously) true is because there is no way to find a counterexample. To prove that a statement is false, all we need is a single counterexample. In the case of A ⇒ B, this means finding a true A where B is false. But if A is always false, then it is impossible to find a true A where B is false. Since no counterexample exists, we call the whole thing (vacuously) true. The "vacuous" qualifier indicates that the truth of the statement carries no information, i.e., it's "true" only in form, not in meaning.

Incidentally, this is why we need to be careful about our logical arguments. You've surely seen "proofs" that 1 = 2 and such. A proof is essentially a sequence of "if, then" statements. Invariably, one of the steps in the given proof involves division by zero or some other proposition that can never be true. So, the "truth" of the 1 = 2 conclusion is vacuous, and the "proof" is meaningless.

I know this is old but I enjoyed reading it. It's a good reminder of why in my day a philosophy class in logic was a prerequisite for discrete math.

Bogosort,

The original question was: If A->B. Is this a propositional logic statement albeit incomplete? As I remember propositional logic doesn't use quantifiers or relationships. By the use of the ">", a predicate logic or relation symbol doesn't this define the question as one of predicate logic?

It's been a long time and I don't know.

 

I know this is old but I enjoyed reading it. It's a good reminder of why in my day a philosophy class in logic was a prerequisite for discrete math.

Bogosort,

The original question was: If A->B. Is this a propositional logic statement albeit incomplete? As I remember propositional logic doesn't use quantifiers or relationships. By the use of the ">", a predicate logic or relation symbol doesn't this define the question as one of predicate logic?

It's been a long time and I don't know.

Generally -> is the arrow operator used for material implication. A->B is merely "if A is true, then B is true". The truth table is false only when A is true and B is false. If A is false, then the truth table is true regardless of the value of B.

Most that particular TS's posts were rather rambling and difficult to decipher what they were actually trying to get at -- that's what a good portion of the thread was trying to accomplish.

 

Generally -> is the arrow operator used for material implication. A->B is merely "if A is true, then B is true". The truth table is false only when A is true and B is false. If A is false, then the truth table is true regardless of the value of B.

Most that particular TS's posts were rather rambling and difficult to decipher what they were actually trying to get at -- that's what a good portion of the thread was trying to accomplish.

WBahn, That makes sense now that you've explained it. I took the statement to read A- > rather than A -> as in a class or assignment grade which would define a relationship and qualify as an event. I haven't looked at anything like this in many years let alone use it. I agree with you on the typos or language problems. They did get in the way. Also, you're pointing out to the OP the problems caused by inferring information not given was spot on. Thank you for pointing out my errors. As I said, I enjoyed reading the thread.

 

The original question was: If A->B. Is this a propositional logic statement albeit incomplete? As I remember propositional logic doesn't use quantifiers or relationships. By the use of the ">", a predicate logic or relation symbol doesn't this define the question as one of predicate logic?

WBahn explained what the original symbolism meant, but we can still address your question. If I understand you correctly, you're wondering if the inclusion of the relation ">" (greater than) in the statement necessarily invokes the extra machinery of predicate logic, and in doing so takes us out of the propositional realm. You're probably thinking that we can't even define ">" without predicate logic, so -- like trying to describe a sphere in two dimensions -- we simply can't express ">" in propositional logic. Great question! It drills right down to the distinction between use and mention, which is a critical (and sometimes confusing) aspect of formal logic.

Recall that a propositional statement is just one or more symbols that can be associated with a single truth value, either true or false. For example, the sentences "Earth is the third planet from the Sun" and "Chuck Norris killed the first king of England" are both propositions. Note that the complexity of the ideas invoked in each sentence has no bearing on whether or not it is a proposition; all that matters, from the perspective of propositional logic, is that we can assign a truth value to them. This becomes evident when we associate each sentence with a single symbol: call the first sentence P and the second Q. Then, the propositional statement "P ∧ Q" is settled solely by whether one of P or Q is false. We don't need to encode the notions of planets, kings, or third/first in propositional logic, only whether the respective sentences are true or not. That's the hallmark of propositional logic: only truth values matter.

So, expressions such as "4 > 2" or "the derivative of x-squared is 2x" are perfectly valid propositional statements, even though the subjects of well-ordered sets and differential calculus far exceed the scope of propositional logic. Indeed, the expression "∃x∀y(x + y = 1)" -- which is clearly a statement of predicate logic -- is also a propositional statement (namely, a false proposition over the domain of integers). The key takeaway is the distinction between use and mention. Any system of logic will have a finite set of symbols that can be used as part of the logic; in the case of propositional logic, we have the familiar ∨, ∧, parentheses, arrows, individual letters from the alphabet, and so on. As long as the only symbols used are from the allowed set, you can make well-formed formulas. You are, however, allowed to mention whatever symbol you want as part of a string, as you can always "collapse" the string into a single symbol from the allowed set. So, P = "A- > B" is a perfectly valid propositional statement.

Bonus question: Over the domain of integers, is the expression "∃x(x + y) = 1" a propositional statement?