Logic, Inference, consequence (propositional logic) - basic question

[Rembrandt]

Hello together
I have a very basic question that would help me to understand logic better.

"It's true that A ⊨ B exactly, if A -> B is logically true."

Why is this sentence always true?
I thought about sufficient and necessary conditions. B is necessary for A, and A is sufficient for B? e.g. Fire and oxygen. Or am I wrong?
Thanks in advance!

 

[Dr.Peterson]

Hello together
I have a very basic question that would help me to understand logic better.

"It's true that A ⊨ B exactly, if A -> B is logically true."

Why is this sentence always true?
I thought about sufficient and necessary conditions. B is necessary for A, and A is sufficient for B? e.g. Fire and oxygen. Or am I wrong?
Thanks in advance!

We need to start with definitions. What definition were you given for the "⊨" symbol? And what does it mean for "A -> B" to be "logically true"?

Also, I think you have punctuated the sentence incorrectly, and meant

"It's true that A ⊨ B, exactly if A -> B is logically true."​

That is, I don't think you mean "A entails B exactly", whatever that would mean, but "exactly if" (that is, if and only if). Am I right?

 

[pka]

Hello together
I have a very basic question that would help me to understand logic better.
"It's true that A ⊨ B exactly, if A -> B is logically true."
Why is this sentence always true?
I thought about sufficient and necessary conditions. B is necessary for A, and A is sufficient for B? e.g. Fire and oxygen. Or am I wrong?

The symbol [imath]\vdash[/imath] appears in and is standard in many books on Logic & Sets.
I means something like follows from as in [imath](P \Rightarrow Q)\vdash(\neg P\vee Q)[/imath]
Now for the symbol [imath]\vDash[/imath] I of course had seen it used but had ever used it myself.
So I guess it makes sense that could not find it in any of the textbooks on the selves of my home-office.
I did come across a book still in its mailing package that I moved home over twenty years ago.
It was LOCIC, SETS, AND RECURSION by Robert Causey a textbook for computer science.
In it is a complete discussion of [imath]\Gamma\vDash_T\phi:\;[/imath] [imath]\;\Gamma[/imath] tautologically implies [imath]\phi[/imath]
[imath][/imath][imath][/imath][imath][/imath][imath][/imath][imath][/imath]

 

[Dr.Peterson]

The place I found the symbol (which I've run across before, but never used) is in Wikipedia, here and here. But that doesn't help a lot in interpreting this problem; I suspect the usage is a little different. That's why I asked for the applicable definition.

In any case, proofs need context, including definitions and axioms.

 

[MaxMath]

I agree context is essential here, particularly for symbols that are not frequently used or well-known.

I'm intrigued by "exactly". Even with the change suggested, it still reads odd to me. If that is the intended meaning, "iif" (if and only if) is much better.

 

[OB1M]

Hello together
I have a very basic question that would help me to understand logic better.

"It's true that A ⊨ B exactly, if A -> B is logically true."

Why is this sentence always true?
I thought about sufficient and necessary conditions. B is necessary for A, and A is sufficient for B? e.g. Fire and oxygen. Or am I wrong?
Thanks in advance!

I must confess to only using the = symbol in a math sense never the double turnstile however, I am aware, as general knowledge, that philosophical logic is an important part of a law degree and is perhaps unique in this sense. I asked the LLM (Language Learning Model) tool aka Bard some questions:

Q1: Give an example of the symbol ⊨ in a legal example

A1: Sure. Here is an example of the symbol ⊨ in a legal example:

In a contract, the double turnstile symbol may be used to denote the relation of implication. For example, the statement "If the customer does not pay the invoice, then the supplier will not ship the goods" means that if the customer does not pay the invoice, then the supplier will not ship the goods. This means that if the first statement is true, then the second statement must also be true.

In this example, the double turnstile symbol is used to denote the relation of implication between the two statements. The first statement is "If the customer does not pay the invoice", and the second statement is "Then the supplier will not ship the goods". The double turnstile symbol implies that if the first statement is true, then the second statement must also be true.

Q2: How do the symbols ⊨ and = differ in their use

A2: The symbols ⊨ and = are both used in mathematics, but they have different meanings. ⊨ is the symbol for implication, while = is the symbol for equality.

Implication means that if the first statement is true, then the second statement must also be true. For example, the statement "If it is raining, then the ground is wet" is true because if it is raining, then the ground must be wet.
Equality means that two things are the same. For example, the statements "2 + 2 = 4" and "4 - 2 = 2" are both true because 2 + 2 and 4 - 2 are both equal to 4.

In some cases, the symbols ⊨ and = can be used interchangeably. For example, the statement "If 2 + 2 = 4, then it is raining" is true because if 2 + 2 = 4, then the ground must be wet. However, this is not always the case. The symbols ⊨ and = have different meanings, and they should not be used interchangeably unless the meaning is clear.

Be careful with these language tools, I deliberately left "both true because 2 + 2 and 4 - 2 are both equal to 4" to demonstrate this but you get the idea I'm sure?

 

[MaxMath]

I must confess to only using the = symbol in a math sense never the double turnstile however, I am aware, as general knowledge, that philosophical logic is an important part of a law degree and is perhaps unique in this sense. I asked the LLM (Language Learning Model) tool aka Bard some questions:

Q1: Give an example of the symbol ⊨ in a legal example

A1: Sure. Here is an example of the symbol ⊨ in a legal example:

In a contract, the double turnstile symbol may be used to denote the relation of implication. For example, the statement "If the customer does not pay the invoice, then the supplier will not ship the goods" means that if the customer does not pay the invoice, then the supplier will not ship the goods. This means that if the first statement is true, then the second statement must also be true.

In this example, the double turnstile symbol is used to denote the relation of implication between the two statements. The first statement is "If the customer does not pay the invoice", and the second statement is "Then the supplier will not ship the goods". The double turnstile symbol implies that if the first statement is true, then the second statement must also be true.

Q2: How do the symbols ⊨ and = differ in their use

A2: The symbols ⊨ and = are both used in mathematics, but they have different meanings. ⊨ is the symbol for implication, while = is the symbol for equality.

Implication means that if the first statement is true, then the second statement must also be true. For example, the statement "If it is raining, then the ground is wet" is true because if it is raining, then the ground must be wet.
Equality means that two things are the same. For example, the statements "2 + 2 = 4" and "4 - 2 = 2" are both true because 2 + 2 and 4 - 2 are both equal to 4.

In some cases, the symbols ⊨ and = can be used interchangeably. For example, the statement "If 2 + 2 = 4, then it is raining" is true because if 2 + 2 = 4, then the ground must be wet. However, this is not always the case. The symbols ⊨ and = have different meanings, and they should not be used interchangeably unless the meaning is clear.

Be careful with these language tools, I deliberately left "both true because 2 + 2 and 4 - 2 are both equal to 4" to demonstrate this but you get the idea I'm sure?

Do you understand what you are talking about yourself?

 

[OB1M]

Do you understand what you are talking about yourself?

It seems not? Please explain the reason for your critique.

 

[MaxMath]

It seems not? Please explain the reason for your critique.

What I'm saying is basically I'm totally confused, to such an extent that I suspect even you may not understand it, since a large part of it seems to come from a machine (computer). If you do, can you please turn it into your language in a way that is understandable to other people, with a view to answering (or helping answer) the original question? That would be great.

To be more specific, below is beyond my comprehension.

For example, the statements "2 + 2 = 4" and "4 - 2 = 2" are both true because 2 + 2 and 4 - 2 are both equal to 4.

 

[OB1M]

What I'm saying is basically I'm totally confused, to such an extent that I suspect even you may not understand it, since a large part of it seems to come from a machine (computer). If you do, can you please turn it into your language in a way that is understandable to other people, with a view to answering (or helping answer) the original question? That would be great.

To be more specific, below is beyond my comprehension.

The replies here by forumites, are in a pure math form. The machine was asked to provide applied math examples which it has done including the error 2 + 2 and 4 - 2 are both equal to 4 which badly demonstrates the interchangeability of ⊨ and = or entailment and equivalence.

In this sense I hope the OP’s question has been answered.

What the machine should have said is:

Equality means that two things are the same. For example, the statements "2 + 2 = 4" and "6 - 2 = 4" are both true because 2 + 2 and 6 - 2 are both equal to 4.

This particular machine is not performing recursion. When this error was pointed out it said that it was wrong and gave up. On the other type of machine ChatGPT it appears to employ recursion as it will often type an answer then immediately go back on itself to correct it. Unlike Bard when it does get it wrong it will give it another go.

These machines are helpful for auto-didactic learning however, they cannot be trusted in the same way that a tutor or approved authority may be trusted. Sorry you were having so much difficulty with the applied math.

 

[WillisRoy]

Hello together
I have a very basic question that would help me to understand logic better.

"It's true that A ⊨ B exactly, if A -> B is logically true."

Why is this sentence always true?
I thought about sufficient and necessary conditions. B is necessary for A, and A is sufficient for B? e.g. Fire and oxygen. Or am I wrong?
Thanks in advance!

The statement you provided is known as the "deduction theorem" or the "entailment theorem," and it is one of the fundamental results in mathematical logic.

To understand why this statement is always true, it's important to define the symbols used in the statement:

A and B are propositions, which are statements that are either true or false.
⊨ is the symbol for logical entailment, which means that A logically entails B, or that B follows logically from A.
-> is the symbol for logical implication, which means that A implies B, or that if A is true, then B must also be true.
With that in mind, the statement "A ⊨ B exactly if A -> B is logically true" can be restated as "A logically entails B if and only if A logically implies B."

This statement is always true because of the way logical implication and entailment are defined. If A implies B, then whenever A is true, B must also be true. And if A entails B, then whenever A is true, B must also be true. Therefore, the two concepts are equivalent.

Your example of fire and oxygen is a good analogy for this concept. If we have fire (A), then we know that oxygen (B) must be present as well, because fire logically implies the presence of oxygen. And if we know that oxygen is present (A), then we can logically entail that fire must also be present (B), because fire logically follows from the presence of oxygen.

 

[OB1M]

...Your example of fire and oxygen is a good analogy for this concept. If we have fire (A), then we know that oxygen (B) must be present as well, because fire logically implies the presence of oxygen. And if we know that oxygen is present (A), then we can logically entail that fire must also be present (B), because fire logically follows from the presence of oxygen.

I agree, however fire can burn in the absence of oxygen e.g. monopropellants. In the Quine McCluskey treatment of boolean logic there are essential prime implicants that are common to all truths, what is demonstrated is that multiple truths can exist by adding non essential prime implicants to create a given truth. Essentially, for every truth there is an alternative one it depends on the context thought to be relevant.

 

[MaxMath]

The statement you provided is known as the "deduction theorem" or the "entailment theorem," and it is one of the fundamental results in mathematical logic.

To understand why this statement is always true, it's important to define the symbols used in the statement:

A and B are propositions, which are statements that are either true or false.
⊨ is the symbol for logical entailment, which means that A logically entails B, or that B follows logically from A.
-> is the symbol for logical implication, which means that A implies B, or that if A is true, then B must also be true.
With that in mind, the statement "A ⊨ B exactly if A -> B is logically true" can be restated as "A logically entails B if and only if A logically implies B."

This statement is always true because of the way logical implication and entailment are defined. If A implies B, then whenever A is true, B must also be true. And if A entails B, then whenever A is true, B must also be true. Therefore, the two concepts are equivalent.

Your example of fire and oxygen is a good analogy for this concept. If we have fire (A), then we know that oxygen (B) must be present as well, because fire logically implies the presence of oxygen. And if we know that oxygen is present (A), then we can logically entail that fire must also be present (B), because fire logically follows from the presence of oxygen.

Interesting to hear. But I am unable to see much sense of it, if the statement is indeed true AND your interpretation is correct (I don't know either).

The reason for saying this is that, to me, this is no different from "if 1+1 = 2 then 1+1 is equal to 2". Do you see what I mean? Am I holding the wrong end of the stick?

 

[OB1M]

Interesting to hear. But I am unable to see much sense of it, if the statement is indeed true AND your interpretation is correct (I don't know either).

The reason for saying this is that, to me, this is no different from "if 1+1 = 2 then 1+1 is equal to 2". Do you see what I mean? Am I holding the wrong end of the stick?

Pure math or applied math which is to be? In applied math you would need to state the base to which the numbers apply. Is it base 10, base 8 or does it indeed matter. If we have (cat + dog) ⊨ (1 + 1) then equally (cat + dog) ⊨ (animal + animal) something that cannot be stated with meaning algebraically?