## What can we say about logical formulas/propositions?

• 2.6k
The thread "Do (A implies B) and (A implies notB) contradict each other?" was long and trailed off into different simultaneous discussions and disputations. One of them was the matter of putting logical formulas into natural language (English in our case) — that matter was essential for the purpose of correctly interpreting some statements.

We can put formulas such as (¬B)∨(¬A∧B) into a proof checker or truth table and get results. But for what? Logic, among many things, helps us think.

"Logical formula/proposition" here means A∧B, (¬A)∨B, ((A∧¬B)∨(¬A∧B))↔¬((A∧B)∨(¬A∧¬B)), ¬(P∨(P→Q)), ¬(P∨(Q→P)), and so on.

There is no mystery as to what A∧B means, it is simply "A and B". Likewise, ¬A is not-A.

However, as soon as we get to A∨B, we have an issue: in English, there is no lexical distinction between inclusive-or and exclusive-or, but A∨B is inclusive-or, meaning the result is also True if both are True. Thus, one might say A∨B becomes "A or B or both". A⊻B, exclusive-or, might be "either A or B (but not both)".

A→B is somewhat straightforward, A implies B. And logic here agrees with our intuition. A→B, A, therefore B.

However, what about ¬(A→B)? What can we say about this in English? The first thought is "A does not imply B". But here is the trouble: if ¬(A→B) is true and B is false, A is true. If we read it as such, we would have it "If A does not imply B, and B is false, A is true". Surely that can't be the case, otherwise obviously false sentences such as "An equation being quadratic does not imply it has real solutions, the equation does not have real solutions, therefore it is quadratic." would follow. So we can't read ¬(A→B) as "A does not imply B".

So, what could one say about ¬(A→B) in English? And what about the following formulas:

• A→(B∧¬B);
• A→¬(B∧¬B);
• ¬(A→(B∧¬B))?

On the flip side, can the English meaning of "A does not imply B" be converted to first order logic formulas?

Relevant: necessary conditions and sufficient conditions.

Edit: So far, the thread has been given a satisfactory by bongo fury in the first page. More contributions are welcome, especially those that submit a reply to the formulas listed above.
• 5.4k
Good OP, Lionino. Switching logic into natural languages was a big handicap in my last thread. It seemed to be a simple riddle for everyone, until I asked to explain it with natural language and whether the concepts of ambiguity and contradictory are similar or not. I only got answers using logic language constantly until RussellA wrote a very good example using natural language.

I have to agree that statements like A∧B are universal, and I guess it helps people use logic quickly and easily. But, again, it is outstanding to see those logic formulas explained in language. It seems they are only allowed to use it with "A" and "B" in the premises.

So, what could one say about ¬(A→B) in English?

I don't get it, but I'm confident I could get it using natural English. Is there a substantial difference?

On the flip side, can the English meaning of "A does not imply B" be converted to logical formulas?

To what extent should it be converted into logic formulas?
• 9k
Two approaches. First, with any negation, to use as needed the locution, "It is not the case that...". Thus for ~(p -> q) I'd say, "It is not the case that p implies q." Or second, relying on the equivalence of (p -> q) <=> (~p v q), I'd say, "Either it is not the case that p, or (it is the case that) q."

In any case, going back and forth between "logical formulas" and natural language" is always going to be problematic. Presumably with natural language the purpose is successful communication, and for that there is no "logical formula." Instead, one simply has to do whatever it takes, which is often not-so-easy.
• 486
Eventually, you will even need to add quantifiers (∀ ∃) and predicates to express in logic something as simple as:

All humans are mortal.
Socrates is human.
Therefore Socrates is mortal.

If you want to express in logic statements about logic itself -- which is a requirement for philosophical statements -- you even need to add support for arithmetic.

The resulting language is full of issues, collectively known as the foundational crisis in mathematics, which is clearly also a foundational crisis in logic.
• 1.8k
A→B is somewhat straightforward, A implies B. And logic here agrees with our intuition.

I'd push back against this - this is one of the most egregious examples of logic disagreeing with our intuitive use of implication.

In classic symbolic logic, a -> b is true, according to its truth table, if, for example, a is true and b is true.

(2+2=4) implies (Kamala Harris is a presidential nominee). These is true in classical logic. But it doesn't really match our intuition at all.
• 486
(2+2=4) implies (Kamala Harris is a presidential nominee).

"(2+2=5) implies (Kamala Harris is prime minister of China)" is also true in classical logic.

But it doesn't really match our intuition at all.
It actually does.

It just means that knowledge as a justified true belief is not only about truth but also about justification.
• 1.8k
let me rephrase: it doesn't match MY intuition, and many other people. To many of us, (2+2=4) implies (Kamala Harris is a presidential nominee) makes no sense even if the classical logic truth table comes out as true, because the left side of the implication at least seemingly has nothing to do with the right side.

Maybe it matches your intuition, and I'm sorry for trying to speak for you. My mistake.
• 486
let me rephrase: it doesn't match MY intuition, and many other people.

It is probably a mixup between the implication, which is just a truth table, and the entailment, a ⊢ b, which means that consequent b necessarily follows from antecedent a.

(2+2=4) ⊢ (Kamala Harris is a presidential nominee)

is false, because the consequent cannot be justified from the antecedent.

So, it is rather about a mixup in vocabulary than about intuition. I guess that many other people do that indeed too.
• 1.6k
here is the trouble: if ¬(A→B) is true and A is false, B is true.

To be fair, if ¬(A→B) is true and A is false, anything is true.

Because, if ¬(A→B) is true, A is true.

Which isn't counter-intuitive, because it's intuitive that

A→B means not(A without B).

So it's intuitive that

¬(A→B) means A without B.

E.g. "An equation being quadratic implies it has real solutions" means not(the equation is quadratic without the equation has real solutions)

So "An equation being quadratic does not imply it has real solutions" means the equation is quadratic without the equation has real solutions.
• 2.2k
So, what could one say about ¬(A→B) in English?

As I alluded to in the other thread, material implication captures English usage only insofar as it guarantees that if the antecedent is true then the consequent will also be true. Similarly, the negation of a material implication says that if the antecedent is true then the consequent will be false, and this is vaguely similar to the denial of an implication in English except for the fact that the falsity of the consequent is not guaranteed in English.

The key is that in English we prescind from many things that material implication does not prescind from, such as the value of the consequent in that denial case. As another example, if an antecedent is false then the material implication is true, whereas this does not hold in English. At the end of the day the English sense of implication simply isn't truth functional. It is counterfactual in a way that material implication is not.

And what about the following formulas:

A→(B∧¬B);
A→¬(B∧¬B);
¬(A→(B∧¬B))?

I think in examining these we are combining two confusing and non-translatable logical concepts: material implication and contradiction. Neither one translates well into English, and their combination translates especially badly.

Further, I am of the opinion that speech about contradictions is always a form of metabasis eis allo genos. Even in English when we say, "If you make that claim you will be contradicting yourself," we are shifting between two different registers: first-order claims and second-order rules of discourse (i.e. Thou shalt not contradict thyself).
• 2.2k
The key is that in English we prescind from many things that material implication does not prescind from

For example, one can assert the material implication (P→Q) for three reasons:

1. P is true and Q is true
2. P is false (and Q is true)
3. P is false (and Q is false)

In English, on the other hand, we only say, "If P then Q," when we believe that the presence of P indicates the presence of Q. The English has to do with a relation between P and Q that transcends their discrete truth values. One way to see this is to note that an English speaker will be chastised if they use the phrase to represent a correlation that is neither causative nor indicative, but in the logic of material implication there is nothing at all wrong with this.
• 486
In English, on the other hand, we only say, "If P then Q," when we believe that the presence of P indicates the presence of Q. The English has to do with a relation between P and Q that transcends their discrete truth values.

Exactly.

It represents an entailment A ⊢ B, and not just a simple implication A→B.

Logic makes all its decisions by only looking at truth values while the English version assumes the existence of a system that also investigates justification.
• 2.6k
To be fair, if ¬(A→B) is true and A is false, anything is true.

Good catch. The premises ¬(A→B) and ¬A together are explosive. But ¬(A→B) and ¬B aren't, yet ¬(A→B) and ¬B entail A. It A does not imply B and B is false, can we really infer that A is true?
I have updated the thread to remove the explosion :)

¬(A→B) means A without B.

We can go with that.
P1: ¬(A→B)
P2: B is true
Concl.: A is true

¬(A→B) means A without B
B is true
Therefore A is true

Does that make intuitive sense to you?

Rain without wetness
Wetness
Therefore rain.
• 2.6k
Thus for ~(p -> q) I'd say, "It is not the case that p implies q."

It is not the case that p implies q
Not q
p
That is valid
Yet it doesn't agree with our intuitions.

Or second, relying on the equivalence of (p -> q) <=> (~p v q), I'd say, "Either it is not the case that p, or (it is the case that) q."

That is viable as well. But I think that the latter phrase just ends up meaning "P implies Q", which is really "Everytime you see P you also see Q", which is essentially "Either it is not the case that P, or it is the case that Q".

In any case, going back and forth between "logical formulas" and natural language" is always going to be problematic.

It is, but proof checkers and logic have helped us check the validity and consistency of many arguments that would be otherwise extremely difficult to verify. So I think it is very much worthwhile to look into how we can bring language into logic.
• 1.6k
yet ¬(A→B) and ¬B entail A.

To be fair, so does ¬(A→B).

If A does not imply B and [regardless of whether] B is false, can we really infer that A is true?

Yes, because it means A without B. Isn't it intuitive that A without B entails A? And isn't it intuitive that A→B means not A without B, i.e. ¬(A ∧ ¬B), so that ¬(A→B) means A without B, and therefore A ∧ ¬B and therefore A?

¬(A→B) means A without B
B is true
Therefore A is true

Does that make intuitive sense to you?

Yep. Even if you add the irrelevant and contradictory P2, which is going to make everything true anyway.

Rain without wetness
Wetness
Therefore rain.

Rain without wetness
Wetness
Therefore rain.

Yes. So?
• 24.1k
Logic isn't a replacement for natural languages. Nor is it a set of rules for how one ought construct arguments. This was part of the subject of my thread Logical Nihilism, and the work of Gillian Russell.

in English, there is no lexical distinction between inclusive-or and exclusive-or, but A∨B is inclusive-or, meaning the result is also True if both are True.
So what logic does in this case is to set out explicitly two ways of using "or" of which we were probably unaware. After understanding this we are able to say clearly whether we are using an exclusive or an inclusive "or". Prior to that logical analysis, we were probably unaware of the distinction, let alone which we were using.

So logic here is setting up a degree of precision that can carry over into natural languages. It's acting as a tool to make clear what it is we are doing with our sentences.

It's a mistake to think that there are laws of logic that have complete generality - and must be obeyed in all circumstances. Rather logic sets out sub-games within language, with their own specific rules. Natural languages permit the breaking of the rules of any of these sub-games.

Take a look at these examples from Russell. ϑ ⊧ ϑ and ϑ & ϒ ⊧ ϑ might seem to be candidates for logical laws one might expect to have complete generality.

Identity: ϑ therefore ϑ;: a statement implies itself. But consider "this is the first time I have used this sentence in this paragraph, therefore this is the first time I have used this sentence in this paragraph"

Elimination: ϑ and ϒ implies ϑ; But consider "ϑ is true only if it is part of a conjunction".

Logic sets up systems in which some things can be said and others are ruled out, but natural language is far broader than that, allowing for the breach of any such rule.

Logic doesn't give us a crystalline replacement for natural languages. But it can set out clearly what it is we are doing with our statements.
• 2.6k
Yep. Even if you add the irrelevant and contradictory P2, which is going to make everything true anyway.

That was also a mistake, that was supposed to be "B is false" like the first quote, but your point stands.

Rain without wetness
Wetness
Therefore rain.

Ok, so your "A without B" is not that "it is possible to have A without B", but that "there is A without B". I guess that can make sense as ¬(A→B) ↔ (A∧¬B).
See the link on the OP. It says "A→B" means B is a necessary condition for A (this doesn't need to be interpreted in a causal one-directional flow of time sense). How would you put ¬(A→B) in terms of conditions?
• 3.2k

"A does not imply B". In English that is ambiguous. It could mean:

There are instances in which A is true but B is false.

It is not the case that A entails B (same as above).

It is not the case that A implies B (where 'implies' means the material conditional).

It is not the case that A implies B (where 'implies' means a connective other than the material conditional).

Probably others.

The rest of this pertains to ordinary symbolic logic:

We have to be careful to distinguish between, on the one hand, mere implication and, on the other hand, and entailment or proof .

A -> B
is not generally equivalent with
A |= B or A |- B.

In ordinary symbolic logic, '->' does not mean 'entails' or 'proves':

A -> B is false in a given interpretation if and only if (A is true in the interpretation and B is false in the interpretation).

A |= B is true if and only if every interpretation in which A is true is an interpretation in which B is true.

A |- B iff and only if there is a derivation of B from A.

Example:

"If Grant was a Union general, then Grant was under Lincoln." True in the world of Civil War facts. But false in some other worlds in which Grant was a Union general but, for example, Lincoln was not president.

"Grant was a Union general" entails "Grant was under Lincoln". Not true, since there are worlds in which Grant was a Union general but, for example, Lincoln was not the president.

"Grant was a Union general" proves "Grant was under Lincoln". Not true, since there are not other premises along with "Grant was a Union general" to prove "Grant was under Lincoln".

/

Also, we need to be careful what we mean by letters such as 'A', 'B', 'P', 'Q', etc.

(In propositional logic, all formulas are sentences, but in predicate logic, some formulas are sentences and some formulas are not sentences.)

In different contexts, such letters are used to represent either:

(1) atomic formulas (atomic sentences)
or
(2) meta-variables ranging over formulas. (Sometimes logic books use Greek letters for this.)

In recent discussions, the letters are being used as meta-variables.

So, for example, when we mention 'A -> B', we understand that 'A' and 'B' range over all sentences, including ones of arbitrary complexity.

/

If you are asking what is the most accurate English translation of the intended meanings in ordinary symbolic logic, just put in:

"it is not the case that" where '~" occurs
"if ____ then ____" where '____ -> ____' occurs
"and" where '&' occurs
"or" where 'v' occurs
• 3.2k
¬(A→B) and ¬B entail A. I[f] A does not imply B and B is false, can we really infer that A is true?

What do you mean by "A does not imply B"? Do you mean?:

"It is not the case that A implies B"
i.e., ~(A -> B)
which is true in any interpretation in which A is true and B is false.

or

"It is not the case that every interpretation in which A is true is an interpretation in which B is true".

P1: ¬(A→B)
P2: B is true
Concl.: A is true

That should be ('M' here is an interpretation):

~(A -> B) is true in M
B is true in M
therefore, A is true in M

or, if M is tacit:

~(A -> B) is true
B is true
therefore, A is true

or, without 'true':

~(A -> B)
B
therefore, A

Rain without wetness
Wetness
Therefore rain.

'rain without wetness', 'wetness', 'rain' are not sentences.

But it does have a nice haiku-like flavor.
• 9k
So I think it is very much worthwhile to look into how we can bring language into logic.
Ah, well, hmm. No doubt in some circumstances it has to be done; writing laws comes to mind. But the caveat being that natural language is about communication while logic is about demonstration. Two very different animals - two different languages - though sometimes they're on the same path and drink from the same stream.
• 2.6k
A -> B is false in a given interpretation if and only if (A is true in the interpretation and B is false in the interpretation).

A |= B is true if and only if every interpretation in which A is true is an interpretation in which B is true.

A |- B iff and only if there is a derivation of B from A.

Example:

"If Grant was a Union general, then Grant was under Lincoln." True in the world of Civil War facts. But false in some other worlds in which Grant was a Union general but, for example, Lincoln was not president.

"Grant was a Union general" entails "Grant was under Lincoln". Not true, since there are worlds in which Grant was a Union general but, for example, Lincoln was not the president.

"Grant was a Union general" proves "Grant was under Lincoln". Not true, since there are not other premises along with "Grant was a Union general" to prove "Grant was under Lincoln".

That is helpful. I think it relates to and clarifies FJ's post:

In classic symbolic logic, a -> b is true, according to its truth table, if, for example, a is true and b is true.

(2+2=4) implies (Kamala Harris is a presidential nominee). These is true in classical logic. But it doesn't really match our intuition at all.

If you are asking what is the most accurate English translation of the intended meanings in ordinary symbolic logic, just put in:

"it is not the case that" where '~" occurs
"if ____ then ____" where '____ -> ____' occurs
"and" where '&' occurs
"or" where 'v' occurs

Yes, I am asking that. I would only detail that ∨ is more appropriately called "__ or __ or both", while ⊻ is "either __ or __".

But let's go with that. Should we read ¬(A→B) as "it is not the case that if A then B"? If so, how should we understand "it is not the case that if A then B"? You said "A does not imply B" is ambiguous in English. Indeed. However, in plain English "it is not the case that if A then B" is also ambiguous:

[1] There are instances in which A is true but B is false.

[2] It is not the case that A entails B (same as above).

[3] It is not the case that A implies B (where 'implies' means the material conditional).

[4] It is not the case that A implies B (where 'implies' means a connective other than the material conditional).

The English phrase "A does not imply B" typically means "There are instances in which A is true but B is false". By your list, that does not mean the same as the material conditional.
If 'it is not the case that if A then B' is to be understood as the third option, we are simply circling back. What is a phrase in English that unambiguously corresponds in meaning to ¬(A→B)?

'rain without wetness', 'wetness', 'rain' are not sentences.

Short for "There is rain without there is wetness". Does that work instead?

See algo bongo fury's proposal that ¬(A→B) can be read as "there is A without B".
• 2.6k
I don't get it, but I'm confident I could get it using natural English. Is there a substantial difference?
Here is the incongruence:
if ¬(A→B) is true and B is false, A is true. If we read it as such, we would have it "If A does not imply B, and B is false, A is true". Surely that can't be the case, otherwise obviously false sentences such as "An equation being quadratic does not imply it has real solutions, the equation does not have real solutions, therefore it is quadratic." would follow. So we can't read ¬(A→B) as "A does not imply B".
• 2.6k
The resulting language is full of issues, collectively known as the foundational crisis in mathematics, which is clearly also a foundational crisis in logic.

lol
• 3.2k
I don't think that "It is not the case that" is usually ambiguous. (It is not the case that "it is not the case that" is usually ambiguous.)

"If A then B" is understood differently by different people in different contexts.

So any ambiguity in "It is not the case that if A then B" stems from "If A then B".

So specify what you mean by "If A then B", then you will have specified what you mean by "It is not the case that if A then B".

The English phrase "A does not imply B" typically means "There are instances in which A is true but B is false". By your list, that does not mean the same as the material conditional.

[EDIT: Dump the strikethrough potion]

Arguably, they are the equivalent:

(1) "If A then B" if and only if "Every instance in which A is true is an instance in which B is true".(material conditional)

is equivalent with:

(2) "If A then B" if and only if "There are no instances in which A is true and B is false"

So:

(4) "It is not the case that every instance in which A is true is an instance in which B is true"

is equivalent with

(3) "It is not the case that there are no instances in which A is true and B is false"

is equivalent with:

(5) "There are instances in which A is true and B is false"

If 'it is not the case that if A then B' is to be understood as the third option, we are simply circling back.

Circling back to what? Choose whichever option you like, or add options such as relevance, or state another option.

What is a phrase in English that unambiguously corresponds in meaning to ¬(A→B)?

Choose which option you prefer for "If A then B", then prefix it with "it is not the case that".
• 3.2k
"There is rain without there is wetness".

First, that is not idiomatic. I've never heard someone say "There is X without there is Y". Second, it could mean at least a few different things. Third, I don't know your point with the example. Fourth, the previous example at least had a nice haiku-like quality.
• 3.2k
if ¬(A→B) is true and B is false, A is true.

It's been pointed out to you at least twice that B doesn't matter:

~(A -> B) -> A

we can't read ¬(A→B) as "A does not imply B".

Then don't read it that way.

It is suggested to read it as: It is not the case that A implies B.
• 2.6k
First, that is not idiomatic

If I had said "There is rain without there being wetness" you'd have complained that "there being wetness" cannot work as a standalone proposition.
• 2.6k
I don't think that "It is not the case that" is usually ambiguous.

We are not in disagreement.

"If A then B" is understood differently by different people in different contexts.

So any ambiguity in "It is not the case that if A then B" stems from "If A then B".

So specify what you mean by "If A then B", then you will have specified what you mean by "It is not the case that if A then B".

Of course.

(1) "If A then B" if and only if "Every instance in which A is true is an instance in which B is true".(material conditional)

The issue is that the material conditional is not just that. A→B is also true whenever A is false. So by stating «"if A then B" if and only if "every instance in which A is true is an instance in which B is true"» you are not making "if A then B" equivalent to A→B. If you decide that «"if A then B" if and only "Every instance in which A is true is an instance in which B is true, or every instance in which A is false"», which would make "if A then B" equivalent to A→B, we are just back to the old problem. If A then B is ambiguous, as you yourself said:
"If A then B" is understood differently by different people in different contexts.
Saying A→B is "if A then B" does not provide a solution to the matter of unambiguously converting A→B to English.
• 3.2k

It is not strictly speaking a sentence, but idiomatically it is understood that it means "There is rain but there is no wetness".
• 2.6k
"but no" does not work because bongo fury's suggestion for ¬(A→B) was "A without B", which I find pretty good, so the connective I had to use must've been "without".
• 3.2k
A→B is also true whenever A is false.

"Every instance in which A is true is an instance in which B is true"

equivalent with:

"There is no instance in which A is true and B is false."

If A is false in an instance, then that is an instance in which it is not the case that A is true and B is false.
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