Yhyjyk

If I say:

If I am paid today I'll go to the party tonight

I am saying that if I receive a payment today I will go to the party tonight.

But if I am not paid, can we conclude that I am not going to the party tonight?

I think not, because I have not said anything about what I am going to do if I am not paid.

But if I say:

To go to the party I need money. At this moment I don't have any
money, but if I am paid today I am going to the party tonight.

In this case, can we affirm with absolute certainty, based only on the proposition, that if I am not paid I will not go the party tonight?

In the former case, there is no relation between being paid and going to the party. In the latter case there is.

That depends on your underlying logic, or how you interpret "If ... then". If understood as material implication/conditional (written A → B in logic), which corresponds to the usual mathematical use of "if" as found in mathematical theorems, then from

If I am paid today I'll go to the party tonight

you can not deduce that

If I am paid not today, I'll not go to the party tonight.

In propositional logic, "If A then B" means that in all the situations where A is true, B will be true as well, i.o.w., there are no situations in which A is true but B is false; but this doesn't exclude the possibility that there might be situations where A is false but B is still true. So from (not A) you can not conclude (not B) when given A → B.

Such an inference would require a stronger statement, namely "if and only if":

If and only if I am paid today I'll go to the party tonight

This is called a bi-implication or bi-conditonal, written A ↔ B and means that the situations in which A is true are exactly the situations in which B is true. So if A is false, this will enable you to conclude that B can not be true either.

The question is now whether the usual mathematical interpretation of "if" is indeed the "if" that is used in an ordinary English sentence like yours, i.e. whether a straightforward translation of "If A then B" into A → B with said logical properties is appropriate. There has been heaps of philosophical discussion and psychological research about this, and the short answer is: Depends on the context, but in general interpreing natural language "if" strictly as material implication is too short-sighted, because there are many scenarios in which people use and understand "if" in different ways, and good reasons why they do so. In particular, there are many real-life contexts (and theories about why this is so) in which a natural language "if" is intended and understood as what a logician would call an "if and only if", in which case the inference "If not A, then not B" is valid and intended. W.r.t. to your example, such an interpretation seems plausible, because you presumably intend to say that you need the money to spend at the party, in which case it would by cognitively reasonable to draw the inference that if you don't receive your payment, you won't show up at the party.

So: From a classical logic point of view, no, this inference is not valid; from a psychological/natural language point of view, depends on context, because a natural language "if" is vague, and there can be reasons in favor of either interpretation as A → B (from which we can't make any conclusions about B given that A is false), A ↔ B (from which we could deduce that you're not going to the party given that you're not paid), or something completely different.

Edit:

BTW, the title of your question is somewhat misleading, in that what you're asking for is not the negation of the statement, but negation of the antecedent or consequent:

The negation (¬) of

If I am paid today I'll go to the party tonight

would be

It's not the case that if I am paid today I'll go to the party tonight

which denies the truth of the implication (¬ (A → B)), but not the truth of the antecedent (¬ A) or the consequent (¬ B).

If interpreted as material implication, which reads "In all the situations in which A is true, B is true too", the negation of this statement would amount to

Not all A-situations are also B-situations

which is equivalent to

There is at lest one situation in which A is true and B is false

So starting from the implication

(1) (A → B)

If I am paid today I'll go to the party tonight

the negation of this statement

(2) ¬(A → B)

It's not the case that if I am paid today I'll go to the party tonight,

i.e. There is a possible scenario where I am paid today and will not go to the party tonight

is neither the same as negating the consequent

(3) A → ¬B

If I am paid today, I'll not go to the party tonight

nor is this the same as negating the antecedent

(4) ¬A → B

If I'm not paid today, I'll go to the party tonight

nor both

(5) ¬A → ¬B

If I'm not paid today, I'll not go to the party tonight

These are all different statements. For each pair of satements from (1)-(5), you could find a scenario where one statement is true while the other is false. In short, you are not asking for the truth of the negation of the staement (¬(A → B)). Instead, what you are trying to do is to conclude ¬B from ¬A given A → B.

But if I am not paid, can we conclude that I am not going to the party tonight

That is not the negation.

As an easy example:

• If it rains, then the ground will get wet.




• If the ground does not get wet, then it does not rain.

Note that one implies the other.

What you instead constructed is the converse and its negation:

• If the ground gets wet, then it rains.




• If it does not rain, then the ground does not get wet.

This is a separate statement from the first one and the ground might for instance also get wet when turning on a sprinkler (or in your example people giving you money or you decide that you would like to go anyways).

What you seem to be asking is if the person meant to say more/ something else than they actually did: "If and only if I am getting paid, I will go to the party". Might be, might not be, but this is unrelated to logic.

The question asks if we are given more information, that is, more propositions describing the situation about whether someone will go to the party or not, can we say "with absolute certainty, based only on the proposition, that if I am not paid I will not go the party tonight?"

There may be other conditions that arise that have not been anticipated that may prevent someone from going to the party or allow that person to go to the party even if that person is not paid. We don't know that we have covered everything.

For example, that person might be very tired and not want to go to the party when it is time to go even if the person has been paid. Or friends may say that they will loan the person the money to go to the party or pay the person's way allowing the person to go even if the person has not been paid.

Even assuming we have covered all of the possibilities that might come up, if we assume the person has free will that person may choose not to go to the party even if the person has been paid because there are two alternate possibilities, (1) go to the party or (2) do not go to the party, and, by assumption, the person still has enough free will to choose to do either.