# Is it problematic to say that falsity has many values?

If a proposition is clear in the context and is indeterminate, then it is true that it is false (not true) since it is not true in the context. If the proposition would be true in the context, then it would not be indeterminate. Is there something wrong with this reasoning?

EDIT: If a proposition is denied it seems that it can take on many values ​​such as the complement of a set, why indeterminacy cannot be inside? For example, in the problem of future contingents.

Some examples i think of false proposition that can take on many values:

1. If an equality between two numbers that can only be natural is denied, then that denial necessarily implies that one can be greater or less than the other, but both possibilities are part of the denial.

2. The proposition "Jack will be 18 years old" (assuming that Jack is 17 and the future is indeterminate) is currently indeterminate if Jack can die before then, but wouldn't that indeterminacy be part of the falsity of the proposition? If the proposition were true, then that future would be determined.

If the proposition is ambiguous or its truth is unknown, then would bivalence be the problem?

• This question is a bit confusing. What does it mean for falsity to have many values? Are you assuming a multi-valued logic and calling all non-designated values false? What does problematic mean in this context? Do you mean that multi-valued logics are problematic? What do you mean by "indeterminate"? Are you referring to queries that are category errors or for that are about things that don't exist? Commented Feb 24, 2023 at 1:24
• @DavidGudeman If a proposition is denied it seems that it can take on many values ​​such as the complement of a set, why indeterminacy cannot be inside? For example, in the problem of future contingents. I'm assuming it's problematic, my mistake, I really want to ask if it's problematic. Commented Feb 24, 2023 at 1:49
• Maybe an example would help. What sort of false proposition can "take on many values"? Commented Feb 24, 2023 at 2:36
• @DavidGudeman Sorry if I say some wrong things, some examples i think of: 1) If an equality between two numbers that can only be natural is denied, then that denial necessarily implies that one can be greater or less than the other, but both possibilities are part of the denial. 2) The proposition "Jack will be 18 years old" (assuming that Jack is 17 and the future is indeterminate) is currently indeterminate if Jack can die before then, but wouldn't that indeterminacy be part of the falsity of the proposition? If the proposition were true, then that future would be determined. Commented Feb 24, 2023 at 2:58
• OK, those are good example. Please add them to the question so people reading the question will know if they find the topic interesting or not. Commented Feb 24, 2023 at 3:21

## 3 Answers

Falsehood itself can be thought of as a value. Modern logic usually treats it this way. We say that 'true' and 'false' are truth values that a proposition may have. To say of some proposition that it is false can be expressed by saying that the proposition has the value 'false', or that the reference of the proposition is 'false'.

So I would rather rephrase your question as, Are truth and falsehood the only truth values? The principle that there are only two truth values and every proposition has one or the other and not both is called bivalence. It is a controversial principle in philosophy, though widely used in elementary logic.

Examples like future contingents and vague statements are often advanced as counterexamples. It is fairly common to treat statements about the future as neither true nor false until they actually happen, or don't. Vague statements such as "John is tall" may be treated as neither simply true nor simply false but just true to some degree. (There are many approaches to handling vagueness.)

In cases where the truth or falsehood of a proposition is unknown, it is common to suppose that the proposition has a truth value, but we are not in a position to say what it is. Being unknown is not a third truth value but an epistemic status.

In cases where the truth or falsehood of a proposition is somehow permanently indeterminable, the issue becomes more difficult. A realist approach would be to treat it like the unknown case and say that the proposition has a truth value, but we don't know it and never will. An anti-realist approach would be to say in such cases that there is no fact of the matter either way. Under the latter option, bivalence fails to hold.

• A good answer to which I have given a deserved vote. Commented Feb 24, 2023 at 8:09

The Jack example seems like the indeterminateness is simply in the knowledge which one is true. Might make sense to reason about, for example (A iff B => (A indeterminate iff B indeterminate)), but doesn't change the nature of LEM.

In particular, in a logic with indeterminateness as logic value, the following is false:

(A indeterminate) => (A false)

Making your first sentence wrong. Being indeterminate means you don't, or cannot, know the truth value.

I think that the problem you are creating for yourself is that you consider a proposition must be true or false, without accepting the possibility that the proposition can be undecided or undecidable.

Suppose I point to a heap of sand on the floor and tell you it contained 214,879,916 grains. Is that statement true or false?

Assuming you have started to think about how you might establish whether the statement was true or false, eg by counting the grains, suppose that having counted some of the grains you are beset by a storm that washes most of the sand away. How can you then assign a true or false value to the statement?

Suppose, now, the heap really does contain 214,879,916 grains. I can rephrase the original statement about the heap to make it in the form of a future conditional, thus: 'If you count them, you will find there are 214,879,916 grains'. Is that statement true or false? You don't know until you count the grains, and then you might find that you have mis-counted or some are blown away by the wind while you are counting them.

The point is that my statement about the number of grains in the heap was either right or wrong at the time I made it, but you have no way of establishing which. You cannot say that it must therefore have been false, simply because you cannot prove or disprove it- you must instead pigeon-hole the statement under another category as being undecided.