# always false vs necessarily false , is it the same?

according to the basic tenets of classical propositional logic, contradictions are 'always false' tautologies 'always true'. When it is extended to modal logic we have the notions of 'necessarily false' for statements that are false in every possible world , 'necessarily true' for statements that are true in every possible world.

how are these notions from classical propositional logic vs modal logic the same/different?

for example: - 1. this statement(1) is always false. - 2. this statement (2) is necessarily false

do (1) and (2) say the same thing? Are they equally paradoxical?

• In propositional logic "always" refers to any choice of truth values for the letters. This already uses possible worlds implicitly, with different worlds being different truth value assignments. So "always" is the same as "necessarily" with the necessity being logical necessity. However, when one uses modal logic explicitly the type of necessity considered is usually weaker than logical, physical, metaphysical, etc. So "necessarily" has a different meaning. But what is logically necessary is still true in every possible world, of course. – Conifold Dec 23 '19 at 23:46
• I would think that even in classical logic, it would be understood that while a tautology is always true, not all propositions which are true in all cases that actually exist are necessarily tautologies. For example, the classic premise of "All men are mortal" may be true for every man that exists, but one could imagine a god granting a man immortality without his ceasing to be a man. – Hypnosifl Dec 24 '19 at 0:09

I'm not an expert, but I'd like to venture a response.

In classical logic, "this statement is always false" is equivalent to "this statement is false" because there's no intermediate truth value -- "true" in classical logic means the same thing as "always true".

There's more nuance here under the model theoretic interpretation of such statements:

A sentence is consistent if it is true under at least one interpretation; otherwise it is inconsistent

and

A sentence φ is said to be logically valid if it is satisfied by every interpretation

The particular statement you selected, "this statement is always false" is undecidable. That is, while the logic asserts it must be either true or false, there doesn't exist a procedure to resolve it, as Goedel demonstrated with his Incompleteness Theorems. In Tarski's Hierarchy of Truth, it would flip-flop between truth values. Note that these undecidable propositions have a lot more baggage to unpack in the difference of grammar (all statements are either true or false) vs. proof (our ability to figure out whether a given statement is true or false from axioms).

If you didn't intend for an undecidable proposition, then the negation of a tautology could be a good example of "always false" in classical logic.

In modal logic, we have more truth values and thus more room to maneuver. In Kripke Semantics, for example:

"possibly A" is defined as equivalent to "not necessarily not A"

If you intended to ask, is "this statement is false" undecidable in both classical and modal logic, then in general I think the answer is yes.

• How would basic modal logic treat a statement as "this statement is necessarily false" or even a statement such as "this statement is not necessarily true" ? they seem also paradoxical to me – Icon Dec 23 '19 at 23:57
• In classical logic, "this statement is false" could thought of as $P := P \Leftrightarrow False$ or $P := ¬P$, but what we really mean is $P := \space \vdash ¬P$ where $\vdash$ means "it is provable that" and $:=$ means "defined as". In modal logic, "this statement is necessarily false" would be $P := \space \vdash □ ¬P$ and "this statement is not necessarily true" $P := \space \vdash ¬□P$ – TCP Dec 24 '19 at 0:59
• Ah dang, the Mathjax formatted doesn't work in comments :( – TCP Dec 24 '19 at 1:01

The use of modal notions ( such as necessity or possibility) is problematic.

So, in ordre to capture the idea of " necessarily true" or "necessarily false" statement without using modal notions, classical propositional logic invented the concept of " interpretaion" and of " true / false in all interpretation" statement.

When one says informally that : (A OR ~A) is always true

one means that : (A OR ~A) is true in all interpretations.

By " interpretation" is meant : a truth value assignment to all atomic propositions of propositional logic.

Whatever value you attribute to the atomic propositions ( A, B, C,D,E, etc....) of your language, you will be in one of these 2 cases

A is true

OR

A is false.

So in all cases ( " always") , the proposition (A OR ~A) will have a true disjunct, and hence will be true.

Note : " always" , here, has no temporal meaning.