## Existential import: in logic, do propositions default to true or false when objects in them do not exist?

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In this hypothetical:

Firefighters always tell the truth, while politicians always tell lies.

Suppose three people, who are either a mix of firefighters and politicians, all politicians, or all firefighters, surround you and say "every politician in this circle will have a firefighter to their immediate right."

So is the instance where all three are firefighters possible? It would seem if "every politician in this circle will have a firefighter to their immediate right" defaults to true if there's no politicians it would be, otherwise it wouldn't. Which is it?

Edit for clarification in comments: The proposition "every politician in this circle will have a firefighter to their immediate right" cannot be proven, as there are no politicians in the circle. So if the proposition defaults to true, the firefighters were telling the truth and this specific scenario is possible, otherwise it is not, as this would mean the firefighters lied.

I would imagine in a truth-functional logic all propositions would be able to be given either a true or false valuation otherwise one would use a different logic for those propositions. Not all English sentences need be propositions in a truth-functional logic. Welcome to this SE! – Frank Hubeny – 2018-10-01T19:33:54.590

I don't understand why the propositions should default to anything. We're given at the start that firefighters always tell the truth. If we weren't told that, we wouldn't know anything about the veracity of firefighters in this problem. – David Thornley – 2018-10-01T20:19:10.940

The proposition "every politician in this circle will have a firefighter to their immediate right" cannot be proven, as there are no politicians in the circle. So if the proposition defaults to true, the firefighters were telling the truth and this specific scenario is possible, otherwise it is not, as this would mean the firefighters lied. This is what confuses. – Josh – 2018-10-01T20:33:55.433

Are unicorns blue or green? - is a well formed proposition but it doesn't have a truth value because it doesn't refer. – Mozibur Ullah – 2018-10-01T21:17:47.953

You are mixing up terms. You have likely been taught or heard all propositions are true or false with no other possibility. YOU are now thinking about well that means I must be Aware of that value, which is WRONG. In science claims are sense verifiable. Philosophy does not require sense verification. Truth values for literal meaningful statements are sense verifiable. That is the statement has to make sense in grammar and follow rules of syntax in a given language. In Mathematical logic you dont know the value of x. We assume x is positive because there is no explicit negative sign. – Logikal – 2019-01-31T21:27:50.303

The way a proposition is written expresses its truth value whether you can test it or not. Just like in math I must blatantly place a minus sign in front of x for x to be negative (-x). If I just write x without any sign the proposition is stated to be true which is NOT SAYING the proposition is literally true. I can say All swans are white and find out later the proposition is false. Do not confuse objective truth values for real world values. Objective truth never changes, but science can. An objective true proposition can never be false. Contingent propositions can change value with time – Logikal – 2019-01-31T21:41:11.140

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"Every politician in this circle will have a firefighter to their immediate right" is typically coded into predicate calculus as ∀x∃y(P(x) → F(y)∧IR(x,y)). If there are no politicians in the circle P(x) is always false, and, by the convention about the material conditional, when the premise is false the conditional is true. So it is not that the proposition can not be proven, it is that the conditional "defaults" to true by the convention about false premises. That firefighters tell the truth will then be true in this case, but there is no need to assume that they always do.

This said, one can instead interpret conditionals existentially, and insist that objects spoken of in the premise must exist. This is related to the question of existential import that was much discussed by logicians in 19th century. Under the (modern reading of) existential import one makes a different translation of the sentence into the predicate calculus, one that explicitly adds the existence claim. Namely, ∃xP(x)∧∀x∃y(P(x) → F(y)∧IR(x,y)). In this case in the absence of politicians the proposition comes out as false. But this interpretation is almost never used today.

Ironically, the historical discussion was not about whether universal propositions (like "all politicians are liars") imply existence, but whether particular ones do (like "some politicians are liars"). On modern view the only way to express such propositions is by using the existential quantifier, so the answer is trivially yes. But in Mill we read

"That the employment of [the word "is"] as a copula does not necessarily include the affirmation of existence, appears from such a proposition as this: A centaur is a fiction of the poets; where it cannot be possibly implied that a centaur exists, since the proposition itself expressly asserts that the thing has no real existence". (System of Logic I.iv.1)

The modern view was introduced by Brentano in 1874, who argued that "sick man" is just a combined concept with no existential import, but it becomes more when "is" turns it into a sentence. So "some man is sick" has the same meaning as "sick man exists". Even before that in private correspondence Brentano managed to convince Mill, who wrote in 1873:

"You did not, as you seem to suppose, fail to convince me of the invariable convertibility of all categorical affirmative propositions into predications of existence. The suggestion was new to me, but I at once saw its truth when pointed out."

But the real reason for the triumph of Brentano's view is that it fits the modern predicate calculus. As for centaurs and other fictions, some artificial devices have to be deployed, like paraphrase or a separate existence predicate, see What are the counterexamples to Kant's argument that existence is not a predicate?

Amazing answer, well written and easy to follow. Thanks. – Josh – 2018-10-01T22:02:51.390

The first part is technically wrong (but doesn't change the outcome). P(X) is never evaluated if there is no X, the all quantor is true if there no (counter-)examples. Think of it as conjunction of all examples: TRUE AND P(x1) AND P(x2) AND ... - if there are no x then this is simply true. You can't say P(x) is false if there is no x. – kutschkem – 2018-10-02T07:01:41.660

@kutschkem Predicate is evaluated on the available universe, which in this case will only include firefighters, so P(x) will be false for all x. In predicate calculus the universe is traditionally stipulated to be non-empty. – Conifold – 2018-10-02T17:54:53.900

@Conifold Ah yes I see, I thought the Universe x is taken from is just the politicians, and didn't read enough to see that politician is not the universe but a predicate. – kutschkem – 2018-10-04T07:39:32.173

@Conifold Re: "By the convention about the material conditional, when the premise is false the conditional is true." It is not merely a convention; it is a provable fact of classical logic. Using the most elementary methods of proof, we can prove that if A is false, then A implies B must be true for ANY proposition B. See my formal proof at http://www.dcproof.com/ImpliesLines3-4.html

– Dan Christensen – 2018-10-05T02:55:43.650

@DanChristensen The convention about the material conditional is one of the things that makes classical logic classical. Of course, one can derive it from other things by assuming something else, that conditional is determined by truth values, for example. – Conifold – 2018-10-05T21:10:30.760

@Conifold The important point in such foundational matters is often how these results are derived, using what principles of logic. See my proof at the above link. There I make use of only conditional (direct) proof, proof by contradiction, introducing & (joining two statements) and removing double negation. I do not make any use of any of the usual definitions of material implication. – Dan Christensen – 2018-10-06T01:38:18.960

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If we take propositions to be true or false of reality, then a true proposition could, at least in principle, be known as a fact, while a false proposition could not. However, we may know propositions as such without knowing whether they are true or not, such as "physical space is three-dimensional". Deductive logic is essentially the science of consequences. As such, it is irrelevant whether any particular proposition is true or false. It is enough to be able to assume propositions either as true or as false. In practice, this is done through the analysis of logical cases, where a case is determined by which of the propositions we are considering are assumed true and which are assumed false.

In your example, to talk of "politicians" is implicitly to assert that there exists something which is a politician, i.e. something which exists and is asserted as having all the properties, qualities, characteristics you have in mind when you talk of a politician, for example that of being a human being or being "in this circle". The logic of this assertion that there exists politicians is independent of the actual existence of such politicians. However, logic is meant to be applied to real situations and should provide meaningful deductions no matter what. If there is no politicians in the situation concerned, then the proposition that there exists politicians will be false.

The phrase "every politician in this circle" in your example can be clarified by adding a conditional making explicit the implicit assumption that the proposition only applies if there is at least one politician in the circle: "If there is a politician in this circle, then every politician...". Then, it is up to the logic you use to produce the kind of deduction you expect, and no contradictions.