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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."
Is the instance where all three are firefighters possible? It would seem as though if the proposition "firefighters always tell the truth" defaults to true, then it would be. Otherwise it wouldn't. Which is it?
logic propositions
edited 2 hours ago
Conifold
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asked 4 hours ago
Dylan
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up vote
2
down vote
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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."
Is the instance where all three are firefighters possible? It would seem as though if the proposition "firefighters always tell the truth" defaults to true, then it would be. Otherwise it wouldn't. Which is it?
logic propositions
edited 2 hours ago
Conifold
33.1k148130
asked 4 hours ago
Dylan
132
New contributor
Dylan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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
4 hours ago
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
3 hours ago
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.
– Dylan
3 hours ago
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
2 hours ago
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
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."
Is the instance where all three are firefighters possible? It would seem as though if the proposition "firefighters always tell the truth" defaults to true, then it would be. Otherwise it wouldn't. Which is it?
logic propositions
edited 2 hours ago
Conifold
33.1k148130
asked 4 hours ago
Dylan
132
New contributor
Dylan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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."
Is the instance where all three are firefighters possible? It would seem as though if the proposition "firefighters always tell the truth" defaults to true, then it would be. Otherwise it wouldn't. Which is it?
logic propositions
logic propositions
edited 2 hours ago
Conifold
33.1k148130
asked 4 hours ago
Dylan
132
New contributor
Dylan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 2 hours ago
Conifold
33.1k148130
asked 4 hours ago
Dylan
132
New contributor
Dylan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 2 hours ago
Conifold
33.1k148130
edited 2 hours ago
Conifold
33.1k148130
edited 2 hours ago
Conifold
33.1k148130
33.1k148130
asked 4 hours ago
Dylan
132
New contributor
Dylan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 4 hours ago
Dylan
132
asked 4 hours ago
Dylan
132
132
New contributor
Dylan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Dylan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Dylan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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
4 hours ago
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
3 hours ago
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.
– Dylan
3 hours ago
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
2 hours ago
add a comment |Â
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
4 hours ago
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
3 hours ago
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.
– Dylan
3 hours ago
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
2 hours ago
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
4 hours ago
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
4 hours ago
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
3 hours ago
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
3 hours ago
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.
– Dylan
3 hours ago
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.
– Dylan
3 hours ago
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
2 hours ago
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
2 hours ago
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
3
down vote
accepted
"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?
answered 3 hours ago
Conifold
33.1k148130
Amazing answer, well written and easy to follow. Thanks.
– Dylan
2 hours ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
"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?
answered 3 hours ago
Conifold
33.1k148130
Amazing answer, well written and easy to follow. Thanks.
– Dylan
2 hours ago
add a comment |Â
up vote
3
down vote
accepted
"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?
answered 3 hours ago
Conifold
33.1k148130
Amazing answer, well written and easy to follow. Thanks.
– Dylan
2 hours ago
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
"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?
answered 3 hours ago
Conifold
33.1k148130
"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?
answered 3 hours ago
Conifold
33.1k148130
edited 2 hours ago
answered 3 hours ago
Conifold
33.1k148130
answered 3 hours ago
Conifold
33.1k148130
answered 3 hours ago
Conifold
33.1k148130
33.1k148130
Amazing answer, well written and easy to follow. Thanks.
– Dylan
2 hours ago
add a comment |Â
Amazing answer, well written and easy to follow. Thanks.
– Dylan
2 hours ago
Amazing answer, well written and easy to follow. Thanks.
– Dylan
2 hours ago
Amazing answer, well written and easy to follow. Thanks.
– Dylan
2 hours ago
add a comment |Â
Dylan is a new contributor. Be nice, and check out our Code of Conduct.
Dylan is a new contributor. Be nice, and check out our Code of Conduct.
Dylan is a new contributor. Be nice, and check out our Code of Conduct.
Dylan is a new contributor. Be nice, and check out our Code of Conduct.
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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
4 hours ago
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
3 hours ago
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.
– Dylan
3 hours ago
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
2 hours ago