Tones' is literally applying the material conditional as an interpretation of English language conditionals — Leontiskos
The OP's question was not about ordinary English at all. — Srap Tasmaner
And, yes, the equivalence is per the material conditional. — TonesInDeepFreeze
English as a meta-language regarding formal logic. In that meta-language, 'if then' is taken in the sense of the material conditional. — TonesInDeepFreeze
It is raining
It is not raining
George Washington is made of rakes
Per our definition, this argument is not valid becasue all the premises are true — Hanover
I mentioned it several posts back, but it seems possible to have an invalid argument with necessarily false premises. — Count Timothy von Icarus
Edit:
This is a matter of different modal levels, so to speak, or different domains or levels of impossibility. Tones is committing a metabasis eis allo genos. He is committing a category error where the genus of discourse is not being respected. Contingent falsity, necessary falsity, and contradictoriness are three different forms of denial or impossibility. The definition of validity that Tones favors is dealing in the first category, not the second or third. The domain of discourse for such a definition assumes that the premises are consistent. It does not envision itself as including the degenerate case where an argument is made valid by an absurd combination of premises. An "argument" is not made valid by being nonsense. — Leontiskos
Another one:
"a major topic in the study of deductive logic is validity. This is a
relationship between a set of sentences and another sentence; this relationship holds whenever it
is logically impossible for there to be a situation in which all the sentences in the first set are true
and the other sentence false." [bold added]
https://logiclx.humnet.ucla.edu/Logic/Documents/CORE/LogicText%20Chap%200%20Aug%202013.pdf — TonesInDeepFreeze
The idea that it is a relationship already excludes your reading. If a relationship between A and B must be established, then one must know something about both A and B. Yet you think that merely knowing something about A—that it is inconsistent—proves validity. If an isolated fact about A proved validity then validity would not be a relationship between A (premises) and B (conclusion). This is another source that excludes your view. The other (single-sentence) sources you presented favor my view but do not exclude your tendentious view. — Leontiskos
. . .The validity relation is a relation in the ordinary formal sense of a set of ordered pairs. That is distinct from any of the ordered pairs themself. — TonesInDeepFreeze
You might want to double-check that. — Srap Tasmaner
so you're talking about the principle of explosion? — Michael
Tones is interpreting English-language definitions of validity according to the material conditional — Leontiskos
the material conditional and the consequence relation do not operate in the same way — Leontiskos
You're giving a different reason for why it's valid versus Tones. — frank
Lots of people are not paying attention to the differentiation of arguments for why the OP might be valid. Three options have been given: modus ponens, explosion, and the definition of validity. TonesInDeepFreeze's is the latter... — Leontiskos
There is a reason we don't need an additional implication operator ― that is, one that might appear in a premise, say, and another for when we make an inference. — Srap Tasmaner
Validity is a relationship between premises and conclusion. — Leontiskos
Three options have been given: modus ponens, explosion, and the definition of validity. TonesInDeepFreeze's is the latter... — Leontiskos
In natural deduction systems, if you assume A and then eventually derive B, you may discharge the assumption by writing 'A → B'; this is just the introduction rule for →, and it is exactly the same as the '→' that might appear in a premise. — Srap Tasmaner
I affirm that it is valid by any of these considerations:
(1) Apply the definition of 'valid argument'. — TonesInDeepFreeze
Your interpretation is mistaken because validity is an inferential relationship between premises and conclusion. You would establish an inferential relationship without examining the inferential structure and relations. To say, "The premises are contradictory, therefore an inferential relationship between premises and conclusion holds," is to establish an inferential relationship without recourse to inferential relations. — Leontiskos
They cannot interpret real English — Leontiskos
I mean your post does use two different operators? — Michael
The standard semantics of a judgment in natural deduction is that it asserts that whenever[11] A 1 , A 2 , etc., are all true, B will also be true. The judgments
A 1 , … , A n ⊢ B
and
⊢ ( A 1 ∧ ⋯ ∧ A n ) → B
are equivalent in the strong sense that a proof of either one may be extended to a proof of the other. — wiki
The sequents
A 1 , … , A n ⊢ B 1 , … , B k
and
⊢ ( A 1 ∧ ⋯ ∧ A n ) → ( B 1 ∨ ⋯ ∨ B k )
are equivalent in the strong sense that a proof of either sequent may be extended to a proof of the other sequent. — same
"There are a number[11] of people voting for me for President on Tuesday — Srap Tasmaner
The reason that there is no interpretation where both premises are true is because the premises are inconsistent — Michael
What does footnote 11 say? Because the whole dispute rides on that single word, "whenever." — Leontiskos
Here, "whenever" is used as an informal abbreviation "for every assignment of values to the free variables in the judgment" — same
Tones is pointing out is that anytime there are no cases where both premises are true, the argument will be valid. The premises don't have to be inconsistent for that. They're just never both true. — frank
reductio? — Leontiskos
The one time he did — Moliere
Checking the validity of one argument using another is done all the time. — Hanover
They can never both be true only if they are inconsistent. If they are consistent then they can both be true.
— Michael
@TonesInDeepFreeze is this true? — frank
Couldn't it be:
1. The present King of France is bald.
2. The present King of France is wise.
Therefore: Cows bark.
It's valid, right? — frank
To be sure, one might use disjunctive syllogism to prove that B is A from the contradiction, but that doesn't make the form of the above valid. — Count Timothy von Icarus
But surely we don't want to claim that the fallacy of exclusive premises is true just in cases it is possible for its premises to be true. — Count Timothy von Icarus
Wrong — TonesInDeepFreeze
TL;DR. If you think of the material conditional as a containment relation, its behavior makes sense. — Srap Tasmaner
Material implication is the way it is for much the same reason that humans are the way they are given Epimetheus' mistake. When the logic gods got around to fashioning material implication they basically said, "Well if the antecedent is true and the consequent is true then obviously the implication is true, and if the antecedent is true and the consequent is false then obviously the implication is false, but what happens in the other cases?" "Shit! We only have 'true' and 'false' to work with! I guess we just call it 'true'...?" "Yeah, we certainly can't call it 'false'."
I haven't thought about this problem in some time, but last time I did I decided that calling the vacuous cases of the material conditional 'true' is like dross. In a tertiary logic perhaps they would be neither true nor false, but in a binary logic they must be either true or false, and given the nature of modus ponens and modus tollens 'true' works much better. It's a bit of a convenient fiction. This is not to say that there aren't inherent problems with trying to cast implication as truth-functional, but it seems to me that an additional problem is the bivalence of the paradigm. — Leontiskos
...Soon after this, Frege expresses frustration that 28 years after he introduced the material conditional mathematicians and logicians continue to resist it as something bizarre! — Leontiskos
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