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65

The issue here, as it often is, is that colloquial English is horribly ambiguous, which makes any sort of precise and rigorous discussion difficult. But with sufficient effort, it is possible to make claims precisely, and once you do that, the problem disappears. Alice states that the sky is blue. Bob states that we live in a simulation. Let's assume for ...


8

Yes, an argument with false premises and a true conclusion can be valid. For example: All cats are human Socrates is a cat Therefore, Socrates is human The argument has false premises and a true conclusion. But the argument is valid since it's impossible for the premises to be true and the conclusion false. In other words, if the premises are ...


6

Consider what the second person said: You cannot prove that we are not in a simulation and that anything is real, hence you cannot prove that your claim was true. This may be an example of an argument from ignorance. Here is how Wikipedia describes it: Argument from ignorance (from Latin: argumentum ad ignorantiam), also known as appeal to ignorance (...


5

First, let's observe (in response to a comment) that we can get infinitely nonequivalent sentences; this isn't what you want, but it demonstrates that the situation is nontrivial. Namely, we can talk about cardinality, via the following statements which are easily seen to be appropriately expressible: N(n): There exist x_1,...,x_n which are all distinct. ...


5

Parmenides introduces an early version of the problem of negative existentials. In modern times, this has been construed as a problem about the relationship between reference and meaning and the linguistic mechanics of singular terms, existential quantification, de re/de dicto distinctions, etc. For Parminedes, of course, the problem appears as a problem ...


4

Throughout, I'm assuming that ZFC is consistent. There are a lot of confusing points here, and I don't really understand what you're setting up with bijections. However, I believe the key mistake you make isn't actually related to sizes of sets at all, but rather a serious misapplication of the incompleteness theorem. The language about infinite sets and ...


4

"Valid" in logic is just a technical term meaning that if the premises are true, the conclusion logically must be true as well. As you pointed out in another question, a special case of this is when the conclusion is a tautological sentence, in which case it doesn't matter what the premises are. For non-tautological conclusions, though, validity is ...


4

No, that argument is not logically correct because knowledge need not be absolute, but can be relative. To know the sky is blue means to connect the perception of the atmosphere that we call the sky with the quality we call blue. If the world is a simulation or not, the two perceptions are relatively the same. This is common in science: The work done ...


4

Yes, the existential quantifier expresses existence. If you assert that Some pegasus are flying then you do assert that pegasuses exist, at least by the classical logical treatment of the existential quantifier, and I would claim also by the intuitive understanding of the sentence. If there are some pegasuses which are flying, then well, there are some ...


4

There is a logic called bi-intuitionistic logic, which combines elements of intuitionistic and dual intuitionistic logics. It includes a strong intuitionistic implication connective and the corresponding strong negation of intuitionism, i.e. that ¬A is equivalent to A → ⊥. It also includes a dual of implication, which is a subtraction or exclusion connective ...


3

There are two questions. True or False? If monkeys can fly, then 1 + 1 = 3. The antecedent of the conditional, "monkeys can fly" is false. So is the consequent, "1 + 1 = 3". In classical truth-functional logic the conditional connecting these two sentences also has a truth-value. Wikipedia describes this "material conditional" as follows: The material ...


3

The answer is suggested by the quote that you provided: Logical equivalence is different from material equivalence. Formulas P and Q are logically equivalent if and only if the statement of their material equivalence (P ↔ Q) is a tautology. There is a difference between being true and being a tautology. Once you see this you can see the difference ...


3

The classical syllogisms consist of propositions which use only simple (ie, one "variable") terms in the subject and predicate, so any argument containing propositions which use complex terms isn't a valid (classical) form, although it can still be handled within term logic by extended inference rules. A complex term can be any Boolean expression. Being a ...


3

Unless one is considering a "free logic", classical first order logic assumes all models that would be permitted to provide countermodels have a non-empty domain. So there exists a member of the domain which one may name t. Let x be a variable representing any member of the domain. Here is a proof of the above using a Fitch-style proof checker: The first ...


3

Irving Copi calls propositions such as "James' son is a man" or "Socrates is mortal" singular propositions. They are nonstandard-form propositions. They need to be translated into standard form categorical propositions which relate classes before being used in categorical syllogisms. He recommends the following: (page 239) To every individual object there ...


3

The first observation is that moving the ∀b quantifier in the second formula to the front is unproblematic since we can safely move a quantifier from the right-hand side of an implication to the outside provided we don't bind any variables that were free before, which is the case here, so ∀a,b [(P(a) ∧ ¬R(a) ∧ S(b)) → G(a,b)] (first formula) is ...


3

This is basically equivalent to solipsism, the idea that the only thing you can be sure really exists is yourself, and everything else could be just a figment of your imagination. It's fundamentally irrefutable: any argument can be dismissed as "That's not real and I could imagine something different tomorrow", "But that's just what the simulator told you ...


2

Here's a skeleton to start you out |_ ~(P ↔ Q) Premise | |_ P Assume | | |_ Q Assume | | | |_ P Assume | | | | Q Reit | | | P → Q Cond. Intro. | | | |_ ... Assume | | | | ... ... | | | ... ... | | | ... ... | | | ┴ Neg. Elim. | | ~Q Neg. Intro | P → ~...


2

One way to approach this is to think of the goal, ¬P ↔ Q, not as a biconditional but as this conjunction of disjunctions: (P ∨ Q) ∧ (¬P ∨ ¬Q). This will allow one to access more inference rules. Then the first step would be to assume the negation of that to derive a contradiction with the premise. (See lines 2-27 below.) Once this contradiction is obtained ...


2

Some of the rules that might be present in another system may have to be derived separately in this proof checker as they are needed. Here is the question: I'm looking for Hypothetical Syll, Constructive Dilemma, Communication, association, distribution, transportation, material implication, material equiv, exportation, and tautology (even though it's ...


2

I take you to be asking why the validity of an argument is defined in such a way that the truth of its premises is irrelevant to its validity. That is a great question. It is natural to think that an argument at least purports to reveal something as true, so that, if it succeeds, that is just what it does. After all, it is natural to mark the conclusion ...


2

A truth table would show this is a tautology, so one can try deriving this without premises. Here is a proof using the proof checker associated with forallx. Something similar should work with Fitch: On line 1, I assume the antecedent of the conditional I would like to derive. The consequent of that conditional is also a conditional so on line 2 I make ...


2

Just so it's said, Most of the things people refer to as (informal) fallacies — No True Scotsman and Strawman fallacies included — are not actually fallacies, but are simply mistakes in language. A fallacies is an error in the structure of the logic involved; a language mistake is derived from semantic content. Apples and oranges... But that caveat aside, ...


2

Here is a forward proof that gets by without contradiction, constructed with the Natural deduction proof editor and checker: Your problem presumably was that you can't just eliminate the existential quantifiers directly to replace them by an individual constant -- this would be unsound, and an incorrect application of the ∃E rule -- but rather the rule ...


2

From the comments: I tried to use Existential Elimination but I can't figure out how to do it properly. Existential Elimination: When given that an existential statement (eg ∃z P(z)) holds, and show that a statement (eg Q) may be derived when we assume a witness for the existential (eg P(c), where c is a term that does not occur within P(z) or Q), then we ...


2

The problem seems to be more a linguistic than a logical one; in particular, the crucial phenomenon here is that of presupposition. In "James' son is a man", the possessive construction "James' son" can be seen as acting as a so-called definite description where one particular individual is identified by the description "that individual which is the son ...


2

Expressions are equivalent if they can be related with an if and only if connection. One way to check if they are equivalent is to use a tree proof generator. Putting these expressions into such a tool will also require one to write a well-formed formula eliminating any ambiguity. Here is the result of the tool showing that the expressions are equivalent ...


2

A person who is claiming to know something is asserting that they meet the criteria for claiming knowledge. There are three requirements for a person to "know" something: They must believe the thing they claim to know. They must have an objectively reasonable justification for that that claim. The thing they claim to know must be true. So if a person asks ...


1

I am contributing an answer to this question because although I agree with Frank's answer, there is more that could be said about why the argument in question seems unsatisfactory, even though it is indeed valid. Traditionally, logic arose as a way of codifying the rules of what differentiates a good argument from a defective one. Back in Aristotle's day, ...


1

How does an extensive definition (or maximum possible analysis) of P differ from logical consequence of P. Logic, as a performance of human beings, does not operate on "full definitions" or on Wittgensteinian "state of affairs". It operates on the form, or ostensive structure, of the argument. a is F; All y that are F are G; Therefore, a is G. ...


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