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13

In general □□ p and □ p are very different. Thinking in terms of Kripke frames, they only obviously coincide if the accessibility relation is transitive. This is true for Kripke frames validating S5, but not in general. Consider a frame with three worlds a,b,c where a sees b and b sees c but a doesn't see c: then given a valuation making p true at b but ...


5

You said in a comment that you were referring to the material conditional, not other notions of if/then like the antecedent being a cause of the consequent, or the antecedent logically implying the consequent. So let's get rid of the if/then structure and write them explicitly as material conditionals: A) "I stay" -> "I eat fish" B) "I didn't stay" -> "I ...


5

According to the research of the Natural semantic metalanguage project, the problem of circular definitions is solved through the identification of semantic primes, the basic blocks of meaning which are shared by all languages and cannot be meaningfully subdivided. Any attempt to define a prime will end up circular or more convoluted than the word itself. ...


5

There's a "purely algebraic" approach, albeit one of dubious success: namely, we take the view that negation is characterized by explosion (avoiding non-contradiction since the latter is negative). We work in the context of abstract deductive systems. There are many different notions of such floating around in the literature; what I'll mean in this answer ...


5

To add to the existing answer, there's also epistemic logic, where □ is interpreted as knowledge (relative to a given subject). Possible worlds in such a system are worlds that are consistent with the subject's current information. Since necessity is truth in all possible worlds, it means in epistemic logic that if something is necessary then it just ...


4

Reductio ad absurdum is a valid argument form and inference rule in classical logic. It says: given some background assumptions, Γ, to show that some proposition P is false, we can show that Γ and P lead to a contradiction. Symbolically, for any propositions P and Q, and any set of propositions Γ (including the empty set): If Γ, P ⊨ Q ∧ ¬Q, then Γ ⊨ ¬P ("⊨...


3

In standard propositional logic, where “if…then…” is the material conditional, “if A then B” and “if not-A then B” can certainly both be true together. Indeed, “(if A then B) and (if not-A then B)” is equivalent to simply “B”. Your everyday example with staying vs. eating fish is a bit misleading, because it pushes us towards thinking of the subtler ...


2

I think it is useful to stop thinking about time as a quantity. Time is simply cause-effect relationships. Yes, time slows down in a mathematical way as you approach speed of light, but this can be explained by a slowing of the physics giving rise to cause-effect relationships. Try this out: time


2

And/or is a rhetorical term used to indicate the “inclusive or”. Rhetorically, what you’re doing is indicating the inclusive use by adding extra redundancy; you’re saying “or, and also possibly and”. “Or/And” is not a term in common rhetorical use, so it seems fair to say that neither of your above options applies if we’re going by either a “sense ...


1

There is no fallacy here since a fallacy involves a logical error in reasoning and I can't see that your friend commits that. More than that, your friend's conduct is capable of defence on grounds of actual consequence utilitarianism. This position is not open to the 'what if everyone did the same?' objection since it rejects the requirement hypothetically ...


1

Assuming SL is Sentential Logic and you may use a version of Natural Deduction rules. You have a nested conditional sentence, which indicated that you should nest Conditional Proofs (CP). Look to Rules for "→" and in particular the rule of →I (conditional introduction).


1

Preface Please consider reading this answer in the voice of a highly motivated and potentially slightly crazy University Professor which talks about the Topics he loves with great passion: https://images.app.goo.gl/bbcQeEtYZxBRrdew7 (That's not me, but it makes my point.) I will not link to external third party evidence that the contained thoughts are true....


1

Aahh.... I see what happened. There is a little known fact which explains the entire argument. Mathematicians are almost the only people in this world who would understand the logical fallacy committed. Humans have more than one "equality relation" in our models of thought. An "equality relation" is a definition of what it means for x and y to be: ...


1

This has nothing to do with reductio ad absurdum. The comparison is made to show that the claim of the existence of god cannot be falsified. I.e. one cannot disprove its existence, similarly as one cannot disprove the existence of fairies, unicorns, the flying spaghetti monster, the many world interpretation and the fact that you only exist in the ...


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