Questions tagged [logic]

For questions about logic, whether it concerns syllogistic logic, mathematical logic or the nature of logic itself.

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Is it okay that vacously true statements can be used to prove apparent contradictions?

I am reading How to prove it by Velleman, and something is bothering me. On page 74, he states that "all unicorns are green" is vacuously true and so is "all unicorns are purple." ...
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Understanding "unless"

We are trying to understand the English word "unless". Consider the following statement: "I will not go to Best Buy unless I need a TV". It appears that I can rephrase the ...
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Erotetic quantifiers?

I can't find it, but I have a distinct memory of there being an SEP article that says something like as if the ordinals were there before the sets in relation to a question about whether ordinals (or ...
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Distribution step in resolution proof

I'm struggling to understand a step in resolution proof at http://www.ale.cs.toronto.edu/docs/ref/ale_trale_ref/ale_trale_ref-node44.html (man(X) --> mortal(X)) AND man(socrates) (man(socrates) --&...
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Given a premise and ten axioms about a world with three shapes T, C, and D, how to prove that either all objects are T, all are C or all are D?

I'd like to prove the following argument: Premise: ∀x SameShape(x,c) Conclusion: ∀x Cube(x) | ∀x Tet(x) | ∀x Dodec(x) From the following axioms The following axioms say that any object can be exactly ...
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How to write a formal proof of Euclid's Theorem, specifying every logic rule used?

Euclid's Theorem is the statement that there is no largest prime. We might put it this way in first-order logic: ∀x ∃y (y >= x & Prime(y)) An informal proof is pretty straightforward: Let n be ...
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Help on proving that there is at most 1 x such that A(x)

I'm looking for someone that can explain this lecture slide to me in a more clear language. I have an exam in a few days and i just cannot wrap my head around why this means that there is only one x s....
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Language, Proof, and Logic: Problem 11.26 How to give both possible translations of an ambiguous English sentence into FOL?

In the book "Language, Proof, and Logic" there is a problem 11.26 about translating ambiguous sentences. Given an ambiguous sentence in English, we should provide two translations, one for ...
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How to understand scope of action of a quantifier in a quantified sentence that is a translation of an English sentence?

Let's try to translate the English sentence: "Everything to the left of every tetrahedron is also to the left of every dodecahedron" Consider the predicates Tet(x): x is a tetrahedron LeftOf(...
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9answers
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What is the idea behind "p or not p" being a tautology?

Most (all?) logic books consider "p or not p" to be a tautology, hence always true, and this is usually stated without any further discussion. (I never gave it a second thought.) In common ...
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Causation with Inductive/Deductive Logic

I'm sofa-philosopher and I have a case I can't quite dismantle. My friend (let's call him John) just made a statement that I find philosophically weak, and I'd like to run it thru you guys. So, John ...
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Does logic really exist [closed]

Does logic apply to everything? I mean, are there some thing that exist outside the realm of logic, and are there things that can be used to explain things other than logic?
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Do individual obligations cease to exist when discharged?

While I was thinking about a (hopefully possible!) system of deontic logic with a prospective and retrospective OB operator respectively(!), I was fairly uncertain, maybe even confused, about how I ...
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Nested Quantifiers Proof - Logic

When I prove this: -∃x.P(x) ⊢ ∀x.-P(x) [True] I did it like that: ∀x.-P(x) ⊢ ∀x.-P(x) because (negative ∃) -∃x.P(x) becomes ∀x.-P(x) so that we can say that it's true. However, I didn't ...
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Language, Proof, Logic - Problem 7.32 solution verification: exclusive disjunction introduction and elimination formal rules [duplicate]

In the book "Language, Proof, and Logic" there is the following problem 7.32 that doesn't have an available solution that I can check. Later on in the book, there is the following comment: ...
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What does it mean, intuitively and then also precisely, that a particular English word is not truth functional?

What does it mean, intuitively and then also precisely, when we say that a particular English word is not truth functional? Let me present some examples. Example 1 As far as I can tell from a book I ...
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Proofs of propositional logic truth tree rules in natural deduction?

It is a great irony of natural deduction that some of the most seemingly obvious inferences are also some of the trickiest to prove! So far, I haven't been able to prove the following, and I'd greatly ...
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Is every argument an enthymeme?

Since any valid argument requires logic, any valid argument requires the unstated premise that logic exists. So isn't every argument an enthymeme?
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Do all true statements express the same proposition?

Do all true statements express the same proposition? I know that, for example, the statements "2=2" and "1+1>1" are distinct sequences of symbols. However, I want to know, do ...
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30 views

How to prove (P → Q), (P ∨ R), ¬Q ⊢ R using Fitch interference rules [closed]

P>Q:PR PvR :PR ~Q :PR -R:AS P:vE2,4 Pv-R:vI5 R:
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How to interpret "P ⟺ Q is true if and only if the first-order logic sentence P ↔ Q is logically necessary"?

I'm learning about the two truth-functional connectives "material conditional" and "material biconditional". I came across this particular snippet in a book: An important fact ...
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How is the translation of ∧ into → done in infinitary logic?

Preamble: at first I thought I had been extra-clever in arriving at the idea of a deontic logical explosion, but I then found out that a very simple (and identically named) example of the phenomenon ...
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Does analysis start with a premise or conclusion?

I'm trying to perform an analysis but I'm unsure which way around the premise and conclusions are. Here is the example: "The circulatory system is able to circulate blood.". Now I'm trying ...
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Proof of the Model Universe Theorem: Proving the invalidity arguments in quantifier logic

I am studying how to prove an argument in quantifier logic is invalid. The textbook I am using by Virginia Klenk claims that you can use a Model Universe that contains a finite number of objects to ...
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Prove that if S tautological consequence of P, S tautological consequence of Q, then S tautological consequence of P | Q

Consider the following argument: S is a tautological consequence of P. S is a tautological consequence of Q. Therefore, S is a tautological consequence of P | Q. I wish to give an informal proof of ...
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What, if anything, is the difference between studying logic mathematically and studying it philosophically?

There seems to be a distinction between studying logic mathematically and studying it philosophically and, in practice, it is reasonably clear which framework one is using when one studies logic. I've ...
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McGee's argument on restriction of consistent instances of T-schema

Could someone help me understand what is McGee's argument on restriction of consistent instances of T-schema about please? This is gonna be messy so please bear with me. Halbach and Holsten's "...
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2answers
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Why is any sentence a logical consequence of a set of inconsistent premises?

If a set of premises is inconsistent, there is no situation that makes all the premises true simultaneously. Given a sentence S, there is no situation in which a conjunction of a set of inconsistent ...
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2answers
114 views

How to understand a proof by contradiction in minute detail?

I am following the course "Language, Proof, and Logic" from Stanford on EdX. I am trying to understand proof by contradiction specifically. I understand the gist of this type of proof, and I ...
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1answer
132 views

Why should universal generalization work for abstract objects?

I am reading a logic book in my free time and usually the inference rule of universal generalization is motivated by real-life examples: Imagine having the statement that all people with brown hair ...
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If A entails C, and B entails C, why doesn’t (A and B) necessarily entail C?

The original question is in Greek letters Γ and Δ, each representing a set of sentences, and φ representing an individual sentence (atomic proposition). The question is from Introduction to Logic by ...
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Is Martin Scorsese guilty of no true scotsman?

Genuine Question: When Martin Scorsese criticised the Marvel Cinematic Universe and directly said that they were theme parks not cinema, was that an example of the no true Scotsman fallacy? ...
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critique of informal logic

I sometimes communicate with a logician X, and today I asked him about a professor Y who works in informal logic, and he told me: Y is part of informal logic, I am not a big fan of that", what ...
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1answer
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Is there a logical fallacy for falsely dismissing criticism as snobbery?

From Richard Nixon to Spiro Agnew to todays Marvel fans, there seems to be a particular penchant for framing light criticism, scepticism or even simple disagreement by others as snobbery/elitism. '...
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1answer
142 views

Non-Deductive methods in philosophical argumentation

I'm a mathematician, and I'm trying to expand my understanding of the philosophical basis of mathematics. Mathematics is very much taught axiomatically establishing deductive theories, but philosophy ...
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Is mathematical induction deduction, and if so, why is it so named?

By induction, I mean this screenshot from this Youtube video Abduction as an Aspect of Retroduction | Chiasson, Phyllis | Commens Induction: The prefix “in,” also from the Latin has to do with ...
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2answers
105 views

Is there a difference between contradictory and the opposite?

Is there a difference between contradictory and the opposite? In natural language, we have the idea of opposite such as 'The opposite of good is evil'. In logic, we can represent that symbolically. ...
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Do the rules of basic logic have to presupposed to gain philosophical knowledge?

First, let me apologize if my question is nonsensical in any sense. I do not have any philosophical training whatsoever, but I am really interested in some philosophical questions. I was thinking ...
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The application of logic to the metaphor of glass half full or half empty [closed]

If there is optimistic, pessimistic, and realistic, what would be the most logical way of defining unrealistic? For example, the optimistic says the glass is half full. The pessimistic says the glass ...
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1answer
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If humans lack non-deterministic free will, is rationalism invalid?

I have an argument regarding the validity of rationalism in a universe without non-deterministic free will. If human being's theories, thoughts and ultimate conclusions are inevitable, (bound by ...
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Is the argument that God can't be omnipotent, omniscient and all good invalid because omnipotence would let God violate logic

I have heard it argued that a god can not be all-benevolent, omniscient and omnipotent at the same time. My question is, is this a valid logical argument. Given the premises I would argue that there ...
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3answers
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Is killing an suffering ant better than to leave her suffering? [closed]

In the morning, my mother accidentally stepped on an ant. Usually, the ant would die instantly, but this time it was suffering for a minute or so. I noticed and asked my mom to kill it so that it didn'...
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How can circles be a subclass of ellipses? Popper [closed]

My question comes from: 36 Levels of Universality and Degrees of Precision from The Logic of Scientific Discovery Excerpt from the chapter 36: We have the following natural laws: p: All heavenly ...
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Help me solve this predicate logic proof theory: -(∀z)(∃y)Tzy ⊢ (∃z)(∀y)-Tzy

-(∀z)(∃y)Tzy ⊢ (∃z)(∀y)-Tzy Using Lande, (AI,EI,AE,EE,->E,->I,-I,--E,^I,^E,vI,vE) I cannot figure out the proof on this sequent.
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How can we know (with absolute certainty) that our logic is correct?

How can we know (with absolute certainty) that our logic is correct? Even statements like Descartes’ “I think therefore I am” relies on our logic: that I exist is a logical consequence of my ...
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Correct wording for analysis? What is the nature of this relation?

I'm trying to analyse the statement "The computer has the capacity to perform long division", but I can't decide whether to use the connector because or therefore. Which one is more fitting ...
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Why is the law of the excluded middle not a exclusive disjunction?

So the law of the excluded middle, as I have read in every logic textbook that I have read, has been ( ϕ ∨ ¬ ϕ ) , but this seems somewhat unintuitive, since I was under the impression that the ...
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131 views

How to say “everyone respects someone” and “someone respects everyone” in quantified logic?

I’m having a hard time with this logic obviously isn’t my thing
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3answers
73 views

Would you recommend the book "Philosophical devices: Evidence, Probability, Possibility and Sets" by David Papineau?

Would you recommend the book "Philosophical devices: Evidence, Probability, Possibility and Sets"? I am not familiar with the author, David Papineau, and would like to ask for an opinion ...
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Laws of logic literature recommendations

What are some books/papers/articles I could read to develop an informed perspective on questions like "where do the laws of logic come from? Do they have a deep connection with the structure of ...

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