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Questions tagged [symbolic-logic]

For questions related to symbolic logic, also known as mathematical logic. Topics might range from philosophical implications of metamathematical results to technical questions.

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How could the Tarski Undefinability Theorem be Refuted?

When we specify that True(x) selects the consequence of the subset of the conventional formal proofs of mathematical logic having true premises then True(x) is always defined and never undefinable. ...
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369 views

Deductively sound formal proofs of mathematical logic? [closed]

When we specify that True(x) is the subset of the conventional formal proofs of mathematical logic having true premises then True(x) is always defined and never undefinable. When true is defined as ...
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1answer
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Do you know of any mathematical theorem whose proof relies on the use of the principle of explosion (ECQ)?

Ex contradictione (sequitur) quodlibet (ECQ) is almost universally recognised in mathematical logic as a valid inference. In symbolic logic, this inference is usually expressed in the following way: ...
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1answer
115 views

Checking the validity of the logical conclusion gleaned from a heated conversation

I have two friends - call them John and Jane. I was recently privy to an argument concerning a book between John and Jane that went like this: John: This book did not make a single coherent, ...
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Fitch-style natural deduction

How to prove the following questions? (a) p from assumption ¬(p → q) (b) ¬¬p → p from no assumptions.
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1answer
75 views

trouble with rules of inference practice problems [closed]

Prove the following symbolized arguments applying the appropriate rules of inference: 1) P ∨ Q = M ⊃ ¬ Q M =conjunction Therefore P 2) (P V Q) ∧ ¬ Q P ⊃ R =...
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3answers
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How should I use the propositional logic rules for → and ↔?

My question is how should I use the propositional logic rules for → and ↔ (although other rules may be required) to prove the following: A → B, B → C ⊢ (AvB) → C A ↔ B ⊢ ¬A ↔ ¬B Please use the ...
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2answers
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Known self-evident unproven logical truths

Is there any authoritative source for all known self-evident logical truths that most specialists would agree are true although they can't be proven? There are many different axiomatic systems, and ...
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1answer
52 views

Does 'until' imply a conditional with a negative consequent?

Suppose a father tells his kid that he can play video games whenever he wants. Then, one day, when the kid fell sick, the father told him that he can play video games until he recovers. Does this '...
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1answer
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Symbolic logic and rules of inference: two questions

Question one: (C>D) & (D>B) (B>D) & (E>C) (D>C) BvE ∴ DvB ? ? ? ? DvB I'm fairly sure this questions has constructive dilemma at the end, but after four hours of working on these two ...
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1answer
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Prove the rule that proves X(P) from X(a) preserves derivability in modal system K

I'm trying to solve a problem which asks me to show that the meta-rule defined by deriving X(P) from X(a) preserves derivability (i.e. if ⊢X(a) then ⊢X(P) in modal system K, where a is a sentence ...
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119 views

S5 proof of ⊢◻(◻P→◻Q)∨◻(◻Q→◻P)

I'm trying to construct an S5 proof of ⊢◻(◻P→◻Q)∨◻(◻Q→◻P). I know that ϕ∨ψ is equivalent to ~ϕ→ψ, and so what I'm really trying to derive is ~◻(◻P→◻Q)→◻(◻Q→◻P) (which is equivalent to ◊~(◻P→◻Q)→◻(◻Q→◻...
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1answer
66 views

Axiomatic proof of ⊢ □P → □◇□P in S4

As the title explains, I'm trying to give an axiomatic proof of ⊢ □P → □◇□P in S4. This is simple to prove in B, but I'm struggling to see how it's done in S4. I'd really appreciate any help you ...
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1answer
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Equivalence of strings of modal operators in modal logic

I'm trying to solve a question which asks me to show that for any two finite strings O₁ and O₂ of □s and ◊s, (e.g. □□◊□◊□), that i) if O₁≡O₂ then OO₁≡OO₂ and ii) if O₁≡O₂ then O₁O≡O₂O where O is ...
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151 views

Semantic expressiveness of modal logic

I am wondering how much of the semantic of basic philosophical questions can be expressed by formal arguments in modal logic. Here is one argument I formalised myself: P1 ◇ ∀a, ∃x // GNB(x, a) ∧ ...
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2answers
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Justification of existing methods of formal logic [duplicate]

What is it that mathematicians, and more likely perhaps philosophers, give as an explicit justification that any method of formal logic, which is actually used by mathematicians, or even by automatic ...
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1answer
190 views

Classical logic, symbolic logic, higher-order logic, First-order logic? Learning from scratch

I'd like to ask you a question about logic. I study philosophy in a Spanish Christian university. In the first year, we study logic but it's the classical one, following Aristotle's Organon, the ...
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4answers
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How would i go about using natural deduction to prove this argument is valid?

How would I use natural deduction to prove this argument is correct? It's always either night or day. There'd only be a full moon if it were night-time. So, since it's daytime, there's no full moon ...
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2answers
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How do I input these statements into a truth table generator?

I have tried inputting my problems into several truth table solvers. I keep getting error messages. Which solver should I use and how do I change my statements on the homework in order to prevent ...
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1answer
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Does anyone know how to prove ~ ∀x (Ax→Bx) from Ǝx(Ax & ~Bx)?

Ǝx(Ax & ~Bx) Premise SHOW: ~ ∀x (Ax→Bx) I really appreciate anyone who could help The instructions for the homework were to Prove that the obverse of a particular ...
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1answer
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How to prove (PvQ) & (RvS) : ((P&R) v (P&S)) v ((Q&R) v (Q&S)) by Natural deduction

Another of Tomassi's exercises I can't solve (Logic, page 109, Revision exercise III, 3) (P v Q) & (R v S) : ((P & R) v (P & S)) v ((Q & R) v (Q & S)) I have to use natural ...
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2answers
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proof for relational predicate logic

I have been working on this problem for over an hour and I think I have simply missed something. I need some help. The rules I am allowed to use are the Basic Inference rules (MP, MT, HS, Simp, Conj, ...
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5answers
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What would be an intuitive understanding of Peirce's law?

Wikipedia describes Peirce's law as In propositional calculus, Peirce's law says that ((P→Q)→P)→P. Written out, this means that P must be true if there is a proposition Q such that the truth of P ...
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language proof and logic chapter 13 question 49 Help

Premises: ∃xP(x) ∀x∀y((P(x)∧P(y)) → x = y) Prove: ∃x(P(x)∧∀y(P(y) → y = x)) I've started it but the end is starting to get super muddy and not work out and I don't know where I went wrong.
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4answers
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Symbolic Logic - Quantifier Proof (w/ Conditionals)

I'm not sure if lines 6 - 7 & 8 - 11 are being done correctly. I feel like it's necessary to prove 12 which proves the rest of the problem. I'm a bit stuck on lines 8 - 11. I initially tried to ...
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1answer
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Seeking clarification of how an argument from Aristotle is found fallacious using Frege's quantification tools

G. E. M. Anscombe writes in An Introduction to Wittgenstein's Tractatus (page 15-16): Again, the following fallacious piece of reasoning is found in Aristotle: 'All chains of means to ends must ...
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2answers
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Fitch Biconditional Proof Help?

Hi, I'm starting to learn formal proofs using Fitch, but I'm having a bit of trouble figuring out my arguments. I've generally mapped out the subproofs I was considering to use, but I'm unsure how to ...
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114 views

Can something proved by contradiction always be proved without a proof by contradiction?

Proof by contradictions work by assuming that something is true, and then using logic (along with other assumptions which you know are true) to show that that leads to a contradiction, thus proving ...
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4answers
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How to prove : (( P → Q ) ∨ ( Q → R )) by natural deduction

Here's another of Tomassi's exercises I can't solve (Logic, page 106): : (( P → Q ) ∨ ( Q → R )) I have to use natural deduction and the only rules I know are: • assumptions, • modus ponendo ...
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1answer
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Conditional IFF - Not sure what's wrong

"Not a valid application of the rule". I don't think 7 - 8 is something that really needs to be proven beyond a reit, but I feel like you should be able to... I'm quite confused on proving Cube(a) ...
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2answers
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Symbolic Conditional Help

Premise: (Tet(a) ^ Tet(b)) v (Cube(c) ^ Cube(d)) Cube(c) -> Dodec(e) Goal: ~Tet(a) -> Dodec(e) Anyone have a clue on where to start with this?
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1answer
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How to solve the derivation?

Derive the following without assumptions: ¬∃xFx↔∀x¬Fx How do I solve this derivation?
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Completeness/Soundess of Second Order Logic

I recently read that Gödel's incompleteness theorem entails that second order logic cannot simultaneously hold the traits of: (i) completeness, (ii) soundness, and (iii) effectiveness. However, I saw ...
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2answers
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How does one prove ‘(B→C)→¬A’ from ‘(A→B)∨C’ and ‘(A→¬C)’ in Fitch?

I am trying to work my way through this Fitch proof, and I am not sure what I am doing wrong, but I keep getting stuck no matter what I try. First attempt: Second attempt:
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Recommendation: Second Order Logic textbook

I'm looking into Universalist Realism, Nominalism, Trope theory and the application of Second Order logic to each of them, however I have little/no experience with Second Order logic. Please let me ...
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How to get proof using proof editor and checker

How can I use http://proofs.openlogicproject.org/ or http://logic.tamu.edu/daemon.html to derive the given conclusion from the given premise: (∃x) ( Fx ∙ (y) (Fy → y = x) ) / (∃x) (y) (Fy ≡ y =...
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How do you prove B v A |- A v B?

I am having trouble with how to use the assumption, which I feel that I will need for this proof. If any one can demonstrate or give hints for this proof, I would greatly appreciate it.
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3answers
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McGee's Counterexample to Modus Ponens

I'd like to start off by saying that I have read the other posts in the Math StackExchange and here about this paper, but I think my question is a bit different from those although it does stem from ...
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2answers
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Are Statements with Existential Quantifiers General or Particular?

Consider the following argument: The number 2 is a prime number and is divisible by 2. Thus, some prime number is divisible by 2. The first statement in this argument concerns a particular, i.e. ...
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How did Fitch's opposition to the Russell-Whitehead theory of types turn out since the 1950's?

In a footnote to Appendix C of Frederic Fitch's Symbolic Logic (page 217), Fitch writes about his article, "Self-Reference in Philosophy": It is reprinted here in order to indicate more fully my ...
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157 views

How to prove ~ (~P & ~Q) : P ∨ Q by natural deduction

Here's another of Tomassi's exercises I can't solve (Logic, page 106): ~ (~P & ~Q) : P ∨ Q I have to use natural deduction and the only rules I know are: assumptions, modus ponendo ponens, ...
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2 simple Formal Fitch Proofs

I'm having difficulty proving these. They seem obvious, but I can't figure how to set up formal proofs for them. Could anyone give me clues on how to start them? ¬(P∧¬Q) from the premise P→Q; ¬Q→(R→P)...
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What is an example of a monadic predicate calculus argument that cannot be represented by the 19 classical Aristotelian syllogisms alone?

While reading Wikipedia's description of the monadic predicate calculus, I read the following: Inferences in term logic can all be represented in the monadic predicate calculus. and Conversely, ...
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Proof from tree to steps

I'm able to get the proof in a tree form (it's invalid). Is there a method where I can transform it to steps method indicating the rules of inference and replacement?
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2answers
534 views

meaning of (r .⊃. s ⊃ r) [the syntax meaning]

I'm trying to to determine whether the following is a tautology, contingency, or contradictory: (p ⊃ q) ∨ (q ⊃ p) .⊃. (r .⊃. s ⊃ r) This is school work. I'm getting that it's a tautology, but only ...
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1answer
57 views

Does this symbolic logic proof work?

So, I have this proof: Let a be an open sentential variable. Let K and C be prepositional variables asserted of sentences. Given K(a) <=> C(a) & a C(a) <=> C( C(a)) C(a & b) <=> C(...
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1answer
169 views

Prove transitivity in Fitch

How to prove transitivity in Fitch. Is it Ok? | 1. a = b | 2. b = c | 3. c = c =Intro | 4. a = c =Elim: 3, 2 | 5. b = c =Elim: 4, 1
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237 views

Derive |- [(P>Q)>P]>P using only primitive rules

I've been having issues trying to derive |- [(P>Q)>P]>P in natural deduction using only primitive rules. Wondering if anyone would have a solution to it. Thanks
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1answer
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Logic Either..Or

In the book: "Elementary Logic" authored by Brian Garrett, he has a few examples, one with solution and one without that conclude the following: 1) Either many people will attend the concert, or it ...
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177 views

~(P&Q) derive to ~Pv~Q

I would be grateful if someone could derive, by showing the proofs that: ~(P&Q) derives to ~Pv~Q. The same derivation would be appreciated for |- [(P>Q)>P]>P