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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Honestly have no Idea how to prove A v ¬¬B from A v B (LPL Q. 6.18)

Premise A v B Goal A v ¬¬B Please help. It seems so self evident but I don't know how to get there.
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60 views

How to define a new logical language

There are a number of logical languages defined that are so called flavors of a certain logic (for example, flavors of First Order Logic). Such new logical languages sometimes extend or restrict the ...
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What is the difference between logical consistency and logical entailment in deductive logic?

I am having a little trouble sorting out two definitions from the first chapter in my logic textbook, The Logic Book by Bergmann, Moor and Nelson. I am under the impression that a set in a sentence ...
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54 views

How to translate “No dolphin sings unless it jumps” into predicate logic?

i have a silly logic question again. How would you translate the following sentence into predicate logic? No dolphin sings unless it jumps. My guess is that it is an E-sentence of the form "no A ...
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67 views

How to translate “Only dogs and dolphins jump if petted” into predicate logic?

The Logic2010 software provides exercises for the symbolization of English sentences. But i'm stuck with one symbolization, which closely resembles another one I solved correctly. The first one ...
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1answer
58 views

What is the meaning of free variables?

In some books I see something like: phi(x1,...,xn) with free variables among x1,...,xn. An example of this is on 2nd paragraph, page 25 of the book Knowledge in Action. I have included a snapshot of ...
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1answer
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Logical, semantic and self-referential paradoxes: The Truth teller and the Liar (draft) can an expert on the matter give feedback?

Title: Logical semantic and self-referential paradoxes: The Truthteller and the Liar (draft, informal) (major) assumption: A statement is either true or not true (law of excluded middle, classical ...
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1answer
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Fitch Proof - Arrow's logic of preferences

I've been stumped on this one question in particular for several days now and I'm hoping to get some help on where I'm going wrong. Given the following premises: ∀x∀y(StrongPref(x,y)→ ¬StrongPref(y,...
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What exactly is a first-order logic?

Can someone explain in simple terms what exactly is a first-order logic? From my amateur standpoint, I think that first-order logic is a some kind of a system of symbols and general logical rules and ...
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1answer
64 views

Do computer languages instantiate only predicate calculus?

All the computer languages I'm familiar with, be they imperative or declarative have the same core mechanics (arithmetic and logic).They have the same loops, conditionals etc. Whatever the language it ...
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A question about possibility

If A, then B ~ A So, possible that B Valid or not? My take: Not valid. Reason: Valid means if all the premises are true, the conclusion must be true That means adding new information should not ...
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1answer
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Proof Tree to Fitch Proof

I was wondering if anyone could help me on a proof I've been working on: I was able to check that it is valid with a proof tree generator (prooftools): However, I still haven't figured out the proof....
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Fitch Question Please Help Me [closed]

I'm having trouble understanding writing out a proof. The proof I'm trying to work with is : [![enter image description here][1]][1] How do I reach this goal? Which rules do I use and with which ...
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66 views

Fitch Questions Please Help Me

I'm having trouble understanding writing out a proof. The proof I'm trying to work with is : How do I reach this goal? Which rules do I use and with which support steps to each rule (proofs to prove ...
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3answers
673 views

How is Wittgenstein’s “notorious paragraph” about the Gödel's Theorem not obviously correct?

Timm Lampert quotes from Wittgenstein's "notorious paragraph" (§8 of Remarks on the Foundations of Mathematics, Appendix 3) in http://wab.uib.no/agora/tools/alws/collection-6-issue-1-article-6....
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Why do we need model theory to express semantics?

https://en.wikipedia.org/wiki/Model_theory Why can't semantics be directly expressed in the formal language? This is the key part of model theory that I don't understand: https://www.lesswrong.com/...
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Hard Predicate Proof Help

I have been working on this proof for over a week now, and I can't seem to figure it out: Pd ⟷ (Hj & Mj), Gsd, ∀x∀y∃z(((Gxy & (Py ➝ Pz)) & Rxyz) ➝ Gxz), Pe ⟷ ∀x(Hx ➝ Mx), Rsde |- Gse I ...
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Can classical logic have deduction with infinite steps

I've been reading the Stanford Encyclopedia of Philosophy article on classical logic, and I've been confused about Theorem 9, and the preceding statement. They mention how (*), the clause which ...
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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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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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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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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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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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55 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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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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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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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
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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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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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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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125 views

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
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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: