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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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Selection of logical connectives {¬,∨,∧,⇒,⇔} in set theory?

Nearly every treatment of set theory, whether Paul Halmos' Naive Set Theory, Herbert Enderton's Elements of Set Theory, Patrick Suppes' Axiomatic Set Theory, etc. introduce a common set of logical ...
7
votes
3answers
651 views

Is it true that (P∧Q≡P)⇔(Q≡⊤)?

Consider the statement (P∧Q≡P)⇔(Q≡⊤) Where P and Q are statements, and ⊤ denotes the tautology (true) statement. It seems intuitively true that the above biconditional statement is true. But I ...
6
votes
8answers
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What is the explicit reasoning behind proof by contradiction?

From my understanding, proof by contradiction consists of the following steps. 1. Show that p -> q, where "->" is the conditional. 2. Show that q is false. 3. Deduce from a truth table that p must be ...
5
votes
3answers
947 views

What exactly are the identity rules in logic?

In first order logic, I have read that there are a couple of identity rules. If I have "a=b" does it mean that I can also write it as "b=a"? Is it true one-way or both? And if I have two ...
5
votes
3answers
518 views

In Fitch, how does one prove “(P → Q)” from the premise “(¬P ∨ Q)”?

It's all in the question really. I am working on a proof in Fitch for a class, but I am very much stuck. I am proving the tautology that "(P → Q) ↔ (¬P ∨ Q)", and I have already finished half of it, ...
4
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5answers
579 views

How to show that (P & Q) v (~P v ~ Q) is a theorem in SD

I’m working on a practice question on my logic textbook. And I’m stuck at this question. This is what I have so far: 1. ~(P & Q) Assumption/ Negated Eelimination 2. P Assumption/...
4
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5answers
141 views

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 ...
4
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3answers
540 views

Symbolic Logic Proof: Leprechauns Exist?

I am reviewing a study guide for an introductory logic course (basic predicate, syllogistic etc.). The problem asks me to symbolize that "leprechauns exist" and prove that it is a logical truth and ...
4
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5answers
363 views

Is there a logical symbol for “why”?

Is there a formal logic symbol for "why"? For example how would you formulate "Why is 2^4 > 4^2?" Could that be formulated in pure symbols of logic if possible? Also, the phrase "what is" can it be ...
4
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2answers
312 views

Subformulas of the WFF (∀x) ((∀y) ((x ∈ y) ∨ (y ∈ x )))

Consider the well-formed formula in set theory (∀x) ((∀y) ((x ∈ y) ∨ (y ∈ x ))). I believe there are 5 subformulas: (x ∈ y) (y ∈ x) ((x ∈ y)∨(y ∈ x)) (∀y) ((x ∈ y)∨(y ∈ x)) (∀x) ((∀y) ((x ∈ y)∨(y ∈ x)...
4
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1answer
801 views

What, at present, are the major unsolved problems of logic?

In the 1900s, Hilbert published a list of 23 (later 24) unsolved problems in mathematics, which sparked increased research into each of them and the subsequent resolution of several of these problems. ...
4
votes
2answers
98 views

need some guidance for this easy symbolic logic question [closed]

Every dog and cat who is well trained is a good pet. (F: a is a dog; G: a is a cat; H: a is well trained; I: a is a good pet.) Here are my options: a) ∀x((Fx∨Gx)∧Ix→Hx) b) ∀x((Fx∨Gx)∧Hx→Ix) c) ∀x(...
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3answers
2k views

Propositional Logic: How to prove the contraposition in the Fitch system?

Given that: p ⇒ q prove that: ¬q ⇒ ¬p using the Fitch system. (This being the proof of the Contraposition)
4
votes
3answers
280 views

P <=> (Q v R), P, -Q⊢R Propositional Logic Question

<=> is bi-conditional, "-" is negation, "v" is disjunction. I can't figure out where to take it from line 4. Negated Q is throwing me for a loop. P <=> (Q v R), P, -Q ⊢ R P <=> (Q v R) P ...
4
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2answers
1k views

Fitch style disjunction elimination

I am having difficulty in formally proving a simple argument. Consider P(x) v Q(x) not P(x) ---------- Q(x) It is easy to see that the argument is indeed valid, but I cannot seem to prove it ...
4
votes
2answers
87 views

help with deductive proof

∀x (Fx ∨ x=c), ¬Fb ∧ Gb |- ¬Fa → Ga So far I don't understand how to switch variables around to prove the result. I've got a subproof set up assuming "¬Fa" in order to derive "Ga". In that proof I ...
4
votes
2answers
2k views

Fitch Proof - LPL Exercise 8.17

I am currently finding the third part of this exercise (Conditional 3) difficult to prove. I was sure that my proof was correct, but the Fitch program is saying otherwise. I am finding it ...
4
votes
1answer
416 views

What it the relationship between Type theory and logic?

I am aware that a similar question was asked about the type theory in the principia, but I'm more interested in what the relationship between, say Martin-Lof Type theory and intuitionistic logic is.
4
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1answer
126 views

Prove ∀w(∀v((v=w∧φ(v))⇔φ(w)))

In this math question of mine, an answer pointed me to this theorem: ∀w(∀v((v=w∧φ(v))⇔φ(w))) which in turn, the answerer stated, implies another theorem: ∃v(v=t∧φ(v))⇔φ(t) which was the fact I ...
4
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2answers
1k views

Disjunctive Syllogism in a Fitch Style System

I'm trying to prove an argument of the form: B ~(C & B) Therefore: ~C. I can expand out ~(C & B) into ~C OR ~B, and with the premise B, it is clear that ~C is the case. ...
4
votes
1answer
134 views

Is there a formalized logic for adpositional connectives?

Certain words in natural language are more amenable to logical formalization. The conjunction "and" or weak conditional "unless" are easily applied to break statements into their constituent atomic ...
4
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1answer
56 views

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 ...
3
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4answers
125 views

How to give proof for Q ∧ R with the premisse ¬(¬¬¬P ∨ P)?

I'm trying to use Fitch to get to an answer, but I'm really confused right now. Can someone help?
3
votes
2answers
308 views

Is there a uniform way of differentiating sufficient and necessary conditions?

I am struggling to formulate symbolic conditional logic rules from basic sentences (studying for the LSAT). It seems that subtle differences in syntax are throwing me off. Is the conditional ...
3
votes
3answers
2k views

Prove P v ~P using most basic rules?

Is there a way to prove P v ~P in basic inference rules? I can't think of where to start because nothing applies to this. I was thinking about usinig Conditional proof, but I don't know what should I ...
3
votes
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 ...
3
votes
2answers
312 views

Step by step natural deduction: (T > E) ^ (A > L) /… (T v A) > (E v L)

I'm having trouble proving the following using natural deduction: (T > E) ^ (A > L) /... (T v A) > (E v L) I checked the answer but I didn't quite understanding the reason why the proof progressed ...
3
votes
2answers
388 views

Implication Introduction formulated as a theorem?

While making a list of the rules of inference for my math students, I came across this list on Wikipedia: I noticed a pattern: for every introduction rule, there seems to be an elimination rule, and ...
3
votes
2answers
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 ...
3
votes
3answers
1k views

Prove A ∨ D from A ∨ (B ∧ C) and (¬ B ∨ ¬ C) ∨ D ( LPL Q6.26) without using --> or material implication

This is a repeated question: Language Logic and Proof Q. 6.26 Using the natural deduction rules, give a formal proof of A ∨ D from the premises A ∨ (B ∧ C) (¬ B ∨ ¬ C) ∨ D ...
3
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3answers
311 views

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)...
3
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3answers
143 views

Symbolic Logic - Negation Introduction

I am working on a problem for an online class that I'm struggling to figure out. I'm given these premises: 1. (H > (A > B)) (The > sign here represents conditional) 2. (~K & ~B) 3. (~A > K) The ...
3
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3answers
596 views

What does the truth-value of a material implication represent?

This question comes from my attempts to understand what the truth value for a material implication with a false antecedent represents. I have seen several justifications for this convention, usually ...
3
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2answers
244 views

In modern logic, why does “All S is P” contradict “Some S is not P”?

In modern logic, the existential import is removed from universal statements. So All S is P may still be true if there is no S at all. Contradictory statements must have opposite truth values. Why ...
3
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4answers
907 views

Anyone can help me with proving ~(AvB) |- ~(BvA) via natural deduction?

~(AvB) ㅡㅡㅡㅡ ~(BvA) I have to provide a derivation to establish validation of this argument. First of all, can I first change ~(AvB) into ~A&~B by using the De Morgan rules? And the second is:...
3
votes
2answers
241 views

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, ...
3
votes
2answers
264 views

How would one go about proving the following statement in predicate logic?

I need to prove this: ⊢(∀x)((Fx→Gx)∨(Gx→Fx)) Not entirely sure how I'd go about this.
3
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2answers
305 views

Formal proof : predicate logic

This is what I need to prove formally: 1.∃x Cube(x) ∧ Small(d) . . . . Goal :∃x (Cube(x) ∧ Small(d)) I have already tried different ways, but I still can't prove the goal. 1. ∃x Cube(x) ∧ Small(d) ...
3
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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 '...
3
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2answers
211 views

Getting started with reading papers in (philosophical) logic

I have worked through some textbooks thus far, but would like to get started reading papers in the field of logic. Can anyone recommend any papers or how I could get started? I found that many papers ...
3
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2answers
447 views

How to prove ¬¬(A ∨ B) leads to ¬¬(B ∨ A)?

Using laws of natural deduction, how can one prove that the single premise ¬¬(A ∨ B) leads to ¬¬(B ∨ A)? I have tried solving the problem for some time but to no avail.
3
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1answer
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→◻...
3
votes
1answer
119 views

What are the rules for a zero-premise derivation involving disjunctions?

I'm having trouble with the following zero-premise deduction that involves two disjunctions: The solution seems simple, but I'm unsure of how to proceed with the two disjunctions. If it were just ...
3
votes
4answers
248 views

Classical logic derivation question

Premise 1: R∨T Premise 2: ∼P↔(∼P→Q) Prove: (R∨S)∨(T∧Q), using only R, DN, MP, MT, S, ADJ, MTP, ADD, BC, CB, CDJ, DM. Here's what I got so far: Show (R∨S)∨(T∧Q) R∨...
2
votes
4answers
245 views

In fitch, S → (R ∨ P), P → (¬R → Q) ∴ S → (Q ∨ R)

Construct a proof for the argument: S → (R ∨ P), P → (¬R → Q) ∴ S → (Q ∨ R) I have gotten to the point in the illustration, but I am unable to figure out where to go from here. I get tricked up on ...
2
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4answers
3k views

Conditional disjunction equivalence proof using FItch

Prove P v Q ⇔ ¬Q → P So far I have the obvious things... 1. P v Q _ | 2. ¬Q | _ | 3. | 4. | 5. | 6. | 7. | 8. P 9. ¬Q → P → Intro 2-8 I think the problem here is that I do not ...
2
votes
4answers
146 views

Negation of a statement

The question asks us to negate the following statement. Jackie eats sweets, if she is not hungry. This is a basic if (p), then (q) statement whose negation will simply be p and ~q, but the solution ...
2
votes
3answers
189 views

Logic question regarding a logical truth

Is the following logically true? ∃x[Cube(x) →∀yCube(y)] I think that it is logically true. When translated into truth functional form we have: A→B. A truth table shows that it is not a tautology but ...
2
votes
2answers
673 views

Verum, Falsum, Atoms

I have been somewhat confused about the definition of atoms, or atomic formulae. Some sources say that verum (⊤) and falsum (⊥) are atoms, some not. Is there any consensus within the community or is ...
2
votes
1answer
464 views

Problems with Existential Instantiation [duplicate]

Why is it required to use a "fresh name/variable"? And because of that requirement, Existential instantiation always precedes universal instantiation. What I am thinking is, If we are picking elements ...