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1answer
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Could someone help me prove (P → Q) ↔ (~P ∨ Q) follows from (P ∨ Q) ↔ (~P → Q) in sentential logic?

I need to prove that (P → Q) ↔ (~P ∨ Q) follows from (P ∨ Q) ↔ (~P → Q).
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3answers
53 views

Is the reiteration rule in formal logic begging the question?

Wikipedia defines "begging the question" as To "beg the question" is to put forward an argument whose validity requires that its own conclusion is true. I assume this is something Aristotle's ...
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1answer
69 views

How to prove ¬(p→q) ⊢ p &¬q

This is the first time I have posted anything on this forum. I am using Tomassi's Logic. Unfortunately I have been unable to solve some of the problems. One I can't solve is this one: ¬(p → q) ⊢ p &...
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2answers
175 views

Can inductive arguments be made in first order logic and, if not, why not?

After reading a question by rus9384 Why is faulty generalization called an informal fallacy? I wondered whether induction can be part of any argument in first order logic (FOL). rus9384 symbolized ...
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5answers
403 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/...
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1answer
22 views

Need help translating a formal definition into FOL

I need help formally translating the following definition into FOL: "a property F is essential to an object x if and only if x could not have been the object it is without possessing the property F." ...
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1answer
34 views

Predicate Logic - Universal Introduction

Another question I'm struggling on with predicate logic: Premises: (There exists x) ~Fx (For all x) Ox Desired conclusion: ~(For all x)(Ox > Fx) (if Ox, then Fx) My thoughts are to start by ...
2
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1answer
24 views

Predicate Logic - Existential Elimination

I am working on a predicate logic proof given the following premises: (For all x)(Fx > Vx) (There exists x)(Fx & Bx) Desired conclusion: (There exists) (Vx & Bx) My instinct here says to ...
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3answers
63 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 ...
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1answer
46 views

Monadic Predicate and Polyadic Predicate?

Consider the following sentences; how would each be symbolized?: "Kate loves John" vs. "Kate loves cheese" ^(here, I'm trying to understand the difference between a relation between two individuals ...
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3answers
72 views

“Kids shout at plants” (relation vs implication?)

I suspect "Kids shout at plants" can be represented as an implication: (x)(y)[Kx --> (Py --> Sx)] or as a relation Skp Is this correct? Why or why not?
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1answer
40 views

Applying rules of inference in natural deduction

When applying a rule of inference, is it okay to "skip" a step (i.e, apply a same rule to multiple parts of a statement)? For example: (A > B) ^ (C > D) (~A v B) ^ (~C v D) 1, Impl. As opposed ...
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2answers
72 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 ...
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2answers
102 views

Do mathematicians take Modern Logic to be an appropriate representation of our sense of logic?

What examples do we have of mathematicians who explicitly and publicly expressed their personal confidence that mainstream modern logic, as used in mathematics, either as object of study in itself or ...
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1answer
73 views

What is the proof for the Reductio (in a derivation)

In deductive logic, we may make the following step: ( {Γ,P}⊨Q & {Γ,P}⊨¬Q ) ⇒ {Γ}⊨¬P I've been trying to find examples of a proof that this inference follows, but I've struggled with my search. ...
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1answer
311 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. ...
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2answers
74 views

Formalizing negations

I'm working through a problem now that asks whether it is possible to logically formalize two sentences: "Say of each of the following pairs of English sentences whether there is a sentence φ of L1 ...
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2answers
54 views

Is the Completeness of a logical system considered an integral part any 'good' logical system?

Most logical systems will have two distinct forms of entailment, one is system-based entailment (logical consequence), and the other is proof-based entailment (derivability). In the former, an ...
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2answers
74 views

An example for (∃y)(Fy→(∀x)Fx)?

I am very confused how this can be possible. Could someone give me a substitution instance? If this is not correct, is there anything wrong with this proof ? ├ (∃y)(Fy→(∀x)Fx) [1] 1. (∃x)~Fx ...
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3answers
78 views

Deriving “(p.q) v (p.r) from ”p.(q v r)"?

I am new to logic. and here are my tryouts for deriving deriving "(p.q) v (p.r) from "p.(q v r)", and further I want to show that ”p.(q V r)” is equivalent to ”(p.q) V (p.r)”, by using natural ...
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1answer
81 views

Predicate logic - Symbolizing sentence

K_ = _ is a Kiwi, M_ = _ is a Moa, F_ = _is flightless If something is a moa only if it's flightless then if all kiwis are flightless, some kiwis are moas. Ax( (Mx -> Fx) -> ( Ax(Kx & Fx) -> ...
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1answer
73 views

FOL and Tarski's world logic connectives question

I'm trying to solve the following five problems where I'm asked to translate these English sentences into FOL by using Tarski's World symbols. I'll appreciate it very much if anyone can help me ...
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2answers
65 views

Question about logic syntax

I am trying to symbolize the sentence "If Alma paints a square, then Alma paints a rectangle" using the dictionary: S1 : is a square R1 : is a rectangle a : alma P2 : Paints My question is is it ...
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4answers
87 views

Negation of a statement.

I have just started learning logical reasoning and hence was studying logical connectives when I stumbled upon a question. The question asks us to negate the following statement. Jackie eats sweets, ...
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0answers
149 views

Logic symbol meaning: ⩙

So there is the logical conjunctive (True ∧ True ⇔ True) and logical disjunctive (True ∨ False ⇔ True). However, unicode defines an overlapping logical and, ⩙, found here. My question is this: What ...
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0answers
33 views

how to prove from ~ (A ∩ B) that ~A∩~B? [duplicate]

How to prove ~ (A ∩ B) ⊢ ~A ∩ ~B ? my thought is to assume A, find a contradiction, so I can use "~I" to get ~A; do the same for B to get ~B; then use "∩I" to get ~A ∩ ~B. But I struggle to infer ...
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4answers
172 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 ...
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5answers
216 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 ...
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2answers
118 views

Predicate logic proofs - how to split a disjunction bound by two quantifiers

I need to complete the following proof using only primitive rules (the introduction and elimination rules for each connective and quantifier). (∃x)(∀y)(Py ∨ Qx) ⊢ (∀y)Py ∨ (∃x)Qx I've only been able ...
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3answers
222 views

Prove (¬P ∨ Q) ↔ (P → Q)

How can one use a standard logic proof to prove this without using any premises? I've tried doing subproofs and splitting up ¬P and Q to try to get to P → Q but I'm very stuck!
2
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1answer
53 views

If it is not all wrong, then I have a problem in line 25. Can anybody help, please?

∃x ∃y (Cube(x) ∧ Cube(y) ∧ ¬x = y ∧ ∀z (Cube(z) → (z = x ∨ z = y))) a∃y (Cube(a) ∧ Cube(y) ∧ ¬a = y ∧ ∀z (Cube(z) → (z = a ∨ z = y))) bCube(a) ∧ Cube(b) ∧ ¬a = b ∧ ∀z (Cube(z) → (z = a ∨ z = b)) ...
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2answers
184 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 ...
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1answer
98 views

Disjunction elimination proof

I'm having trouble making assumptions in this exercise. Can someone point me in the right direction? premise: P OR Q conclusion: R → (P OR Q) AND R My attempt so far: 1. P OR Q ...
2
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3answers
217 views

Is it logically correct to say that if A implies B then not A implies not B?

Here is an argument: The government has announced that it wants to reduce the level of ill-health due to workplace stress. Ministers could learn a lot from a recent study of 8000 white-collar workers ...
2
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3answers
130 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 ...
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4answers
100 views

Language Logic Proof Question: ¬∃x∀y[E(x,y) ↔ ¬E(y,y)]

I am wondering if I have completed this proof properly. I don't think I have it right. It's tricky! Conclusion: ¬∃x∀y[E(x,y) ↔ ¬E(y,y)] ¬¬∃x∀y[E(x,y) ↔ ¬E(y,y)] ∃x∀y[E(x,y) ↔ ¬E(y,y)] ¬E,1 ...
3
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1answer
72 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
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2answers
243 views

Proof for the Rule of Absorption in Propositional Logic?

I know there is a "formal proof" for the "rule of absorption" that employs the "law of excluded middle". It is presented in Wikipedia (and I think it is Russell's): https://en.wikipedia.org/wiki/...
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1answer
60 views

Can anyone help me solve this (p → r) → (¬a v b), p → q, b → s, q → r, ¬a → s // (r v s)

I have been working almost three days on this problem and I can't to this answer: (p → r) → (¬a v b) p → q b → s q → r ¬a → s // (r v s)
2
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1answer
142 views

Why do “L” and “M” name the strong and weak modal operators in modal logic?

Though the box and diamond are the more common representations of the strong and weak modal operators in modal logic, “L” and “M” are also used. I suspect that those letters were chosen because they ...
2
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1answer
301 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 ...
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2answers
291 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 ...
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2answers
463 views

Can a true statement also imply the opposite of itself?

It's unlikely that there could be a thesis that also is its own antithesis. Similarly, a formula usually isn't the "opposite" of itself if we use well-defined terminology. Somehow I have a notion ...
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3answers
134 views

How is the correct way to read out negation in symbolic expression?

I am not sure does the following parts of symbolic expressions read the same way or not when being the first part of the expression: [~(p v q)] -> .... If it is not p or q, ... or perhaps If it is ...
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2answers
125 views

I am stuck on how to prove the contradiction of R(b,a) can anybody help me?

Here are some well-known properties of dyadic (2-place) relations: ∀xR(x, x) (Reflexivity) ∀x¬R(x, x) (Irreflexivity) ∀x∀y(R(x, y) → R(y, x)) (Symmetry) ∀x∀y(R(x, y) → ¬R(y, x)) (Asymmetry) ∀x∀y∀...
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1answer
50 views

Is this a valid move in a proof or does this create a contradiction?

If I have something like the following can I use the add inference rule to add ~A. Does that cause a contradiction, or am I fine since it's if A and not A being directly declared? 1. (A ⊃ B) ⊃ C 2. ~...
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2answers
226 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.
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4answers
341 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, ...
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2answers
95 views

Can someone help me understand how to symbolize?

There are jackals on the stairs and in the elevator and Tom is scared. If there are jackals on the stairs, then they are not on the elevator and Mary is happy. Either it is the case that, if there ...
4
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2answers
198 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) ...