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.
307
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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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1
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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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1
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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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2
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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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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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3
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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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1
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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
user34868
2
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1
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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 ...
- 123
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2
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~(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
user34868
2
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2
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119
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Is it possible to define argument validity as a formula?
Let A, B and C be propositions. Define ARG(A, B, C) as the following argument:
A.
B.
Therefore, C.
My goal is to create a formula whose truth value is equivalent to "ARG(A, B, C) is ...
- 206
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3
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How to prove P ∨ Q : ~ (~P & ~Q) with natural deduction
Here's another Tomassi's problem I can't solve (Logic, Exercise 3.9.1.17, page 106):
P ∨ Q : ~ (~P & ~Q)
I have to use natural deduction and the only rules I know are:
assumptions,
modus ...
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1
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560
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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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3
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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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1
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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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2
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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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6
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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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1
answer
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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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1
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214
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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 ...
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2
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1
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615
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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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3
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4
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801
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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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1
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374
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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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3
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85
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"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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1
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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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2
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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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2
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How do I prove: 1. A v (B & C) 2. (A v C) > ~(G & O) / ~G v ~O
This is a question for my philosophy.
Prove this valid using any of the rules we've studied so far:
A v (B & C)
(A v C) > ~(G & O) / ~G v ~O
user33343
0
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2
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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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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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1
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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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2
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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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2
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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 ...
2
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2
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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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3
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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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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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1
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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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2
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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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4
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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 ...
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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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5
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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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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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3
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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!
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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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2
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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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2
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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 ...
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3
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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 ...
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3
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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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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
...
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1
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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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4
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1k
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Proof for the Rule of Absorption in Natural Deduction?
I know there is a "formal proof" in "natural deduction" for the "rule of absorption" that employs the "law of excluded middle". It is presented in Wikipedia (...
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2
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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)
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1
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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 ...
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