# 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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### 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)...
156 views

### 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, ...
727 views

### 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?
640 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 ...
1 vote
80 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(...
1 vote
973 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
1 vote
921 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 136 views

### 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 ...
2k 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 119 views

### 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 ...
695 views

### 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 ...
1 vote
560 views

### 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).
1 vote
439 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 ...
1 vote
7k 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 &...
1 vote
464 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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### 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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### 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." ...
214 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 ...
615 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 ...
801 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 ...
374 views

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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### "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?
1 vote
141 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 ...
946 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 ...
1 vote
2k views

### 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 247 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 ...
1 vote
92 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. ...
2k 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. ...
1 vote
96 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 ...
1 vote
105 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 ...
458 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. (∃x)~Fx ...
2k 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 ...
212 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) -> ...
902 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 ...
83 views

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 ...
311 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 ...
513 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 ...
930 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 ...
1 vote
478 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 ...
5k 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!
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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)) ...
284 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 ...
1 vote
1k 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 ...
6k 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 ...
262 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 ...
1 vote
494 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 ...
301 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 ...
1k views

### 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 (...
1 vote