# 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.

142 questions
1answer
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### Deductively sound formal proofs of mathematical logic? [closed]

How can this possibly fail to partition True(x) from Untrue(x) for every formal system? When we specify that True(x) is the consequences of the subset of the of conventional formal proofs of ...
0answers
213 views

### How could the Tarski Undefinability Theorem be Refuted? [on hold]

When we specify that True(x) is the subset of the conventional formal proofs of mathematical logic having true premises then True(x) is always defined and never undefinable. In this case the Tarski ...
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 ...
3answers
95 views

### 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 ...
1answer
131 views

### 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: ...
1answer
116 views

### 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, ...
2answers
72 views

### Fitch-style natural deduction

How to prove the following questions? (a) p from assumption ¬(p → q) (b) ¬¬p → p from no assumptions.
1answer
76 views

### 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 =...
1answer
191 views

### 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 ...
2answers
142 views

### 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 ...
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 '...
1answer
66 views

### 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 ...
1answer
39 views

### 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 ...
4answers
112 views

### 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 ...
1answer
41 views

### 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) ...
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→◻...
1answer
66 views

### 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 ...
2answers
212 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 ...
2answers
66 views

### 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 ...
1answer
81 views

### 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 ...
2answers
121 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 ...
0answers
151 views

### 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) ∧ ...
4answers
75 views

### 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 ...
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 ...
4answers
837 views

### 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 ...
1answer
76 views

### 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 ...
1answer
97 views

### 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 ...
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∨...
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, ...
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 ...
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 ...
2answers
135 views

### 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.
3answers
942 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 ...
2answers
49 views

### Are Statements with Existential Quantifiers General or Particular?

Consider the following argument: The number 2 is a prime number and is divisible by 2. Thus, some prime number is divisible by 2. The first statement in this argument concerns a particular, i.e. ...
3answers
310 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)...
1answer
45 views

### 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 ...
2answers
317 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 ...
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. ...
2answers
70 views

### 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 ...
2answers
117 views

### 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?
1answer
101 views

### How to solve the derivation?

Derive the following without assumptions: ¬∃xFx↔∀x¬Fx How do I solve this derivation?
2answers
86 views

### 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 ...
3answers
218 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 ...
2answers
124 views

### 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:
0answers
34 views

### Recommendation: Second Order Logic textbook

I'm looking into Universalist Realism, Nominalism, Trope theory and the application of Second Order logic to each of them, however I have little/no experience with Second Order logic. Please let me ...
2answers
404 views

### How to get proof using proof editor and checker

How can I use http://proofs.openlogicproject.org/ or http://logic.tamu.edu/daemon.html to derive the given conclusion from the given premise: (∃x) ( Fx ∙ (y) (Fy → y = x) ) / (∃x) (y) (Fy ≡ y =...
2answers
241 views

### How do you prove B v A |- A v B?

I am having trouble with how to use the assumption, which I feel that I will need for this proof. If any one can demonstrate or give hints for this proof, I would greatly appreciate it.
3answers
93 views

### McGee's Counterexample to Modus Ponens

I'd like to start off by saying that I have read the other posts in the Math StackExchange and here about this paper, but I think my question is a bit different from those although it does stem from ...
2answers
178 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
0answers
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### How did Fitch's opposition to the Russell-Whitehead theory of types turn out since the 1950's?

In a footnote to Appendix C of Frederic Fitch's Symbolic Logic (page 217), Fitch writes about his article, "Self-Reference in Philosophy": It is reprinted here in order to indicate more fully my ...