# 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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### 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 ...
610 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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1 vote
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### 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 ...
1 vote
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### 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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### 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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### 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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### 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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### 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 ...
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### 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) ...
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### 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 ...
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### How to prove 1. ~(KvF) 2. ~F=>(KvC) 3. (GVC)=>~H / ~(KvH) using natural deduction

I need help with this question using the first 13 rules of inference. Here is what I have so far: ~(KvF) ~F=>(KvC) (GVC)=>~H / ~(KvH) ~Kv~F DM 1 ~Fv~K Com 5
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### 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)
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### Symbolizing an argument and deciding on a conclusion

Consider the following conversation: "Gerda," said Hans, "we must know if Petra went to Berlin." "Well," said Gerda, "we know that if she didn't then she went to Cologne. And we know that she didn't ...
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### 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 ...
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### How to express possession in predicate calculus

all! I was wondering if I could get help translating this phrase in to first order logic. I'm trying to say: There exists a u such that u is Russian and there exists a b such that u shot b. Would it ...
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### 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 ...
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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 ...
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### Why does what I've written fail to define truth?

(Also posted in mathstackexchange prior to this). I stumbled across a set of axioms for first order logic a bit ago. Intrigued, I decided to try to write it all down and organise what I read. After I ...
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### Suppose A is a set of premises of an argument and B the conclusion of that argument. Prove that if A U {¬B} ⊢ ⊥, then A ⊢ B

Suppose A is a set of premises of an argument and B the conclusion of that argument. Prove that if A U {¬B} ⊢ ⊥, then A ⊢ B. (Use Fitch) I have no idea where to start, can someone help?
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### Is there a symbolic formulation of modal realism?

Is there a symbolic formulation of modal realism, i.e. the doctrines of modal realism captured in some formal system?
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### 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 ...
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1 vote
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### Prove by induction that no terms in FOL of arithmetic begins or ends with 2 plus signs (++)

Prove by induction that no terms in FOL of arithmetic begins or ends with 2 plus signs (++) I have no idea how to start this proof, can someone help me?
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### 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 ...
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### Fitch Formal Logic Help 6.26

6.26 Premise: A v (B ^C) Premise: ~B v ~C v D Goal: A v D Prove it formally without using DeMorgan's Law.
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### Is there a difference between 'only if' and 'if and only if'

So am reading a book titled 'an introduction to logic', and the topic at hand is Sentential logic > Biconditionals. At one point, the author gives examples of 3 sentences that draw upon three similar ...
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### can you express universal quantification ∀x(Px ≡ Qx) simply as Px ≡ Qx?

Is it permissible and normal to express the prop. ∀x(Px ≡ Qx) simply as Px ≡ Qx? That is, to treat the univ. quantifier as implicit if its scope is all the rest of the prop.?
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### 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 ...
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### How would I deduce a≠c from a≠b and b≠c in Fitch?

How would I deduce a≠c from the premises a≠b and b≠c in Fitch? This is what I've done so far. b=b (=Intro) b≠a (Ana Con) b≠c (Reit) And then for some reason I get stuck here? I know this sounds ...
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### 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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### 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 ...
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### 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∨...
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### 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 ...
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### 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.
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### 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.
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### 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 ...
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### 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 ...
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### 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)...
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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 ...
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### Concerning the logical projection: How to express nullary and unary operations as binary operations?

Let P and Q be two statements, each having two possible truth values: true (T) or false (F). Then there are exactly 16 unique compound statements involving P and Q with corresponding truth tables of ...
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### 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 ...
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### 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 ...
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### Representation of Categorical Syllogism in Symbolic Logic

How would I represent a AAA categorical syllogism with symbolic logic?
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### 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 ...
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### How to derive ( E v C ) using sentential derivations?

I'm having a really hard time trying to derive (E v C) from {~A > ~B, A > C, B v D, D > E}. Where '>' is the material conditional, so that 'A > C' is read as "If A then C". I used negation ...
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### What's the difference among the logical relations :=, =, and ≡?

I understand that ≡ is logical equivalence, "iff". '=' is a symbol for numerical equivalence. And ':=' is an identity claim. I often only see '=' and ':=' used with variables and names, ...
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### 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?
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### 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:...
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### 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 ...
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