# 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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### formalization: definite description (narrow reading)

I am not sure which formalization is right  or : 'The teacher of Plato does not exist.'  ∃x(Tx,p ∧ ∀y[Ty,p → y=x] ∧ ¬∃y[y = x])  ∃x(Tx,p ∧ ∀y[Ty,p → y=x] ∧ ¬∃z[z = x]) Is it possible to ...
1k views

### What's the difference between "iff" and "=df"?

Just a quick question I stumbled upon from my readings. When some philosophers write A ↔ B and others write A =df B, is there supposed to be a difference?
82 views

### Is symbolic logic just a non scientific way when it comes to interpret human natural language?

Let me ask you a thing it is about implication: when I say, if I go to London, I will talk to Paul, I mean an implication, or S=>P. Well, implication means it is necessary that S belongs to P, ...
1 vote
54 views

### Are "A ∧ A" and "A ∨ A" degenerate expressions?

Although some time ago I had become somewhat familiarized with the notion of degeneracy in mathematics and physics, in my musings on the trivial/nontrivial distinction I found that both Wikipedia and ...
335 views

### Origins of the syntactic form for rules of inference in modern presentations

I have been wondering where the form originates from. The turnstile ⊢ famously comes from Frege, but I haven't been able to find where the vertical notation was introduced. In the field of ...
458 views

### Does the existential quantifier express existence?

Does the existential quantifier express existence? The existential quantifier is a symbol of symbolic logic which expresses that the statements within its scope are true for at least one ...
53 views

### Does quantifier dependence involve putting ∃ before ∀ (or vice versa)?

I don't know why I'm having such trouble getting the gist of the SEP article on independence-friendly logic, but I am. I also remain perplexed about a comment I received on the MathOverflow about the ...
1 vote
5k views

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

How can I use Natural deduction proof editor and checker or The Logic Daemon to derive the given conclusion from the given premise: (∃x) ( Fx ∙ (y) (Fy → y = x) ) / (∃x) (y) (Fy ≡ y = x) It tells me ...
57 views

### Can you help me with the inference: if ¬( P & ¬Q ) and Q, then P

I'm taking my classes of symbolic logic, so my question is a bit naïve, but: If this expression is correct: ¬( P & ¬Q), P then Q. Why not the following is not: ¬( P & ¬Q), Q then P. Thank you.
43 views

### How would demi-conditionals work?

Let 𝒜 = an actuality operator and √→ be demi-if. Which, if any, of the following conversions would go through? 𝒜A √→ 𝒜B = √𝒜A → √𝒜B 𝒜A √→ 𝒜B = √𝒜A → 𝒜B 𝒜A √→ 𝒜B = 𝒜A → √𝒜B 𝒜A √→ 𝒜B = √�...
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### Do computer languages instantiate only predicate calculus?

All the computer languages I'm familiar with, be they imperative or declarative have the same core mechanics (arithmetic and logic).They have the same loops, conditionals etc. Whatever the language it ...
1 vote
965 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
239 views

### fitch proof. P v Q, Q→ ¬ R, ¬ P, ¬ R → ¬ S GOAL: ¬ S

Need help exercise using the FITCH program format. I'm stuck on where to start. The following 4 steps must be used to prove the goal. P v Q Q→ ¬ R ¬ P ¬ R → ¬ S GOAL: ¬ S Now I know: ¬ P and P v Q ...
63 views

### Help with formalization of argument (ignore premises) in FOL

I am trying to formalize the following argument: Every Moral theory is equally valid. There always can get a new moral theory from another one. For something to be metaphysically real/exists it must ...
52 views

### Correct way to write statement using symbols?

I would like to write the following using logic symbols but am unfamiliar with the practice. Here is the statement: If it is accepted that life will arise from matter given the right conditions and if ...
220 views

### Help with natural deduction by introduction and elimination rules

This is where I’ve gotten so far. I’ve proven it from left to right but I’m getting some trouble proving it from right to left. I’m trying to reach the conclusion by double negation.
20k views

### What is the difference between Law of Excluded Middle and Principle of Bivalence?

Law of Excluded Middle: In logic, the law of excluded middle (or the principle of excluded middle) is the third of the so-called three classic laws of thought. It states that for any proposition, ...
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
117 views

### From English Sentence to Symbolic Logic: "The Happiest Person is not named John"

Suppose that x is over the domain of all things and I have the following predicates: H(x) = x is a person, J(x) = x is named John, F(x,y) = x is happier than y, a = John Smith My interpretation of ...
40 views

### Question regarding the stipulated 'domain of discourse' for models of first-order sentences

Assume 'S' is a first-order sentence about a subject 'Z'. When one stipulates a Model for 'S' with a domain 'D' does one always assume that the domain will contain all the objects within the subject '...
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### Does a function assigning any sentence to some 𝘢th-order logic exist?

I feel like I'm just reinventing Tarski's wheel with this idea, or maybe I'm even remembering what I've looked over with respect to Tarski's undefinability thesis and phrasing it in a way that ...
66 views

### Modal system K - prove ⊢ (□p ∨ □q) → □(p ∨ q)

I am trying to prove the following: ⊢ (□p ∨ □q) → □(p ∨ q) However, I think that I am lacking the knowledge of a tautology in classical logic that would help me prove this. I tried something, but it ...
101 views

### Axiomatically prove □(A ∨ ¬B), ¬□A, ⊢ ◇¬B in modal system K

This time I have a more "complex" problem at first glance. I need to create a direct proof using the axioms of system K and rules of inference, but I have been unable to do so. □(A ∨ ¬B), ¬□...
286 views

### Proof of □P ⊢ □¬¬P in modal logic system K

I need to prove the aforementioned formula in modal logic system K, which I am having trouble to do. Of course, this should be easy to prove if I had access to axiom T, but since it's system K, we can ...
484 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 ...
122 views

### Why are undefined references and variables not specifically differentiated?

In my opinion, this topic is more philosophical than mathematical, but if it is not, I will ask it on another forum. My understanding I'm talking about non-reserved symbols here. Not about 0, 1 or π. ...
2k views

### Why does the Principle of Explosion not make Mathematical Logic inconsistent? [closed]

Step Proposition Derivation 1 ------P --------- Assumption 2 ---- ¬P --------- Assumption 3 ----- P ∨ Q ----- Disjunction introduction (1) 4 ----- Q --------- Disjunctive syllogism (3,2) https:...
1 vote
44 views

### Is Nozick's Experience Machine self-defeating?

Nozick's experience machine is usually described as able to bring about any desired experience. If it can't do that, then it's not a suitable object for the thought experiments Nozick and others build ...
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### What are the arguments of philosophers against the reasoning which justifies the horseshoe from truth-functionality?

There is a reasoning in mathematical logic which is meant to prove that the horseshoe is the only logical operation which fits our notion of conditional. The reasoning starts from the idea that the ...
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I'm having trouble understanding writing out a proof. The proof I'm trying to work with is : How do I reach this goal? Which rules do I use and with which support steps to each rule (proofs to prove ...
538 views

### Is it possible to construct infinitely many non-equivalent formulas in predicate logic?

In the language of predicate logic with only identity and no predicates, function symbols, or constants, is it possible to construct infinitely many non-equivalent formulas?
39 views

### Zero-one laws Model Logic, question regarding significance of domain size

Wikipedia informs me that: Essentially (correct me if I'm wrong) the result states that as the domain of objects (domain of discourse) grows (n->inf), a static first order sentence (S) will be ...
1 vote
220 views

### Is Norman Megill's view of Gödel's incompleteness theorem compatible with what philosophers have said about it?

Here is one recent and seemingly expert appreciation on the consequences of Gödel’s incompleteness theorem for mathematics: Gödel’s incompleteness theorem showed that it is impossible to achieve ...
3k views

### What did Russell mean when he wrote that the null-class, the class having no members, did not exist?

I am not quite sure I interpret the following sentence correctly in Bertrand Russell's paper on existential import: and among classes there is just one which does not exist, namely, the class having ...
65 views

### Questions about Feature Placing Languages/Predicate Functor Logic

About a year and nine months ago, I poses a question here about Quine's predicate functor logic and ontological nihilism. I'm still having trouble wrapping my head around these ideas. I hope someone ...
1 vote
182 views

### Willard Van Orman Quine: Elementary Logic Exercises 1: Which of the following are statements?

I am currently self-studying formal logic via Quine's "Elementary Logic." The first exercise is to declare which of the following sentences are statements and re-write the sentences that are ...
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### Is it a rule of formal languages that all occurences of a symbol must 'refer' to the same object?

A rule of subsitution is that we replace all free occurences of a symbol x with free occurences of a symbol y to subsitute y for x in a formula φ. Hence the sentence 'x=x' is inherently true for all x ...
66 views

### What is the 'meaning' of an unassigned formula with free variables?

What does a variable refer to in a formula? If it is a free variable, it has no reference, yet it exists as an element of the formula. In an unassigned formula, what is the semantic meaning of a ...
427 views

### Predicate logic proof solve

Provide a proof for the following using FOL in forallx Use the natural deduction system and proof strategies in forallx to provide a formal proof for the following . Please provide a picture of your ...
152 views

### Philosophy books for mathematicians

Are there any books on philosophy that make relatively heavy use of math? I'm not looking for anything on formal epistemology, logic, or philosophy of math. Two examples of books that fall in the ...
1 vote
68 views

### How does 'use-mention' apply to formulas?

When we use 'terms' such as words it is generally clear however, if we have a formula: And I write: 'x+1=2 is true for x=1' is this 'using' or 'mentioning'? If a formula contains variables, it has no ...
1 vote
112 views

### How can sequences/expressions occur in other sequences/expressions?

I know I specifically wrote a question about Wetzel, however I do not want to invalidate previous answers. In Quine's 'Mathematical Logic' he discusses occurences of 'expressions' in other '...
1 vote
95 views

### Wetzel's 'occurences'

I was reading this often quoted article by Linda Wetzel (1993) where she discusses the 'occurence' of expressions in others and Quine's issues with the idea, she describes an expression as a sequence ...
1 vote
254 views

### Is '=' a relationship between the objects or their expressions?

The Wikipedia definiton of equality gives it as a 'relationship between two expressions' This confuses me as when we define mathematical expressions like 2+2=4 it makes no sense to say that '=' or '...
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### How do we arrive at stronger theories in mathematics/logic?

A reasonable aim of formal mathematics/logic is to build systems which can "interpret" many things. As an example, ZFC can interpret a number of things. Incompleteness Theorems provide us ...
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### Question from predicate logic exam: Given model with the domain D = {a,b}, say whether the formulas listed below are true or false

I've got a logic exam coming up and one of the question types is puzzling to me. If anyone could help me by explaining what this is about to me, I would appreciate it greatly. Note: I was unable to ...
198 views

### Is a variable simply a symbol?

If a 'variable assignment' function maps from a set of symbols, would it be correct to formulate a variable as simply a particular symbol that performs the role of a variable in my language? So when ...
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### Are there only two levels in languages, meaning and symbols?

Say in my language I have a 'variable x', in my language the symbol x represents a (variable) number, so at a level of meaning it is an object, and at a level of symbols 'x' is simply a set of lines ...