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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Would an erotetic operator be equivalent to its own demi-operator?

"Recap": demi-operations are e.g. "the square root of negation" in experimental(?) logic. (The association of demi-negation with using imaginary numbers as truth values is a little ...
2k views

Is Russell's Paradox a semantic paradox or a syntactic paradox? I ask because of the following: Let P be a predicate Let SEP be the property of being a set of things that satisfies P Let SP be the ...
38 views

If X is a statement, is the collection of all interpretations of X a set?

Let X be a statement Let SI be the predicate set of interpretations of X Let IX be the predicate interpretation of X Let NA be the predicate not contained in A ∃A∃B(SI(A)∧IX(B)∧NA(B))→∀A∃B(SI(A)→...
216 views

Is equality necessarily transitive?

Consider the equation b2 - 1 =0 If b is a variable, it's neither true nor false. So let the symbol b be a constant, thus the equation denotes a proposition. The symbol b is a referrer, and the ...
70 views

Importance of Logical Notation

Does better notation lead to ease of abstraction and shorter proofs? I ask because I tried translating the following from Euclid’s Elements into my own idiosyncratic notation: Prime numbers are more ...
103 views

Why doesn't one assert in metamathematics that a sentence S is a logical consequence of the conjunction of a set of sentences?

In other words, why isn't there -- at least in standard textbook presentations of things like the deduction theorem and the compactness theorem -- a conjunction connective that is applied to sets of ...
50 views

What is the significance of the Coincidence Lemma?

There is not a Wikipedia article about the coincidence lemma. I will try to explain the proof and then ask why it is important. The coincidence lemma is meant to show that the satisfaction relation ...
1 vote
100 views

How can I formalize the argument that morality cannot exist, in FOL?

I am trying to formalize the following argument: Every moral theory is equally valid. One can always get a new moral theory from another one. For something to be metaphysically real or to exist, it ...
62 views

Which is correct, "the implication A → B" or "the implication ‘A → B’"?

Which is correct? The true (or false) implication A → B. The true (or false) implication ‘A → B’. What are the arguments for saying that it is wrong to say: the implication A → B and the we should ...
75 views

What difference between the truth of a conditional* and its logical validity?

I am confused . . . Here is a remark on the "classical analysis" of the implication: On the classical analysis, logical implication is the same, not as the truth of a conditional statement, ...
1 vote
56 views

Did Russell had something like the notion of domain in the sense defined now by mathematics textbooks?

The expression ∀x(ϕx → ψx) is supposed to mean that, in Russell's parlance, ϕx → ψx is true "for all values of x". However, what are those values that Russell is referring to? At some point ...
58 views

Can assumption in Hilbert style proof system be contradictory?

⊢（¬A→A）→A I don't know how to solve this proof with the Axiom, Theorem and Inference rule in Hilbert-style proof system so I ask my classmate and he show me his answer. After viewing his proof, I was ...
88 views

Unusual change of meaning of word "any" in negative sentences form "for all" to "there exists". Predicate logic

Question. Why does the word "any" in negative sentences changes its meaning from "for all" to "there exists"? Origin of the question. I have a question about translating ...
246 views

At what point in the history of mathematics, and why, did mathematicians come to say "A implies B" to mean "not A or B"?

Here is what one respondent to my previous question says: A big part of the problem here lies with interpreting the word ‘implies’, which is ambiguous in English. Unfortunately, mathematicians get ...
132 views

Why not just give up on the idea of truth-functionality?

I understand that today only a minority of academics who are specialised in formal logic accept the horseshoe (aka "Classical Logic" or "First-Order Logic") as an accurate, or even ...
65 views

stuck! first order logic - identities (specifically "only")

Please correct me on why these may be wrong(identities). I've tried many times but it seems I'm missing something. for they key: M(x) = is a moon, O(x,y) = x orbits y, and m = mars, e = earth Only ...
261 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.
3k views

What does Tarski mean when he says "variables do not posses any meaning by themselves"?

This is an excerpt from Alfred Tarski's Introduction to Logic and the Methodology of Deductive Sciences: As variables we employ, as a rule, selected letters, e.g. in arithmetic the small letters of ...
79 views

How is this logic valid?

An excerpt from Logic 2010: In particular, what is confusing is that it permits assuming the conditional but then reaching a contradiction to prove the conditional. In my experience, that is not a ...
53 views

Is it possible to stick to one of these viewpoints of variables?

It has been a struggle to find a precise account of the concept of variables. There are however two viewpoints that I've seen authors convey in several logic textbooks. Variables as placeholders for ...
1 vote
171 views

What is meant by the expression ∃xHx, if H stands here for “is a human being”?

How academics would go about explaining in everyday English, so without any philosophical or mathematical jargon, what is meant by the expression ∃xHx, if H would stand here for “is a human being”. On ...
1 vote
61 views

A question on contrapositives and predicates

So I am a freshman taking an intro class to logic. And the question started off from a class exercise we've got which asked us to identify the covering generalization for the following conditional ...
40 views

Extending the use-mention distinction to account for variables and predicates

When we talk about the use-mention distinction, often the following is said: To use an expression means to refer to its meaning, to mention an expression means to refer to the expression itself. I ...
2k views

What did Bertrand Russell mean exactly when he said that *such that*, while fundamental both to formal logic and to mathematics, is "undefinable"?

Bertrand Russell in Principles of mathematics (1903) presents the notion of such that as fundamental to logic and mathematics, and states that it is “undefinable”: The Indefinables of Mathematics ...
260 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 ...
97 views

Treating truth as a predicate

It is interesting to me that in some conventions of logic I have seen (generally, common ones), the form of logical language is designed to make “truth” implicit. For example, merely to write: P(x) is ...
306 views

Why is the law of the excluded middle not a exclusive disjunction?

So the law of the excluded middle, as I have read in every logic textbook that I have read, has been ( ϕ ∨ ¬ ϕ ) , but this seems somewhat unintuitive, since I was under the impression that the ...
50 views

I am not sure which formalization is right [1] or [2]: 'The teacher of Plato does not exist.' [1] ∃x(Tx,p ∧ ∀y[Ty,p → y=x] ∧ ¬∃y[y = x]) [2] ∃x(Tx,p ∧ ∀y[Ty,p → y=x] ∧ ¬∃z[z = x]) Is it possible to ...
2k 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?
107 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
65 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 ...
346 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 ...
507 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 ...
62 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
6k 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 ...
59 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.
45 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 = √�...
156 views

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
1k 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
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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 ...
21k 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
118 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 ...
41 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 '...
23 views

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 ...
148 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 ...
112 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), ¬□...
331 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 ...