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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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Can modal logic be used to prove transitivity of equality isn't true in general?

EDIT - During the course of this post, I discovered that I had to rethink universal generalization. I didn't foresee I needed the possibility operator of modal logic inside the universal quantifier. ...
lee pappas's user avatar
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0 answers
76 views

Why is the proof of Godel's incompleteness theorem wrong? [closed]

As for the proof of the incompleteness of the formal system, it is obvious that Mr. Godel has committed a low-level logical error, which is highly undesirable. First, he constructs a self-...
Zhang Hong's user avatar
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0 answers
40 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 ...
Julius Hamilton's user avatar
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0 answers
60 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 ...
Speakpigeon's user avatar
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-1 votes
2 answers
69 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, ...
Speakpigeon's user avatar
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4 votes
1 answer
56 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 ...
san zhang's user avatar
1 vote
1 answer
54 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 ...
Speakpigeon's user avatar
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2 votes
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130 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 ...
Speakpigeon's user avatar
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3 votes
3 answers
52 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 ...
acey's user avatar
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2 votes
4 answers
242 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 ...
Speakpigeon's user avatar
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13 votes
6 answers
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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 ...
Harshit Rajput's user avatar
-1 votes
2 answers
77 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 ...
user129393192's user avatar
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0 answers
50 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 ...
Harshit Rajput's user avatar
4 votes
1 answer
84 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 ...
Alex Alex's user avatar
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1 vote
3 answers
161 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 ...
Speakpigeon's user avatar
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1 vote
1 answer
60 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 ...
Alex Li's user avatar
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1 answer
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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 ...
Harshit Rajput's user avatar
5 votes
3 answers
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 ...
Speakpigeon's user avatar
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0 votes
2 answers
95 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 ...
Julius Hamilton's user avatar
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1 answer
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formalization: definite description (narrow reading)

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 ...
Gion's user avatar
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5 votes
3 answers
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?
John Smith's user avatar
1 vote
0 answers
64 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 ...
Kristian Berry's user avatar
2 votes
4 answers
104 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, ...
Danyel 80be's user avatar
5 votes
1 answer
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 ...
Iain's user avatar
  • 153
3 votes
0 answers
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 ...
Kristian Berry's user avatar
2 votes
2 answers
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.
Danyel 80be's user avatar
0 votes
0 answers
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 = √�...
Kristian Berry's user avatar
0 votes
1 answer
63 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 ...
user avatar
2 votes
0 answers
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 ...
Kristian Berry's user avatar
3 votes
1 answer
136 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 ...
john doe's user avatar
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0 votes
2 answers
111 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), ¬□...
l0ner9's user avatar
  • 133
3 votes
2 answers
326 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 ...
l0ner9's user avatar
  • 133
1 vote
0 answers
55 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 ...
Corbin's user avatar
  • 1,239
3 votes
1 answer
130 views

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 ...
Speakpigeon's user avatar
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1 vote
3 answers
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 ...
gradual_gradient's user avatar
2 votes
1 answer
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 '...
help-me's user avatar
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0 votes
0 answers
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 ...
help-me's user avatar
  • 69
10 votes
7 answers
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 ...
Speakpigeon's user avatar
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2 votes
0 answers
71 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 ...
GhostRocket's user avatar
3 votes
3 answers
100 views

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 ...
Confused's user avatar
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1 vote
1 answer
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 ...
Rieke's user avatar
  • 115
2 votes
2 answers
97 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 ...
Confused's user avatar
  • 1,181
0 votes
2 answers
71 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 ...
Confused's user avatar
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1 vote
0 answers
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 '...
Confused's user avatar
  • 1,181
1 vote
1 answer
103 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 ...
Confused's user avatar
  • 1,181
1 vote
4 answers
275 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 '...
Confused's user avatar
  • 1,181
0 votes
1 answer
237 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 ...
Confused's user avatar
  • 1,181
0 votes
0 answers
79 views

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 ...
Confused's user avatar
  • 1,181
0 votes
0 answers
83 views

Need help with this Symbolic Logic Proof please

I am having trouble solving this proof. Line 5 is wrong, I know it's Demorgan's Law, but the proof machine doesn't accept that as an answer. I think it only accepts ~Elim, vElim, vIntro, ~Intro, &...
GarretBobbyFerguson's user avatar
0 votes
2 answers
105 views

First use of exportation/importation in formal logic?

Who is the logician who first used exportation/importation, namely, ((p ∧ q) → r) ⇔ (p → (q → r))? Gödel used it in his 1939 Logic lecture, but it doesn’t seem to have been known from the Aristotelian ...
Speakpigeon's user avatar
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