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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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
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3 answers
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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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2 answers
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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 ...
D J Sims'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
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58 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
95 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
342 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
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4 votes
0 answers
57 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
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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
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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
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1 answer
56 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 ...
NetCentric's user avatar
2 votes
0 answers
21 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
92 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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2 answers
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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
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3 votes
2 answers
305 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
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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
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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 ...
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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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
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9 votes
7 answers
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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
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 ...
GhostRocket's user avatar
3 votes
3 answers
99 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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3 votes
1 answer
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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 ...
Rieke's user avatar
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2 votes
2 answers
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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
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0 votes
2 answers
69 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
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1 vote
1 answer
101 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
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1 vote
4 answers
266 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
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0 votes
1 answer
220 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
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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 ...
Confused's user avatar
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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
104 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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-1 votes
2 answers
265 views

Is there a proof of exportation/importation from more obviously true implications such as Modus ponens?

Is there a proof of exportation/importation, namely, ((p ∧ q) → r) ⇔ (p → (q → r)), from more obviously true implications such as the Modus ponens, Transposition, de Morgan etc. I don’t believe that ...
Speakpigeon's user avatar
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0 votes
1 answer
72 views

Phrases such as 'x is an unspecified object' [closed]

Would a phrase like 'x is an unspecified object' be part of my meta-language? As x is a variable, such an expression is not meaningful in relation to any object in my interpretation, however we ...
Confused's user avatar
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0 votes
1 answer
172 views

Has anyone ever really constructed a countable model of set theory that falls in the trap of the Skolem's Paradox? [closed]

In an article named 'Skolem’s Paradox' on SEP, there is a description of the Paradox I'm asking about here: Skolem's Paradox arises when we notice that the standard axioms of set theory can ...
Michael's user avatar
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1 vote
1 answer
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Why does this conversion rule need ∃xT?

In the wiki page of Prenex normal form, there is a rule for conjunction as follows: (∀xφ)∧ψ is equivalent to ∀x(φ∧ψ) under (mild) additional condition ∃xT or, equivalently, ¬∀x⊥(meaning that at least ...
Michael's user avatar
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2 votes
2 answers
134 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. ...
Alexander Chaikov's user avatar
1 vote
2 answers
227 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 ...
Speakpigeon's user avatar
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3 votes
3 answers
187 views

How do you prove that a logic system is sound?

I am aware of the fact that a logic system must be sound, in order to be useful. However, I am not sure, about how, after setting up or coming up with the basic logic axioms that make up my system, I ...
Joselin Jocklingson's user avatar
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0 answers
105 views

Some Questions About A Truth Tree

The logical system used in this post is first-order logic. I’ve been reading Introduction to Logic: Predicate Logic, 2nd edition by Howard Pospesel, and I have some questions concerning a truth tree. ...
C. Frick's user avatar
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1 answer
172 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 ...
user236343's user avatar
2 votes
1 answer
65 views

Sentential Interpretation in P. Suppes (1957)

Patrick Suppes gives a working definition of sentential interpretation, based on a sentence maintaining its form. By working definition, I mean an incomplete definition that is needed for someone to ...
Then-Brief-864's user avatar
-4 votes
1 answer
142 views

Hi! I'm 99% sure my formal argument is valid, but can you check? [closed]

I wrote this argument, and while i'm sure it is valid, it has been awhile since I've done basic logic.Thanks!
Anon1313's user avatar
0 votes
1 answer
78 views

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 ...
Zane A.'s user avatar
1 vote
2 answers
171 views

When does a mathematical predicate have a truth value?

Say we have a predicate in a domain of real numbers, P(x), 2x+10=20 we know that we can existentially quantify this and say that the value x=5 makes this true, but we cannot talk about P(x) being ...
Confused's user avatar
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0 answers
190 views

What is mathematical analysis?

Hilbert's aim to reduce all mathematics to finite logical system was shown impossible by Goedel. He did mathematical analysis of logic itself (Goedel numbering). Turing defined algorithms, and ...
Ajax's user avatar
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1 vote
1 answer
228 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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