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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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 ...
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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 ...
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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 ...
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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), ¬□...
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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 ...
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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 ...
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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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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 ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 '...
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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 ...
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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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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 ...
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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, &...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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.
...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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!
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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 ...
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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 ...
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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 ...
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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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Semantic consequence and Sound Argument
Is that correct to say that semantic consequence is equivalent to the concept of sound argument in classical propositional logic?
If it is the case, arguments or theories with contradictory premises ...
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What context do quantifiers make valid expressions?
Say I have a formal language such that x is an individual constant and symbolically has a particular value (say 2) a proposition such as x+1=3 already has the value of true, and I cannot define a ...
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What is the nature of the term 'variable', and is it used differently in math, computer science, and logic?
Say I'm given an expression and talk about x changing what do we really mean by this linguistically? What inferences can be drawn about the nature of variables from their practical usage?
When we talk ...
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Proof for "⊢ (A → ¬¬A)"
I've spent 4.5 hours on this, with no exaggeration. I clearly have no idea what I'm doing here, and it's become a serious time sink. If any of you could help in proving this, I would be eternally ...
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Help with proving: P, ¬(Q ∧ P) ⊢ ¬Q
Here's the issue, there's no usage of derived rules allowed. So no DeMorgan's Law. All that's allowed is the basic TFL elimination/introduction rules, IP, (e)X(plosion), and ⊥. I'm absolutely lost on ...
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Nested Quantifiers Proof - Logic
When I prove this: -∃x.P(x) ⊢ ∀x.-P(x) [True]
I did it like that:
∀x.-P(x) ⊢ ∀x.-P(x) because (negative ∃) -∃x.P(x) becomes ∀x.-P(x) so that we can say that it's true.
However, I didn't ...
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Why aren't Kripke semantics "syntax in disguise"?
The Wikipedia article on Kripke semantics suggests that they were considered a major breakthrough in part because algebraic semantics were seen as merely "syntax in disguise". But Kripke ...
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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 ...
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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 ...
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Are contradictory propositions in the propositional logic still contradictory in the predicate logic?
There is one seeming issue I happened upon that bothers me to no end.
Take a proposition like “Snow is white”. “Snow is white” and its negation “Snow is not white” are obviously contradictory. However,...
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How to prove, in modal logic, that □A→A is valid (T axiom) iff R is reflexive?
How to prove, in modal logic, that □A→A is valid (T axiom) iff R is reflexive? I'm not sure how to prove axiom in reverse?
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How to show disjunction can be expressed as a conditional
How to show that formula "A ∨ B" can be constructed from A and B using only the conditional connective (→).
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