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 [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 ...
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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?
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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.
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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 ...
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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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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 ...
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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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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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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.
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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, ...
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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 (...
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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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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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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 ...
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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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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:...
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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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Fitch Question Please Help Me [closed]
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 ...
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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?
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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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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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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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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 ...
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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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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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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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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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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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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 ...
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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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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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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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