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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Was Tarski the first to discuss the logically of the truth predicate?
Tarski famously discussed, formally, the logically of the truth predicate, in The Concept of Truth in Formalised Languages (1935).
Was he the first to do so?
Thank you for any scholarly reference.
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What is a definition, written in symbolic logic, for a person living nearby?
Students often need some axioms and/or definitions to play with in order to learn formal logic.
What is a definition of a neighbor written in the style of symbolic logic?
By neighbor, we mean a person ...
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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 ...
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Is Russell's Paradox a semantic paradox or a syntactic paradox?
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 ...
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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)→...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
...
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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 ...
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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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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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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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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 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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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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