For questions related to symbolic logic, also known as mathematical logic. Topics might range from philosophical implications of metamathematical results to technical questions.
Questions tagged [symbolic-logic]
298 questions
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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, either that proposition is true,
or its negation is.…
Tames
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Selection of logical connectives {¬,∨,∧,⇒,⇔} in set theory?
Nearly every treatment of set theory, whether Paul Halmos' Naive Set Theory, Herbert Enderton's Elements of Set Theory, Patrick Suppes' Axiomatic Set Theory, etc. introduce a common set of logical connectives, namely "not" ¬, "inclusive or" ∨, "and"…
EthanAlvaree
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How to prove (A v B), (A → C), (B → D) therefore (C v D)
Obviously since A → C and B → D then if A v B one of C or D must be true.
My only idea is v must be introduced, but how would I use subproofs to show one of A /\ C or B /\ D is never false if A v B?
sumsum2
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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 no members, which is called the null-class.
This…
Speakpigeon
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What's the difference among the logical relations :=, =, and ≡?
I understand that ≡ is logical equivalence, "iff". '=' is a symbol for numerical equivalence. And ':=' is an identity claim. I often only see '=' and ':=' used with variables and names, while ≡ only appears with predicates. I'm not sure if this is…
RECURSIVE FARTS
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How do proofs about logic fit into a logical framework?
I'm learning logic from Michael O'Leary's A First Course in Mathematical Logic and Set Theory. In chapter 1 he carefully explains the meaning of logical implication (p ⊨ q), logical inference (p ⟹ q), proof (p ⊢ q), and "star" proof (p ⊢* q). (The…
WillG
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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 frames strike me as very algebraic in flavour, even if they…
jdonland
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How did symbolic logic show that Heidegger's assertions about the nothing were illogical?
In his inaugural address at Freiburg University in 1929, Heidegger
explicitly challenged the central place given to logical principles in
neo-Kantianism, on the basis of a radical account of ‘the nothing’.
Two years later, Carnap used the tools of…
Sayaman
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What exactly is a first-order logic?
Can someone explain in simple terms what exactly is a first-order logic?
From my amateur standpoint, I think that first-order logic is a some kind of a system of symbols and general logical rules and operations defined on that set of symbols in such…
Grešnik
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Is it true that (P∧Q≡P)⇔(Q≡⊤)?
Consider the statement
(P∧Q≡P)⇔(Q≡⊤)
Where P and Q are statements, and ⊤ denotes the tautology (true) statement. It seems intuitively true that the above biconditional statement is true. But I would like to prove it.
One direction is easy enough…
EthanAlvaree
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How to model "forget about" in first order logic?
The other day, my housemate said "Don't forget to not leave the spoon at the bottom of the container". I understood what he meant: "Do not leave the spoon at the bottom of the container due to forgetfulness".
Turning the words over in my head, I…
Steven Gubkin
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In Fitch, how does one prove "(P → Q)" from the premise "(¬P ∨ Q)"?
It's all in the question really. I am working on a proof in Fitch for a class, but I am very much stuck.
I am proving the tautology that "(P → Q) ↔ (¬P ∨ Q)", and I have already finished half of it, but now I must prove that "(¬P ∨ Q)" implies "(P →…
Zenreon
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Propositional Logic: How to prove the contraposition in the Fitch system?
Given that:
p ⇒ q
prove that:
¬q ⇒ ¬p
using the Fitch system.
(This being the proof of the Contraposition)
Am95
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What does the truth-value of a material implication represent?
This question comes from my attempts to understand what the truth value for a material implication with a false antecedent represents. I have seen several justifications for this convention, usually through example, but each one seems to imply that…
IgnorantCuriosity
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What is the explicit reasoning behind proof by contradiction?
From my understanding, proof by contradiction consists of the following steps.
1. Show that p -> q, where "->" is the conditional.
2. Show that q is false.
3. Deduce from a truth table that p must be false.
My problems with this are the…
IgnorantCuriosity
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