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- Does every proof need an axiom saying it works? June 20, 2024I am wondering whether for every (valid) proof $P$ done in mathematics, at least one of the following statements are true: There is an axiom guaranteeing that its schema indeed gives us license to conclude the truth of what it purports to prove (an axiom saying the proof works, like how the axiom of induction […]Princess Mia
- Can conjunction be expressed via negation and equivalence? June 20, 2024Is it possible for conjunction to be expressed via negation and equivalence? I am studying for my Logic exam and I can't find answer to this anywhere. If it's not possible could you give me explanation why?Ива Милошевић
- $A \cup C \subseteq B \cup C$ iff $A\setminus C\subseteq B\setminus C$ June 20, 2024I have attempted to prove the theorem, but I'm not sure whether I can invoke proof by contradiction on the converse the way I did or not. $A \cup C \subseteq B \cup C$ iff $A\setminus C \subseteq B\setminus C$. Proof. $(\rightarrow)$ Suppose $A \cup C \subseteq B \cup C$, and suppose $x \in A\setminus […]Approxiz
- Truth of a statement maintained when performing equal operations June 20, 2024If we have two expressions, a+b, and b+a, and we assume that: a+b=b+a if you subtract b from both sides, giving you: a=a Knowing that a=a is a true statement, does this prove that a+b=b+a is also true? Generally speaking, can a true statement only come about from another true statement? Is it possible for […]Lucas
- What is a predicative system? June 20, 2024In mathematical logic, a number of theories such as PA (Peano arithmetic) are called predicative. See for example the table at the bottom of this page: https://ncatlab.org/nlab/show/ordinal+analysis Does this mean that one cannot write an impredicative definition in them? If not, what does it mean? I use predicative as defined here: https://ncatlab.org/nlab/show/predicative+mathematics A definition is […]Anserin
- What is wrong with this in mathematical logic? June 20, 2024Consider this: ¬A ⇒ [ A ∨ ( T ⇒ R) ] ¬R ⇒ [R ∨ (A ⇒ R) ] (T ∨ D) ⇒ ¬R T ∨ D / D Now when solving it: 1 ¬A ⇒ [ A ∨ ( T ⇒ R) ] (imposition) 2 ¬R ⇒ [ R ∨ (A ⇒ R) […]ilikebread
- Right way to prove 'correctness' of encoding in another logic? June 20, 2024Let us assume that I am using ZFC set theory as my metalogic. Within my metalogic, I think it is possible to encode ZFC. So, I could have (meta)logical statements like IsZFCProvable(phi) == ∃proof: IsZFCProof(phi, proof). (IsZFCProof should also be computable, but I don't know how to express that...) In the same vein, I think […]Suraaj K S
- Proof in formal theory. June 19, 2024Axiom: $A \leq A$ Rules: $ \frac{A\leq C}{min(A,B)\leq C}$ $\frac{A\leq C; B\leq C}{max(A,B)\leq C}$ $\frac{min(A,B)\leq C}{min(B,A)\leq C}$ $\frac{A\leq max(B,C)}{A)\leq max(C,B)}$ $\frac{A\leq B; A\leq C}{A\leq min(B, C)}$ $\frac{A\leq B}{A\leq max(B,C)}$ $\frac{A\leq B; B\leq C}{A\leq C} $ Alphabet: $\Sigma= ({ F, P, I, x, y, z, /, min, max, \leq })$ Grammar rules: $P \rightarrow x; y; […]Danilo Jonić
- Determine grammar whose language is a set of logic formulas. June 19, 2024I need to find grammar for following set: $$\left\{ p_n, \perp , \neg, \Rightarrow \right\}, n \geq 0$$ Now, how I started this is by defining new symbol $P$ being $P \rightarrow p_n$. Then, I defined new symbol $I$ being $I \rightarrow P, I \rightarrow \perp, I\rightarrow \neg I, I \rightarrow (I \Rightarrow I)$. From […]Danilo Jonić
- Compactness of $2^I$ without Tychonoff theorem [duplicate] June 19, 2024Let $I$ be the set of all atomic formulas and $2=\{ 0,1 \}$. I want to prove the compactness of $2^I$ using Compactness theorem for propositional logic. How do I prove this?(I recently studied the compactness theorem, so I don't know what to do.)Rain
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