# Tag Info

### Why is A implies B true if A is false and B is false?

Humans are bad at logic until they have to employ it to figure out human affairs. Think of "if $A$ then $B$" as a kind of promise: "I promise to you that if you do $A$ then I will do $B$". Such a ...

### Can proof by contradiction work without the law of excluded middle?

You asked (I am making your question a bit crisper): "What formal guarantee is there that it cannot happen that both $\lnot p$ and $p$ lead to a contradiction?" You seem to worry that if logic is ...
Accepted

### Why does soundness imply consistency?

I recommend looking into formal logic beyond vague, hand-wavy descriptions. It's interesting and highly relevant to computer science. Unfortunately, the terminology and narrow focus of even textbooks ...

### Why do ¬, ∀ and ∃ have the same precedence?

Order of precedence is simply a notional convenience. There is no notion of strength here, just notation. All three operators are unary operators with notation "$\circ\ \cdot$", where $\circ$ denotes ...

### Why is A implies B true if A is false and B is false?

It's a convention -- we could use a different one, but this one is convenient. Here's what Terence Tao says: This is discussed in Appendix A.2 of my book [Analysis 1]. The notion of implication ...
Accepted

### How is ‘x + ½ = 2 and x ∈ ℤ’ an open statement?

$$x=1$$ $$∃x: x=1$$ The first is an open statement, since no value for $x$ is given. $x$ is called a free variable here. The second is a closed statement, because it talks about all possible values of ...
Accepted

### Predicate Logic Notation: What does a "dot" mean?

The dot just means "such that"; it's often omitted. The difference between the two formulas is the difference between "everybody has a mother" and "there is somebody who is everybody's mother."
### What is the difference between $x:A$ and $x \Xi A$?
$x:A$ is a statement about objects in the formal system, like, for example, $\vdash 2+4:\texttt{int}$, whereas $x\Xi A$ is an expression in the formal system, like \$\texttt{if}~ 2 + 4 ~\texttt{==}~5 ~...