41
votes
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 ...
- 31k
30
votes
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 ...
- 31k
17
votes
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 ...
- 12.1k
17
votes
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 ...
- 8,323
16
votes
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 ...
- 261
16
votes
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 ...
- 330
14
votes
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."
- 82.1k
10
votes
Why is A implies B true if A is false and B is false?
"A implies B" means (short) "if A is true then B is true".
It means (a bit longer) "if A is true then I claim that B is true; if A is false then I don't make any claim about B whatsoever".
Now ...
- 31.4k
9
votes
Accepted
Algorithm for deciding alpha-equivalence of terms in languages with bindings
There are several ways to do what you want. One of them is to use a different syntax representation under which $\alpha$-equivalent terms are actually equal. Such representations go under the name ...
- 31k
8
votes
Undecidable predicate logic is decidable by people?
The author is incorrect. A consequence of Godel's incompleteness is that any sufficiently complex logic has statements that are true, but have no proof of truth.
If every statement had a proof or ...
- 29.9k
8
votes
Accepted
How to understand quantifier without predication " ∀(λφ. (φ x m→ φ y))"?
The $x$ in $\forall x . P(x)$ is not an argument. It is a bound variable indicating which variable the quantifer is ranging over.
Let us compare the situation to the definite integral, for concretness ...
- 31k
8
votes
How is ‘x + ½ = 2 and x ∈ ℤ’ an open statement?
Whether it's an open statement or not depends on the structure of the statement, no on whether you can prove the truth value.
Look at the second statement. There are three things you could substitute: ...
- 31.4k
7
votes
Accepted
Why k- Vertex Cover is not in PTIME when it can be expressed in FO-logic
Your argument shows that for each fixed $k$, the problem $k$-VC can be solved in polynomial time (indeed, the algorithm enumerates all sets of size $k$ and checks whether they are vertex covers, all ...
- 278k
7
votes
Accepted
Any Non-trivial Logic System Defined with only Equality
If all you have is equalities and uninterpreted function symbols, then you have an algebraic theory a la universal algebra. A singleton set (or a collection of them in the multi-sorted case) is always ...
- 12.1k
7
votes
Can proof by contradiction work without the law of excluded middle?
I think your question boils down to "when doing formal verification with some sort of formal logic, what sort of guarantee do I have that the logic is consistent?". And the answer is: none. That's ...
D.W.♦
- 164k
7
votes
Accepted
Represent there are infinitely many in FOL
The existing answers provide examples of contexts where "there are infinitely many" can be expressed. However, there is an important sense in which "there are infinitely many" cannot be expressed in a ...
- 2,980
6
votes
Accepted
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 ~...
- 20.2k
6
votes
Why is A implies B true if A is false and B is false?
Let's take an example. Suppose that we want to express that $a$ is the only element of the set $S$ that satisfies property $P$. Then we can write
$$
\forall x \in S \;\; P(x) \Rightarrow x = a
$$
This ...
- 278k
6
votes
How would one prove the pigeonhole principle with a SAT solver?
SAT solvers work in the propositional calculus, and usually accept as input a formula in conjunctive normal form. There are several different propositional variants of the pigeonhole principle; they ...
- 278k
6
votes
What is the difference between superposition and paramodulation?
This is late for you, but probably might be of help to others. I myself had asked this question and I wrote the summary of my findings to the acl2-help mailing list on 8 and 9 September 2013: https://...
- 61
6
votes
Expressing 3-SAT in first-order logic
There has been a lot of work on formalizing mathematics, and in all of this work one needs to express definitions, theorems and proofs within the logic that one is using for formalization.
This is ...
- 2,516
6
votes
Accepted
How logic programming (especially ASP) is related to the reasoning in (first-order) logic?
Logic programming is proof search for some logic. Traditionally, this is the Horn clause fragment of first-order logic. Languages like lambdaProlog extend this to (intuitionistic) hereditary Harrop ...
- 12.1k
6
votes
Accepted
Why algorithms calculating non-tirivial zeros can't be used as proofs of Riemann Hypothesis?
The fact of the matter is, if a proof exists, then a Curry-Howard version of the program exists too. That doesn't mean that it's easy to find, though.
Undecidability still holds for Curry-Howard: if ...
- 29.9k
5
votes
Accepted
Why is first-order logic (without arithmetic) VALIDITY only recursively enumerable, and not recursive?
However, I am not seeing why VALIDITY is not recursive as well, because given a formula $\phi$, one could run two Turing Machines for THEOREMHOOD, one on $\phi$ and the other on $\neg \phi$, ...
- 14.6k
5
votes
Accepted
Using existential quantifier within implication
Your statement attempts to be express in formal logic the following sentence:
There exist $\sigma_{opt}$ and $n$ in $R^+$ such that if $0 < n < 1$ then $\sigma_{opt} = n$.
The attempt at ...
- 278k
5
votes
Accepted
Undecidable predicate logic is decidable by people?
Keep in mind that the book was written for undergraduate students, and there are aspects of logic that will demand a considerable level of sophistication, which is often omitted at this introductory ...
- 3,276
5
votes
Accepted
Resolution of Barber paradox
No, you don't need to specially add any such clause.
Here's where I think you might have gone wrong. The result is not 3 clauses (3 conjunctions). The result is a bunch of clauses. We get one ...
D.W.♦
- 164k
5
votes
Accepted
Logical characterization of P versus NP problem (and references for least fixed point logic)
Your second question first. The context of your question is called descriptive complexity theory. In this context, I would suggest the book Finite Model Theory by Heinz-Dieter Ebbinghaus and Jörg Flum....
- 2,308
4
votes
Quantified Boolean Formula vs First-order logic
Firstly, a quantified Boolean formula is a formula in quantified propositional logic (which consists of Boolean variables and quantifiers). A true quantified Boolean formula is a formula that can be ...
- 550
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