120
votes
Accepted
Why isn’t ‘because’ a logical connective in propositional logic?
It is because 'because' is not truth-functional.
That is, knowing the truth-values of $P$ and $Q$ does not tell you the truth-value of '$P$ because of $Q$'
For example, the two statements 'Grass is ...
- 102k
109
votes
Accepted
"Modus moron" rule of inference?
It's true that if $P$ is false then $P\Rightarrow Q$ is true. But the question is not asking if $P\Rightarrow Q$ is true, it's asking you if you can infer $P$ from $P\Rightarrow Q$ and $Q$.
Let's be ...
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92
votes
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Still struggling to understand vacuous truths
I've never been satisfied with the definition of the material implication in the context of propositional logic alone. The only really important things in the context of propositional logic are that $...
- 103k
77
votes
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What would happen if we just made vacuous truths false instead?
Notice that 3=5 is false. but if 3=5 we can prove 8=8 which is true.
$$ 3=5$$
therefore $$ 5=3$$
Add both sides, $$8=8$$
We can also prove that $$ 8=10$$ which is false.
$$ 3=5$$
Add $5$ to ...
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68
votes
Accepted
How do I make proofs with long formulae more readable without sacrificing clarity?
Main suggestion
Instead of
We want to show $\def\p{\phi}\def\q{\psi}\def\s{\vDash_{\tiny\text{PL}}}\def\ns{\nvDash_{\tiny\text{PL}}}\p_1,\,\p_2,\,\ldots,\,\p_n\s\q$ iff $\s\p_n\to(\p_{n-1}\to(\cdots\...
64
votes
Accepted
Why are proofs not written as collections of logic symbols but are instead written in sentences?
You have not translated the pages from Apostol's book into mathematical logic.
What you have done is to transcribe them into your own idiosyncratic shorthand, which may be useful to you but is less ...
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62
votes
What is meant by "logically incorrect"?
Rather than assume what the author means, I consulted the textbook (p. $155$) and examined the excerpt in context...
"Logic is the process of deducing information correctly -- it is not the ...
- 6,144
59
votes
"Modus moron" rule of inference?
This pattern is a logical fallacy called Affirming the Consequent, though I often call it Modus Bogus.
To show it is not a valid inference, here is a simple Refutation by Logical Analogy:
If I have ...
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56
votes
Still struggling to understand vacuous truths
Given that we want the $\rightarrow$ to capture the idea of an 'if .. then ..' statement, it seems reasonable to insist that $P \rightarrow P$ is a True statement, no matter what $P$ is, and thus no ...
- 102k
40
votes
Accepted
Does the unique existential quantifier commute with the existential quantifier?
No, they do not commute.
Consider for example the nonnegative reals as a linear order. Then $$\exists x\exists !y(y\le x)$$ is true (take $x=0$), but $$\exists!y \exists x(y\le x)$$ is false since ...
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35
votes
Accepted
Why is “or” (in logic) sometimes equivalent to “and” (in natural language)?
Consider: "All fruits and vegetables are nutritious"
Rather than:
$$\forall x ((F(x) \land V(x)) \rightarrow N(x)) \text{ Wrong!}$$
it translates as:
$$\forall x ((F(x) \lor V(x)) \rightarrow N(x))...
- 102k
34
votes
Why aren't vacuous truths just undefined?
Consider the statement:
All multiples of 4 are even.
You would say that statement is true, right?
So let's formulate that in formal logic language:
$\forall x: 4|x \implies 2|x$
(Here "$a|b$" ...
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32
votes
"Modus moron" rule of inference?
Even if affirming the consequent is not valid, other logical rules still work. Other logical rules still work. Therefore, affirming the consequent is not valid?
- 2,596
30
votes
Accepted
Why is "A only if B" equivalent to "(not A) or B"?
"A only if B" means that you can't have A without B, i.e. $\neg(A\wedge(\neg B))$, which simplifies (via de Morgan) to $(\neg A)\vee B$.
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27
votes
Is the disjunction of these two false statements true?
$$ (\forall x \in \mathbb{N})[x < 3 \implies x \ge 3] \lor
(\forall x \in \mathbb{N})[x \ge 3 \implies x < 3] $$
is not true, and your analysis doesn't imply that it ought to be.
What is true ...
25
votes
Why are proofs not written as collections of logic symbols but are instead written in sentences?
Most humans find it much easier to understand proofs written in a natural language (assuming, of course, it is a language that they are fluent in) with logic symbols kept to a minimum. You may find ...
- 458k
24
votes
What would happen if we just made vacuous truths false instead?
Clearly we want $P\rightarrow P$ to be true, wouldn't you agree?
I mean, if i say:
If Pat is a bachelor, then Pat is a bachelor
do you really dispute the truth of that claim, or claim that it ...
- 102k
23
votes
Accepted
OR in real life vs OR in Mathematical Logic
Both of the English sentences you've written are correct - the word "or" in English is ambiguous, it can mean both the logical operation OR and the logical ...
- 131k
22
votes
Simplify, equivalent for (p ∨ ¬q) ∧ (¬p ∨ ¬q)
$$(p \lor \lnot q) \land (\lnot p \lor \lnot q) \iff (p \land \lnot p) \lor \lnot q \iff \lnot q.$$
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22
votes
Accepted
What exactly is a contradiction and how does it differ from falsity?
Your understanding is correct. Put simply, a contradiction is a sentence that is always false. More precisely,
A statement is a contradiction iff it is false in all interpretations.
In propositional ...
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21
votes
Example of use De Morgan Law and the plain English behind it.
I think your basic problem here is that you expect negation to produce a "complete opposite", whatever that would mean.
The negation of
Miguel has a cellphone and a computer.
ought to be nothing ...
20
votes
Accepted
Logical proposition for "Every positive integer can be written as the sum of 2 squares"
Why would you switch from "forall" to "exists" if you wanted to specify "$x$ is positive"?
You're going to want $$(\forall x > 0)(\exists a \exists b)(a^2+b^2 = x)$$
or, if your language doesn't ...
- 36.5k
20
votes
"Modus moron" rule of inference?
Not stated in the other answers so far is that you misunderstood the meaning of logical validity, which means that, in every situation where the premises hold, the conclusion also holds. Now you may ...
- 59.5k
20
votes
Accepted
If null set is an element of a set then will it belongs to set or subset?
Elements
In the notation $A=\{\varnothing\}$ everything between the curly braces (except possible commas) is considered to be an element of the set, and we can denote this by
$$\varnothing\in A.$$
...
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20
votes
Accepted
Logic - How to say "Not only but also".
It is in fact the other way around. If you want to say "$\psi$ is necessary for $\varphi$", then this translates to $\varphi \rightarrow \psi$ (see https://en.wikipedia.org/wiki/...
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20
votes
19
votes
Accepted
Why aren't vacuous truths just undefined?
This is done so that classical propositional calculus follows some natural rules. Let's try to motivate this, without getting into technical details:
The expression "$P\Rightarrow Q$" should be read "...
- 18.7k
19
votes
What's the difference between biconditional iff and logical equivalence?
In short, $P \leftrightarrow Q$ is statement that could be either true or false. $P \equiv Q$ means that $P \leftrightarrow Q$ is always a true biconditional (so, $P$ and $Q$ have the same truth ...
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18
votes
What is the difference between implication symbols: $\rightarrow$ and $\Rightarrow$?
Usually, $\Rightarrow$ denotes implication in the metalanguage, whereas $\rightarrow$ denotes implication in the formal language that you want to talk about. For example, $$M \models \sigma \...
- 779
18
votes
Accepted
What is the logical flaw in this reasoning? Abusing $T \equiv T \vee F$.
I will focus on your problem with $T\iff T\lor F$.
Sure, under assumption that some formula $A(x)$ depending on a parameter $x$ is true (independent of the value for $x$), you may use that to deduce ...
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