Choosing between the 'and' and 'implies' connectives

Question

All politicians are liars : $\forall x(\operatorname{pol}(x) → \operatorname{liar}(x))$

Some politicians are liars : $\exists x(\operatorname{pol}(x) \land \operatorname{liar}(x))$

No politicians are liars : $\forall x(\operatorname{pol}(x) → ¬\operatorname{liar}(x))$

Some politicians are not liars : $\exists x(\operatorname{pol}(x) \land ¬\operatorname{liar}(x))$

So, do we just use 'and' for $\exists$ statements and 'implies' for $\forall$ statements?

Not necessarily. For example $\forall n \in \mathbb{N}( \lnot Even(n) \land Odd(n))$ is true. Or “there is a man who can’t speak whenever he is sick” is $\exists x ( sick(x) \Rightarrow silent(x))$. In general you just want to use $\land$ to express two things occurring together and $\Rightarrow$ to express one thing following from another. These may or may not occur with certain quantifiers. — Joa, May 29 '24 at 04:59
@Joa Your example with the man is slightly misleading. You are implicitly universally quantifying over “time” in that implication. There is pretty much no scenario where using the implication without universal quantification is the clearest way of describing the situation. — Malady, May 29 '24 at 05:22
I am voting to reopen : There may be other Duplicates available , though I think the Current Choice is not the apt Duplicate. More-over , the Current Choice is unsuitable for the Current Question. — Prem, May 29 '24 at 05:28
See also Restricted quantifiers - Logic as well as Difference between Universal Quantification and Existential Quantification (Restricted Domain) and Restricing domains for universal and existential quantifiers. — Mauro ALLEGRANZA, May 29 '24 at 09:17
Remember that the negation of $\forall x : P(x)\implies Q(x)$ is $\exists x : P(x)\land \neg Q(x)$ — Lorago, May 29 '24 at 10:13
Note that if $\exists x: \neg P(x)$ ($P$ would not be very useful otherwise), then $\exists x: [P(x)\to Q]$ is always true for any $Q$ (see Drinker's Paradox). Similarly, $\forall x: [P(x) \land Q]$ would always be false. — Dan Christensen, May 29 '24 at 16:50
It's not like what kind of wine goes with what kind of cheese. Use "and" when you mean "and" and use "implies" when you mean "implies". — user14111, Jan 16 '25 at 07:33

Lieven · Answer 1 · 2024-05-29T10:29:53.433

There is no such requirement from formal logic, so I propose to look at the question semantically.

A universal quantifier ("for all") applies to all entities in the universe. You would normally not apply a predicate $\hbox{liar}(x)$ to all entities, but only to a well-circumscribed subcollection, like politicians. Otherwise the predicate itself becomes meaningless. There are two ways to formalize this restriction to a subcollection: as an "if/then", or as an "or not" (to exclude all non-politicians). While both approaches are logically equivalent, the if/then speaks more directly to the human reader's natural sense of language.

An existential quantifier guarantees that we are already talking (within the brackets) about one specific entity: no more need to restrict to a subset.

Of course, every existential quantifier is equivalent to the negation of a universal quantifier, and vice versa. Negating an existential "and" will automatically provide a universal "or" which can then be converted into a universal "if/then".

Existential quantifiers need not include an "and" to be semantically meaningful. It could be useful just to know that there exists a politician. The reason why you encounter these "and" operators in your textbook examples of existential propositions is that they can be used to deny a related universal proposition.

ryang · Answer 2 · 2025-01-16T07:29:43.957

So, do we just use 'and' for ∃ statements and 'implies' for ∀ statements?

For the canonical categorical propositions (as illustrated by your examples), yes.

But in general, no:

$$\forall x (Ax \land Bx)\tag1$$ $$\forall x (Ax \to Bx)\tag2$$

Sentence (1) is simply a stronger assertion than sentence (2): the universes that satisfy sentence (1) is a proper subset of the universes that satisfy sentence (2).
$$\exists x (Ax \land Bx)\tag3$$ $$\exists x (Ax \to Bx)\tag4$$

Similarly, sentence (3) is simply a stronger assertion than sentence (4).

Dan Christensen · Answer 3 · 2024-05-29T20:52:20.973

So, do we just use 'and' for $\exists$ statements and 'implies' for $\forall$ statements?

No rule says you must, but weird things can arise otherwise, things you may want to avoid.

Suppose, for example, that like many predicates, there exists $x$ such that $P(x)$ is false. ($P$ may not be of much use otherwise.) Then there exists $x$ such that "$P(x)\implies Q$" for absolutely any proposition $Q$.

$~~~~~~\exists x: \neg P(x) \implies \exists x: [P(x) \implies Q]$

Similarly...

$~~~~~~\exists x: \neg P(x) \implies \neg \forall x: [P(x) \land Q]$

Moral of the story: Proceed with caution when confronted by:

$~~~~~~\exists x: [P(x) \implies Q]~$ or $~\forall x: [P(x) \land Q]$.

(Formal proofs available on request.)

Choosing between the 'and' and 'implies' connectives

3 Answers3