0

To introduce the basics of logic, one ordinarily begins with propositional logic and then proceeds to predicate logic. Unfortunately, the examples of propositions typically either use "real-world sentences" (e.g., "George Washington lived in Mt. Vernon") or, when strictly mathematical, actually use quantifiers implicitly (e.g., "$6$ is an even number" -- which of course really means "there exists an integer $k$ for which $6 = 2 k$; or "an integer that is a multiple of $6$ is even" which similarly involves quantifiers implicitly).

For a very brief introduction to logic that I'm writing -- part of a textbook that is not about logic or set theory -- I'm looking for simple examples of propositions that deal only with mathematical objects, not real-world entities and that do not implicitly involve quantifiers. And I want to use these in order to provide instances of conjunction, disjunction, implication, and equivalence.

I am willing for the examples to involve terms such as "integer", "natural number" along with $=$, $<$, etc. Thus I take as understood such propositions as "$0 = 1$" and "$3 < 4$". But not expressions such as "$2 \in \mathbb{N}$" that explicitly involve sets of objects.

Can you provide any more such examples (other than just changing the particular numbers in those)?

I know this is asking a lot!

murray
  • 778
  • 3
  • 16
  • How about $3 < 4$? for an implicatoin/conjunction: $3<4 \wedge 4 < 5 \Rightarrow 3 < 5$ – G. Chiusole Aug 27 '19 at 13:37
  • @G. Chiusole "$3 < 4$" just means "There exists a number $n$ such that $3+s(n)=4$". So implicit quantification again. – Natalie Clarius Aug 27 '19 at 13:44
  • In that case you need to resort to set theory, since all the usual propositions are build upon sets. And in that case $4 \in \mathbb{N}$ is your best bet – G. Chiusole Aug 27 '19 at 13:46
  • @lemontree: Yes, I'm away of the actual meaning of "3 < 4", and so in a strictly rigorous exposition of logic and math, one would have to introduce quantifiers, not to mention the axioms of set theory, etc., before giving the definition of "<". But one has to begin somewhere with "simple" examples. – murray Aug 27 '19 at 13:48
  • @G. Chiusole: Yes, that about the only such simple example I've been able to come up with so far. (Of course, there's the instantiation there of the transitive law, which of course is a quantified statement: "For all $a, b, c \in \mathbb{Z}$, if$ a < b$ and $b < c$, then $a < c$".) – murray Aug 27 '19 at 13:51
  • 1
    @murray Indeed, if you want to start from the very very very start, then you cannot use transitivity and such. However, then you also cannot use natural numbers or functions, as they are themselves defined in the theory of sets. In a nutshell: if you need to be this strict on formalism you have to use sets. – G. Chiusole Aug 27 '19 at 13:54
  • Also, depending on how much time (of your course) you want to spend on propositional logic, you might want to formally introduce ZFC and give examples of propositions within that theory – G. Chiusole Aug 27 '19 at 13:56
  • I am not being strict on formalism; quite the opposite! This is for a very short, first, subsection of "chapter 0" in an introductory textbook on topology. – murray Aug 27 '19 at 13:59
  • 1
    @murray Ah I see. The way I see it you have 2 options. Either rigorously define propositions and then given an example of it such as $Q \wedge P \Rightarrow P$ (see for example Shoenfield subsection 2.4) or give examples of mathematical statements, assuming readers are familiar with the interpretation of $3 < 4$. – G. Chiusole Aug 27 '19 at 14:12
  • 1
    $0=0$ and $\lnot (0 = 1)$. Then, cook them together with connectives. – Mauro ALLEGRANZA Aug 27 '19 at 14:12

1 Answers1

0

I think "6 is an even number" works just fine as a propositional logic claim ... to treat it as an existential seems unnecessarily complicated. And you can still represent it using something like $Even(6)$ ... that involves a predicate and a constant, which we typically only introduce in predicate logic, but it has no quantifiers. And, you can do propositional logic with such claims just fine.

Bram28
  • 100,612
  • 6
  • 70
  • 118
  • But I need to defer such formalities about predicates until after simple propositions. – murray Aug 27 '19 at 13:48
  • @murray Well, that's your choice. I am using a book that covers propositional logic using predicates. So you don't dig into the predicates (i.e. something like $Even(6)$ would still be treated just like $P$ or $Q$), but at least you can make it more meaningful .. and that seems to be what you are looking for. And it seems like you are already ok with statements like $0=1$ and $3<4$, which are predicates too; they're just short for $=(0,1)$ and $<(3,4)$. So ... maybe do this in 3 steps: start with $P$ and $Q$'s, then introduce predicates and constants, and finally dovariables and quantifiers. – Bram28 Aug 27 '19 at 14:55