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I don't understand why propositional logic isn't enough. Can't any first order statement be encoded in the form of a propositional logic? What does first order logic do for us that we cannot do in propositional logic?

James Prim
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    What about the quantifiers? – MJD Dec 27 '18 at 00:02
  • @MJD What do you mean? Aren't those encodable in propositional logic? – James Prim Dec 27 '18 at 00:03
  • If they were, why would we bother with first-order logic? – MJD Dec 27 '18 at 00:15
  • @MJD This seems to be at the heart of the problem. Why not write a short answer with some details about why quantifiers cannot be encoded? – Arthur Dec 27 '18 at 00:20
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    @MJD That's precisely what I am asking though – James Prim Dec 27 '18 at 00:32
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    You write "Can't any first order statement be encoded in the form of a propositional logic?" I'm not sure why you think this is the case. How would you encode "$\forall x \exists y(x\not=y)$" in propositional logic? – Noah Schweber Dec 27 '18 at 00:47
  • @NoahSchweber disjunction of if/implies, like if $x=1$ then $y^3 = 1$. If $x=2$ then $y^3 = 8$. If $x=3$ then $y^3 = 27$, and so on. $(a \to b) \lor (c \to d) \lor (e \to f) ...$ and so on. – James Prim Dec 27 '18 at 00:50
  • @JamesPrim That's not a propositional sentence, though, since it's infinitely long. Also, who said every element of our domain was definable (that is: we might not have a way to refer to every specific element)? – Noah Schweber Dec 27 '18 at 00:50
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    What do you mean? Who says it can't be infinitely long? We have infinitely repeating decimals and that seems to be okay. Why are infinities okay there but not here? – James Prim Dec 27 '18 at 00:51
  • I also don't know what you mean by the elements not being definable? If something is undefined how can we even talk about it? – James Prim Dec 27 '18 at 00:52
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    @JamesPrim "Who says it can't be infinitely long?" It's part of the basic definition of propositional logic, see e.g. the sources mentioned here. As to definability, that's a specific term in the context of first-order logic, and it's perfectly possible for a structure to have undefinable elements. I think one of the issues here is that you don't have a sufficiently clear picture of each of the logics in question, and should clear that up first. – Noah Schweber Dec 27 '18 at 01:27
  • @NoahSchweber From your link it seems that infinite formulas aren't allowed because the definition of wffs propositional logic more or less prevent them by saying everything is finite? – James Prim Dec 27 '18 at 01:34
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    @JamesPrim Yes, like I said, it's part of the definition. It's an important part too: drop it and you get something very different. Something quite interesting, but not propositional logic anymore, and with importantly different properties (e.g. compactness fails once you allow infinitely-long formulas). – Noah Schweber Dec 27 '18 at 01:35
  • @NoahSchweber From what I can tell the compactness theorem says that if all finite subsets of your set of sentences is consistent, then so is your original set, correct? Is proving compactness normally an inductive proof, which would call for the finite cases and such? I am asking because it seems like we're "sidestepping" the issue by sort of encoding this concept of infinity into these quantifiers. – James Prim Dec 27 '18 at 01:57
  • (I'm not trying to be an infinity-crank here just trying to understand why one is allowed but the other isn't) – James Prim Dec 27 '18 at 01:58

1 Answers1

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Propositional logic cannot account for, amongst other things, the validity of such arguments as

Socrates is a man.
All men are mortal.
$\therefore$ Socrates is mortal.

In propositional logic, we cannot do any better than to translate this argument as (e.g.)

$S$
$M$
$\therefore R$

which is plainly invalid.

In first-order logic, we can formulate this intuitively valid argument in such a way that it does turn out valid, thanks to the universal quantifier:

$Ms$
$\forall x(Mx \to Rx)$
$\therefore Rs$

To generalise, first-order logic allows us to get at the internal structure of certain propositions in a way that is not possible with mere propositional logic. The possession or non-possession of important logical properties turns on the precise nature of these internal structures. So it is important that we have adequate tools at hand to analyse them. First-order logic gives us many of these tools.

solisoc
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