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Is propositional logic a logic system in the sense on p261 in Ebbinghaus' Mathematical Logic,

Is it correct that propositional logic, if it is a logic system, has no $$? (I think so, because of Does "the alphabet of the language of propositional logic" have no function symbols, relation symbols, and constants?)

Thanks.

Tim
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  • See my update... – Tim Sep 06 '20 at 23:59
  • At first sight, the definition is quite simple and seems to apply also to prop logic: we have a consequence relation for it. But if you re-read it, you can find that the def refers to a "symbol set $S$". Now, go back to page 14, where the language of FOL is defined: "Thus, in addition to the "logical" symbols such as "$\lnot$," and "$\land$", we shall need a set $S$ of relation symbols, function symbols, and constants which varies from theory to theory. Each such set $S$ of symbols determines a first-order language." – Mauro ALLEGRANZA Sep 07 '20 at 06:20
  • But in prop logic we have no "symbols" except the connectives... – Mauro ALLEGRANZA Sep 07 '20 at 06:20
  • If satisfied, please accept the answer below. – Mauro ALLEGRANZA Sep 09 '20 at 14:17

1 Answers1

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No, propositional logic is not a logical system.

The "point" of a higher logic like first-order logic is to analyze structures. The structures come first - we care about groups, rings, fields, etc. before we've formulated first-order logic. Propositional logic on the other hand isn't about structures at all; rather, it's about the basic relationships between properties (or sets, or similar). Think of it as a super-high-level, low-information perspective on something more complicated like first-order logic: first-order logic does have "propositional structure" in that the propositional connectives are present in FOL, but it's much more than that.

Admittedly, we can try to shoehorn propositional logic into the logical systems framework. For example, a truth valuation $\nu$ of a set of propositional atoms $\{p_i: i\in I\}$ can be conflated with a one-element structure $Val_\nu$ in a language $\{U_i:i\in I\}$ consisting of one unary relation symbol for each propositional atom - the definition being that $U_i$ holds on the unique element in $Val_\nu$ iff $p_i$ is true according to $\nu$. So we can conflate propositional logic with a logical system restricted to one-element structures and languages with only unary relation symbols. But this - and every other twist I can imagine - is highly artificial, and I see nothing gained in so doing.

The connection between propositional logic and logical systems is that propositional logic tells us exactly how sentences in sufficiently nice logical systems relate to each other via the propositional connectives (note that "having connectives" isn't part of the definition of logical system - it shows up in the definition of regular logical system, which also has the rather subtle relativization requirement which we don't care about right now).

Noah Schweber
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  • Thanks. To recognize propositional logic as a logic system, I wonder if it is necessary that "we can try to shoehorn propositional logic into the logical systems framework.... So we can conflate propositional logic with a logical system restricted to one-element structures and languages with only unary relation symbols." In https://math.stackexchange.com/questions/3811997/does-propositional-logic-have-structures-and-domains, I thought that propositional logic doesn't have structures because it has no $S$, so its interpretations have only assignments. – Tim Sep 08 '20 at 17:25
  • @Tim That's why I said "conflate" - they're not the same thing, but they're equivalent in a precise sense (we can convert from propositional sentences to sentences in this new proto-logical system - I say "proto" since it only applies to some $S$s - and back in an entailment-preserving way). – Noah Schweber Sep 08 '20 at 17:27