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I'm trying to learn sentential calculus now and I'm very confused with the following thing.

In many books on logic I've found out that before working with sentential calculus itself we need the FINITE SET of symbols to define the syntax of the sentential calculus. So the concept of set and even of finite set is supposed to be already known.

But for me this fact contradicts with the fact that axiomatic set theory and construction of natural numbers (for example with Peano axioms) supposes that logic is already known: so it appears that we need to know what is a finite set to use the concept of the set itself.

Please, help me understand how these things shoul be learned.

Thanks in advance for any help.

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  • $\begingroup$ For example, English Wikipedia writes that "The language of a propositional calculus consists of a set of primitive symbols, variously referred to as atomic formulae, placeholders, proposition letters, or variables, and a set of operator symbols, variously interpreted as logical operators or logical connectives. " $\endgroup$
    – Igor
    Apr 23 '13 at 13:41
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    $\begingroup$ There are many informal notions which are best left informal, or given a vague definition, when starting with mathematics. Later you learn how to formalize things. It's a crutch, but it's a good thing. $\endgroup$
    – Asaf Karagila
    Apr 23 '13 at 13:50
  • $\begingroup$ Maybe this question and some links given there might be of interest. $\endgroup$ Apr 25 '13 at 10:45
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Two points.

  1. Do note that the use of set talk is entirely dispensable in e.g. Wikipedia's "The language of a propositional calculus consists of a set of primitive symbols, variously referred to as atomic formulae, placeholders, proposition letters, or variables, and a set of operator symbols, variously interpreted as logical operators or logical connectives." It would do exactly as well to use plurals: "The language of a propositional calculus consists of some primitive symbols, variously referred to as atomic formulae, placeholders, proposition letters, or variables, and some operator symbols, variously interpreted as logical operators or logical connectives." There is no need at all to think of e.g. the propositions (plural) as forming a set (singular), no need to suppose ourselves committed to anything over and above the propositions and operators and truth-values when doing propositional calculus. It makes for a certain grammatical ease to use low-commitment set talk to enable us to talk of many things at once, but it is a façon de parler, and you shouldn't read too much into it.
  2. But waive that point. Even if you do take the set talk more seriously, that's fine: for note that it here belongs to the informal mathematical background which is to be presupposed when investigating formal systems as mathematical entities. There is no circularity in helping ourselves to some unproblematic informal notions as we get on with our informal mathematics when standing outside a formal system (whether Propositional Calculus or ZFC) and investigating its properties.
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In the mathematical application of formal logic, it has to be developed at the same "level" as the things you want to use it to study. For example, if you want to study groups as they are constructed in your ambient mathematical 'universe', then if you want to talk about the "theory of a group", it had better be in a formal logic developed within the ambient mathematical universe, so that you can use the ambient mathematics to connect the results of formal logic and apply them to group theory.


If you want to do philosophical applications, then you have a whole other set of issues to worry about; e.g. to approach foundations epistemologically, you're going to have to choose some response to the infinite regress argument.

I like a variant of coherentism: e.g. among your prior notions is some form of set theory, which is used to develop formal logic, which in turn is used to develop a formal version of set theory, which develops its own internal formal logic, and work there. Or maybe go round the spiral another time or two for technical reasons.

There are variants, of course; you might start with a prior notion of logic instead of a prior notion of set theory. Or you might just suppose it's turtles all the way down, and as long as you can show each loop of the spiral is sufficiently similar to the next loop, you're happy.

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