Is there no propositional letter in first order logic?

Question

In Kaye's math logic book, in propositional logic, there is a set of propositional letters, but there is no symbols for formulas or sentences in first order logic. Does the book miss it?

Strangely, the book uses $\phi$ and $\psi$ to represent formulas later enter image description here

Thanks.

\xi is $\xi$ (and not ksi), you're looking for \psi which is $\psi$. — Asaf Karagila, Jul 23 '14 at 21:22

score 4 · Answer 1 · answered Jul 23 '14 at 21:19

4

In the usual formulations of first-order logic, there are no proposition letters. So it is definitely not an oversight, it is absolutely standard.

The $\phi$, and $\psi$ that are used later are not proposition letters, they are a way of referring to arbitrary formulas. In particular, they are not part of the formal language.

answered Jul 23 '14 at 21:19

André Nicolas

514,336

thanks. Why are there propositional symbols in propositional logic? – Tim Jul 23 '14 at 21:36
You are welcome. There are propositional symbols because without them, the language would be empty. – André Nicolas Jul 23 '14 at 21:40
Are True and False atomic formulas? The book doesn't say so, but I think so. – Tim Jul 23 '14 at 21:42
@Tim: The quote in your question plainly lists $\top$ and $\bot$ not among the "atomic formulas", but as "composite formulas" -- so it considers them not atomic. In most cases this choice is quite unimportant. – hmakholm left over Monica Jul 23 '14 at 21:44
Some people include True and False, which are atomic formulas. All the logic books I have used leave them out in the presentation of first-order logic. – André Nicolas Jul 23 '14 at 21:45
why are they left out? – Tim Jul 23 '14 at 21:46
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Because they are not needed. – André Nicolas Jul 23 '14 at 21:47
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@Tim There's no practical difference between a zero-arity relation symbol, e.g., as in $P() \land Q()$ and a propositional variable, e.g., as in $P \land Q$. – Joshua Taylor Jul 24 '14 at 18:22

score 2 · Answer 2 · answered Jul 24 '14 at 12:22

About the question regarding the purported absence of symbols for formulae (or statements) :

but there is no symbols for formulas or sentences in first order logic. Does the book miss it?

we have to note that the "basic convention" (alas! left implicit in Kaye's book) is that lower case letters from Greek alphabet are meta-variables (i.e. variables in the meta-language) standing for formulae of the propositional and first-order languages, while upper case letters stand for set of formulae (see page 65 : a "derivation from assumptions $Σ ⊆ BT(X)$ is a derivation of finite length where each statement in it is an element of $BT(X)$ and which uses only the following proof rules [...]").

Please, note that in the propositional logic, formulae are called boolean terms, and $BT(X)$ is the name of the set of boolean terms over the set $X$ of propositional letters [see page 64].

See also page 119 for the :

Definition 9.4 : A statement or formula of a first-order language is [...]

and see page 120 :

Definition 9.6 A formula $σ$ is closed or is a sentence if every occurrence of every variable $x$ in it is in the scope of a matching quantifier [...].

Thus, a sentence is a (closed) formula and we use the same symbols (lower case Greek letters) for them.

Compare with Dirk van Dalen, Logic and Structure (5th ed - 2013), page 7, where the "basic convention" is made explicit :

Definition 2.1.2 The set $PROP$ of propositions is the smallest set $X$ with the properties :
(i) $p_i \in X (i \in \mathbb N), \bot \in X$,
(ii) if $ϕ,ψ \in X$, then $(ϕ ∧ ψ), (ϕ ∨ ψ), (ϕ→ψ), (ϕ↔ψ) \in X$,

(iii) if $ϕ \in X$ then $(¬ϕ) \in X$.

[...]

A warning to the reader is in order here. We have used Greek letters $ϕ,ψ$ in the definition; are they propositions? Clearly we did not intend them to be so, as we want only those strings of symbols obtained by combining symbols of the alphabet in a correct way. Evidently no Greek letters come in at all! The explanation is that $ϕ$ and $ψ$ are used as variables for propositions. Since we want to study logic, we must use a language in which to discuss it. As a rule this language is plain, everyday English.

We call the language used to discuss logic our meta-language and $ϕ$ and $ψ$ are metavariables for propositions. We could do without meta-variables by handling (ii) and (iii) verbally: if two propositions are given, then a new proposition is obtained by placing the connective $∧$ between them and by adding brackets in front and at the end, etc. This verbal version should suffice to convince the reader of the advantage of the mathematical machinery.

score 2 · Answer 3 · answered Jul 24 '14 at 14:10

In propositional logic, we usually think of the propositional letters as representing statements. For example if $\phi$ means "It is cloudy" and $\psi$ means "It is raining", then $\psi \rightarrow \phi$ means "If it is raining, then it is cloudy."

In first order logic we take it a step further and start caring what those statements are (along with some other changes). So now we instead put the actual statement, or formula, into our logical sentences. For example instead of saying $(\phi \wedge \psi) \rightarrow \theta$, we might say $(x \leq y \wedge y \leq x) \rightarrow x = y$. Thus we no longer have the need for (or allow) propositional variables.

The problem here is that we still sometimes want to talk about what kind things that are true of arbitrary formulas. In that case we will use letters like $\phi$ to represent arbitrary formulas. $x$ is not a number despite the fact that is often represents numbers. In the same way, $\phi$ is not a logical symbol.

Is there no propositional letter in first order logic?

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