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While reading A tour through Mathematical Logic by Robert Wolfe, while defining the logical connectives in predicate logic, the logical connectives are itself used. For example, "Specifically, P $\wedge$ Q is true if and only if both P and Q are true." Here if and only if is used. Shouldn't the definition be independent of iff as if and only if has to be defined later. It is as though the mathematical definition depends on the linguistic definition. Help?

SHASHANK PATHAK
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  • You are not using the same language that you are defining while you are defining it. You are using some language that is a metalanguage wrt. the language you are defining. it can be assumed to be either set theory, or some informal natural language (depending on context). – Pavel Čoupek Mar 27 '18 at 18:23
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    We do need to take some amount of linguistics for granted - we need a language to make our definitions in, after all. We can't get anywhere without an "ambient language." This will make certain definitions - those corresponding to notions of the "ambient language" we presuppose - appear trivial. But there's no way to do math without assuming something by the way of an ambient language and understanding (for an extreme case of this, consider this parable of Hofstadter). – Noah Schweber Mar 27 '18 at 21:25
  • See this post on the circularity. – user21820 Jul 31 '18 at 17:27

3 Answers3

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T, ~F, ~~T, T^T, ~(T^F), ~(F^T), ~(F^F), TvT, TvF, FvT, ~(FvF)

The only way to avoid using the logical connectives of English in defining formal logical connectives is to simply list truths and omit falsehoods as I started doing above. -- James Richard Spriggs

James Richard Spriggs
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Just as we know what 'and' means in natural language, we also know what 'iff', or 'if and only if' means. So, when we say that "$P \land Q$ is true iff $P$ is true and $Q$ is true", we are providing a meaning (semantics) to this particular logical symbol.

Bram28
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I feel a lot of introductions to logic handle this rather poorly. I've described my issues in detail here.

Instead of saying "$P$ is true", let's write $v(P)=1$. We can think of $v$ as a function on the syntax of formulas mapping them, for classical logic, to either $0$ or $1$. "$\land$" is not a symbolic way of saying "and". It is just a operator whose meaning we're in the process of (indirectly) defining. Or rather, we're defining what $v$ means on formulas of the form $P\land Q$.

Already using this notation makes things a bit clearer: "$v(P\land Q)=1$ if and only if $v(P)=1$ and $v(Q)=1$." Since, by definition, if $v(P)$ is not $1$ then it must be $0$, we can write this fully out as: $$v(P\land Q)=\begin{cases}1,&v(P)=1\text{ and }v(Q)=1\\0,&\text{otherwise}\end{cases}$$

Arguably, this approach jumps the gun a bit e.g. by assuming a functional valuation. So here's another approach. Say we have a set of formulas, $S$. We want to describe which formulas are in this set. The statement from the question can then be described by the two rules: "If $P$ is in $S$ and $Q$ is in $S$, then $P\land Q$ is in $S$. Conversely, if $P\land Q$ is in $S$, then $P$ and $Q$ are each in $S$." Each connective in the syntax of formulas will have its own rules for describing which formulas are in $S$ or not. Altogether, they will describe what is and isn't in $S$, and formulas that are in $S$ will be called "theorems".

So you have a bunch of formal formulas that you are either sorting into $S$ or determining the value of $v$. You reason about these informally using natural language. This informal reasoning is referred to as the meta-logic or meta-language. (You can have formal meta-logics, but then you'll need an informal meta-meta-logic.) Once you've finished defining everything then you can wonder if this formal system matches up with your informal reasoning.

The statement in the question doesn't define what "and" means as symbolized by "$\land$", instead it's part of a definition of a model of reasoning which you can compare to your informal reasoning once you've completed defining the model. You can decide how well this model captures your informal reasoning. Most likely you will quickly realize that this model is a very limited one. For example, it fails to deal with things that are "true" in some cases but not others (this is addressed by predicate logic) and it fails to deal with the phenomenon of uncertainty. You already know what "and" means. What you're in the process of learning is how $\land$ fits into the model of logic that is being built.

Derek Elkins left SE
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