# Universal Quantifier $\forall$ and Generalized Conjunction $\bigwedge$ in intuitionistic logic

I have a question about $$\forall xA(x)$$ and $$\bigwedge\! A(a_i)$$ (= $$A(a_1)\wedge...\wedge A(a_n)$$). In classical logic, the universal statement $$\forall xA(x)$$ can be understood as a generalized conjunction of $$A(a_i)$$ if the universe of discourse is finite, say, $$\{a_1,...,a_n\}$$. I wonder if this generalizes to intuitionistic logic?

The concrete problem that I meet is as follows. I assume that this generalizes, then we have $$\forall xA(x)\leftrightarrow\bigwedge\! A(a_i).\quad\quad(*)$$ However, putting $$\neg\neg$$ before $$\forall xA(x)$$, $$\neg\neg\forall xA(x)\rightarrow\forall x\neg\neg A(x)$$ is intuitionistically valid, but the converse is not. But putting $$\neg\neg$$ before $$\bigwedge\! A(a_i)$$, $$\neg\neg\bigwedge\! A(a_i)\leftrightarrow\bigwedge\! \neg\neg A(a_i)$$ (note: both directions) is intuitionistically valid. Why is there such a difference? Does this signal that $$(*)$$ does not hold in intuitionistic logic? Thanks!

• But (*) is a well-formed formula only in a finite universe. And with finite collections, Intuitionism has no issue with LEM and DN Jan 10, 2021 at 18:05

In general, $$(*)$$ is the wrong way to think about the universal quantifier. In both classical and intuitionistic logic, the power of the universal quantifier $$\forall x$$ is that it allows $$x$$ to range over an arbitrary, unspecified domain. There is no sense in which $$(*)$$ is a valid sentence in either classical or intuitionistic logic.
If you want to restrict your semantics to a fixed finite domain, you need to introduce an axiom. Let $$\varphi$$ be the sentence: $$\forall x\, \bigvee_{i=1}^n (x = a_i).$$ Then we indeed have $$\varphi\vdash (\forall x\,\psi(x)\leftrightarrow \bigwedge_{i=1}^n \psi(a_i))$$ for all formulas $$\psi(x)$$, both classically and intuitionistically. But also $$\varphi\vdash \forall x\, \lnot\lnot \psi(x) \leftrightarrow \lnot \lnot \forall x\, \psi(x)$$ for all formulas $$\psi(x)$$, both classically and intuitionistically. So your concerns vanish in this restricted setting.