Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 16J (Logic & Set Theory)

Using the definition twice gives $P(a,b)=1$ when $a\le b$; when $a>b$, $a\to b=b$, then $b\to a=1$, and $1\to a=a$.

If $b\le c$, then $b\to c=1$ and $T=1\to a=a$. If $b>c$, then $b\to c=c$, so $T=c\to a$, which is $1$ exactly when $c\le a$ and is $a$ otherwise.

The implicational Hilbert axioms are $A\to (B\to A)$, $(A\to (B\to C))\to ((A\to B)\to (A\to C))$, and Peirce's law $((A\to B)\to A)\to A$.

A direct case split in the linearly ordered chain verifies types 1 and 2. Peirce's value is $P(a,b)$ from part (a); for $n\ge 3$, choose $0<a>b$, obtaining $P(a,b)=a<1$. Thus type 3 need not be $n$-valid.

One example isApplying the two case formulas from (a) and (b) checks the three possible values in $H_3$ and always gives $1$. In $H_4$, the valuation $p=1/2,q=1/3$ gives $v(\phi )=1/2$, so it is not 4-valid.