Weak diamond and pcf theory

We obtain bounds on the cardinality of $pcf(\mathfrak{a})$ from instances of weak diamond. Consequently, under mild assumptions there are many singular cardinals of the from ${\mathrm{\aleph }}_{\delta }$ for which $2^{{\mathrm{\aleph }}_{\delta }}<{\mathrm{\aleph }}_{(\mathrm{\mid }\delta {\mathrm{\mid }}^{+3})}$. For example, if every limit cardinal is a strong limit cardinal then this bound holds at a class of singular cardinals.