It appears that Josiah/Silas were one person perhaps using a middle name. The marriage record in the Holliston Vital Record states:
"Silas [int. Josiah] Jr. and Martha Bullard of Holliston, Nov. 27, 1788, in Sherborn.".[4]
Sources
↑ "Massachusetts, Town Clerk, Vital and Town Records, 1626-2001," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:FHPP-NQ3 : 6 November 2017), Isaac Bullard in entry for Martha Bullard, 05 Oct 1757; citing Birth, Holliston, Middlesex, Massachusetts, United States, , town clerk offices, Massachusetts; FHL microfilm 763,666.
↑ "Massachusetts Marriages, 1695-1910," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:FCHL-1Q4 : 9 February 2018), Josiah Burnam and Martha Bullard, 30 Jul 1788; citing reference ; FHL microfilm 0873746 IT 3.
↑ "Massachusetts Marriages, 1695-1910," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:FCHP-32Y : 9 February 2018), Silas Burnam and Martha Bullard, 27 Nov 1788; citing reference ; FHL microfilm 0547197 IT 1.