I would go further than checking the content of /var/run/syslogd.pid and seeing whether there's a process running with that id. After all, if syslogd went down, another process could have taken PID syslogd was using.

Furthermore, to check whether a PID is in use, and what program is using it, I would use neither kill, nor ps. I'd run something like (untested):

    my $pidfile = "/var/run/syslogd.pid";
    if (-e $pidfile &&
         qx "cat /proc/`cat $pidfile`/cmdline" =~ /^syslogd$/) {
        ... syslog is running ...
    }
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Notes:

• On my systems, you must be root to be able to read /var/run/syslogd.pid. Your milage may vary.
• It's not foolproof. Something may have removed the PID file for instance.
• This is for a 2.4 Linux kernel. Different OSses (or Linux versions) may have a different /proc file system layout, or no /proc filesystem at all. The information from the file cmdline may be elsewhere, or not available at all.