1c) Let c represent "the cat fiddled", let j represent "the cow jumped over the moon", let d represent "the little dog laughed", and let r represent "the dish ran away with the spoon." Now write each of the premises and the conclusion in symbolic form.
The first premise (p1) is "If the cat fiddled or the cow
jumped over the moon, then the little dog laughed." This can be written symbolically
as (c V j) 
d.
The second premise (p2) is "If the little dog laughed, then the dish ran
away with the spoon." This can be written symbolically as d
r.
The third premise (p3) is "But the dish did not run away with the
spoon." This can be written symbolically as ~r.
The conclusion (q) is "Therefore the cat did not fiddle." This can be written
symbolically as ~c.
Remember that an argument is written as a conjunction of the premises
implies the conclusion. So this argument can be represented as
[(c V j) d) L (d r)
L (~r)] (~c).
To determine if it is possible for an argument to be invalid, we are looking for truth values for the variables which make the conclusion false but all of the premises true.
For the conclusion ~c to be FALSE, we need c to be TRUE.
For the premises to be TRUE we need:
d to be true (1); r to be true (2);Since premise (3) has the least number of variables, start with that premise.
For premise (3) to be true we need r to be FALSE.
For premise (2) to be true, and knowing that r is false, we need d to be FALSE.
For premise (1) to be true, and knowing that d is false, we need (c V j) to
be FALSE, which means that c
is FALSE and j is
FALSE.
But now c must be TRUE (from the conclusion) and FALSE (from the premises). This is impossible so we can conclude that there is no set of truth values which make the conclusion false and the premises true, so the argument cannot be invalid.
Therefore, the argument is valid.