Question

Write the following in symbolic notation (be sure to provide the dictionary) and determine whether the argument is valid using a truth table If it rains, then I take an umbrella. If I take an umbrella, then I am not in fashion. Therefore, if it rains, then I am not in fashion.

Answer 1

Let A: Rains B: Umbrella and C: fashion
If it rains,then I take an Umbrella ==> A $\rightarrow$ B
If I take an Umbrella,then I am not in fashion ==> B $\rightarrow$ ~C
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if it rains then I am not in fashion ==>A $\rightarrow$ ~C

A $\rightarrow$ B $\wedge$ B $\rightarrow$ ~C $\equiv$ A $\rightarrow$ ~C

Take  A $\rightarrow$ B $\wedge$ B $\rightarrow$ ~C

(~A $\vee$ B)  $\wedge$ (~B $\vee$ ~C )

( ~A $\vee$ B  $\wedge$ ~B ) $\vee$ ( ~A $\vee$ B  $\wedge$ ~C )

( ~A $\vee$ F ) $\vee$ ( ~A $\vee$ B  $\wedge$ ~C ) { We know that ( B  $\wedge$ ~B ) = F }

( ~A ) $\vee$ ( ~A $\vee$ B  $\wedge$ ~C )

( ~A $\vee$ ~A $\vee$ B  $\wedge$ ~C )

(~A $\vee$ B  $\wedge$ ~C ) { We Know that  ~A $\vee$ ~A =  ~A }

~A $\vee$ B  $\wedge$ ~C { We Know that ( B  $\wedge$ ~B ) = F }

A $\rightarrow$ ( B  $\wedge$ ~C )

if it rains then I take an Umbrella AND I am not in fashion

The Given argument is Invalid

TRuth Table:

0 0011 011 0 11 011 1 1 1 1

If we Observe the last two columns in above Truth table they are having different Truth values

So, The Given argument is Invalid