How to prove A → (B ∨ C) given A → B

How to prove A → (B ∨ C) given A → B

I know this is a valid argument, I'm just terrible at fitch-style proofs and have no idea how to start, let alone finish.

I'm just terrible at fitch-style proofs and have no idea how to start, let alone finish.

Use a conditional proof to introduce that conditional statement. Assume $$A$$ and then use the premise to derive the conclusion of $$B\lor C$$.   I do believe you can fill in the rest of the details.

$$\def\fitch#1#2{\quad\begin{array}{|l} #1\\\hline #2\end{array}}$$ $$\fitch{1.~~A\to B}{\fitch{2.~~A}{3.~~:\\4.~~B\lor C}\\5.~~A\to(B\lor C)\qquad\text{Conditional Introduction (2-4)}}$$

Use truth table.

A B C A->B A->(BVC)
0 0 0  1     1
0 0 1  1     1
0 1 0  1     1
0 1 1  1     1
1 0 0  0     0
1 0 1  0     1
1 1 0  1     1
1 1 1  1     1


You can see that when A $$\rightarrow$$ B is true, A$$\: \rightarrow$$(B$$\: \vee \:$$C) is true.

Or A$$\: \rightarrow$$(B$$\: \vee \:$$C)=(A$$\rightarrow$$B)$$\: \vee\:$$C $$\Rightarrow$$ A$$\rightarrow$$B.

$$A\to B$$

$$\lnot A\lor B$$

$$(\lnot A\lor B)\lor C$$

$$\lnot A\lor(B\lor C)$$

$$A\to(B\lor C)$$

Is this the kind of proof you want?

This result is known as 'Weakening the Consequent'.

$$\def\fitch#1#2{\quad\begin{array}{|l} #1\\\hline #2\end{array}}$$ $$\fitch{1.~~A\to B\qquad \text{Hypotheses}}{\fitch{2.~~A \qquad \text{Introduction/Conditional Proof}}{3.~~B \qquad \text{Modus Ponens 1, 2} \\4.~~B\lor C \qquad \text{Disjunction Introduction 3}}\\5.~~A\to(B\lor C)\qquad\text{Conditional Proof(2-4)}}$$

• You can use the Natural deduction proof editor and generator or the LaTeX fitch package to typeset natural deduction proofs. – lemontree Apr 30 at 11:30
• You cannot use those on this site. @lemontree – Graham Kemp May 1 at 7:09
• @GrahamKemp I always screenshot the result and insert it as an image, I consider that method simple enough. Besides, I assume Rob might be interested to know how to typeset ND proofs in general, not just for this site. (A previous version of their post said "I am not sure how to format this as fitch diagram in here. If someone knows, feel free to edit this", hence my comment.) – lemontree May 1 at 7:13