# Why are the two streams connected in a circuit for P ∨ ¬P?

Source: p 67. Exercise 3.6.7. Sweet Reason: A Field Guide to Modern Logic (2010 2 ed) by Henle, Garfield, Tymoczko.

Design a circuit to function in the same way as each statement.
7. P ∨ ¬P (Is the result what you would expect?)

[p 354 :]

When I attempted this exercise, I drew the red distinct line, without the green line. But why is the green correct and necessary, and the red wrong?

• Both the upper-left black line and your original red line represent P, right? I think the canonically correct answer just more explicitly indicates a single P input, whereas two separate input lines suggests two separate inputs. Sometimes such circuit diagrams have a dup symbol that takes a single input and produces two outputs both identical to the input. And I'd guess you'd have used that if available, rather than the separate red line. Note that a resource-aware logic like linear logic wouldn't typically (unless !exponentiated) let you use the same premise twice, without some "costly" dup. – John Forkosh Jul 31 '16 at 6:25
• @JohnForkosh `Both the upper-left black line and your original red line represent P, right?` Yes; I would infer this, but the answer does not state this. Thank you, and welcome! – Greek - Area 51 Proposal Aug 1 '16 at 4:52
• Oh, okay. When you said above "I drew the red distinct line", I inferred you'd decided on your own to draw it, and therefore knew what you intended it to represent. Or, more logically: "I drew it ==> I knew it" :) – John Forkosh Aug 1 '16 at 7:48
• @JohnForkosh Sorry for the confusion. I did draw the red line myself; but I did not know that it would mean a new proposition (Q) instead of the same one (P). I did not draw the green line originally. Does this clarify? – Greek - Area 51 Proposal Aug 2 '16 at 3:56
• Yeah, that's what I thought -- you drew it yourself, intending both lines to mean P. Take a look at, for example, drstienecker.com/tech-332/… and scroll down a little to "Topic 3". See that B input? It's used twice, but only has one input line that forks into two. That's how such stuff is usually drawn. Drawing two different input lines usually means two different inputs. Just convention. – John Forkosh Aug 2 '16 at 5:33

Each of the input lines represents a different proposition. This should be intuitive because different lines are independent of each other, just as different atomic propositions are. So what you originally wrote would be `p v ~q`. To get `p v ~p` you have the use the same line (the same `p`) twice.