A truth-function, such as the horseshoe, can be defined using other…

Question A truth-function, such as the horseshoe, can be defined using other… A truth-function, such as the horseshoe, can be defined using other logical connectives in various ways.  For example, p ? q can be defined as ~p v q.   (If you do the truth-tables for the two, you will find that the main columns are identical.)  So, anything we could say using ?, we could say using ~ and v instead.  And so, we don’t really need ?, if we have ~ and v.  Similarly, the tri-bar could be defined using ? and &.   Instead of p ? q, we could write (p ? q) & (q ? p), or using v, &, and ~, we could write (p & q) v (~p & ~q).   So, you might wonder, how many truth-functions do we really need?  Well, it turns out, not very many.  Any truth-function, of any number of variables, can be defined using just v, &, and ~.  (The proof for that isn’t too hard, and if you are interested, I can show it to you.)  Well, you might wonder, can we get by with even fewer?  The answer is “Yes!”.  For starters, let’s get rid of &.   p & q is equivalent to ~(~p v ~q).  (Think DeMorgan’s.)   Ok, now we can express any possible truth-function, of any number of variables using only ~ and v.  Can we make do with even fewer?  Again, the answer is “Yes!”.   Let’s let p|q mean ‘neither p nor q’.  So, p|q is T only when p and q are both F, and p|q is F otherwise.  Now, we could define p|q using ~ and & as ~p & ~q.  Or, using v and ~, as ~(p v q).  But, I want to go the other way.  I want to define ~ and v in terms of just |.  If we can do those two things, then that means that any truth-function, of any number of variables, can be defined just using that one connective 1. Define ~ using |.  Find a sentence in our symbolic formal language, using | as the only logical connective, that is logically equivalent to ~p.  You can have multiple occurrences of p, or of |, and you can use parentheses. But, no other logical connectives besides | are allowed.   2. Define  v using |.  Find a sentence, using | as the only logical connective, that is logically equivalent to p v q.  (The sentence will have p’s and q’s in it.)   Perhaps you prefer ‘and’ to ‘or’.  We could get rid of v, by defining pvq as ~(~p&~q). 3. Define & using |.  Find a sentence, using | as the only logical connective, that is logically equivalent to p & q.   This can also be done with a connective meaning ~(p & q) (“not both” or “nand”).  For the next two questions, let p|q mean ‘p and q are not both true’.  So, p|q is logically equivalent to ~(p&q). 4. Define ~ using |.  (Like question #1, but this time, | means ‘not both’.) (2 points)5. Define v using |.  (Like question #2, but this time, | means ‘not both’.)        Math Logic CRN 20195 Share QuestionEmailCopy link Comments (0)