An elementary precept of logic says that where there are two propositions, P and Q, there are four possible "truth values", P&~Q, ~P&Q, P&Q, ~P&~Q, where ~ means "not."   Do people ever apply this to pairs of significant propositions? For example, has anyone applied it to positive and negative liberty, or to equality of opportunity and equality of condition, or to just process and just outcome? On these topics I can find treatments of the first two truth values but none of the second two. Given that P and Q are not mutually exclusive (which no one seems to say for these pairs) why not explicate P&Q and ~P&~Q as well?

  • 1
    These are not truth values; they are formulas, and of course there are many more than four formulas that can be constructed from P, Q and connectives.
    – WillO
    Apr 11, 2015 at 21:15
  • Thanks, WillO. Formulas: hmm, okay. But are there many more? I give the four AND connectives. The OR connective is not applicable in these topic pairs because the P is understood in contrast to the Q. To have P on its own would probably not be comprehensible, not able to be discussed. They are not propositions where IF...THEN would come into question as a connective, so the four AND options would be it—no? My question remains: why consider only two of them?
    – MikeP
    Apr 12, 2015 at 5:44
  • Under the usual interpretation that P means P is true, these are four possible COMBINATIONS of truth values. In general, with n propositions that can be independently true or false, you have 2^n (2 to the n'th power) possible combinations. The technique of writing out the formulas the way you have done is used in introductions to the theory, e.g. for digital electronics. May 11, 2015 at 9:55

2 Answers 2


I think the issue is a selection bias. Generally speaking, people spend most of their political effort dealing with difficult things. Accordingly, most situations of merit come in the form of tradeoffs, P&~Q or ~P&Q. Any situation where P&Q or ~P&~Q becomes the easy situation often does not warrant discussion... we just stop talking about it and do it!

Solutions which lend themselves to P&Q also often lend themselves to being decomposed into merely P or ~P and Q or ~Q. Thus we rarely get to a point where discussing P&Q as a whole moves things forward.

We do see a resurgence of P&Q in diplomacy. In diplomacy, there is a fine art to wording a treaty such that it serves as a win-win for both countries.

  • Thank you. Yes. But the easy situation should be shown. A purpose of philosophy is to pick up on unthinking bias. In these instances, for my named topic pairs, P&Q does not look that easy—more of a balancing act, like your diplomacy. However, my own finding is that with those political binary pairs, P&Q and ~P&~Q don't concern win-win but give results hardly resembling the other two AND options. That is why I am curious to know why they are not discussed, to know, before I run down the road crying Eureka, whether I am overlooking something.
    – MikeP
    Apr 12, 2015 at 5:43
  • @MikeP There is something interesting about those two overlooked pairs. However, as you run down the road crying Eureka, consider that those are the situations where models break down, and can get tricky. As an example, take a look at the breakdown of Quantum Mechanics regarding fast small particles. QM is easy for slow small particles. Relativitiy is easy for fast big particles. Newtonian physics works fine for slow big particles. However, that last quadrant turns out to be quite frustrating to model. Sometimes, what we end up showing with P&Q is simply the edges of our models.
    – Cort Ammon
    Apr 12, 2015 at 5:58
  • There are many situations or topics where P and Q are contraries or mutually exclusive so yes, models break down. BUT— you don't know till you try. I have political matters in mind, not quantum theory. For 200 years thousands of students have learnt in Logic 101 that two propositions yield four truth values and in the moral philosophy seminar they have discussed Kant's Price and Dignity—and apparently never put two and two together for nowhere do I find P&D or ~P&~D discussed.
    – MikeP
    Apr 15, 2015 at 3:19
  • @MikeP You're welcome to add discussions on those two. When it comes to discussions like that, we have to remember that the mere act of discussing has a cost. That cost is very much medium and approach dependent. It is entirely possible that, for a seminar, the cost of discussing them outweighted the value, and, to steal a much hated phrase, "they were left as an exercise to the reader." I was not involved in said seminar, so I cannot claim this is true in this particular case, but I have found an ENORMOUS body of such cases in many disciplines.
    – Cort Ammon
    Apr 15, 2015 at 4:01
  • We may find that you have the opportunity to discuss it in a lower cost medium, and can provide answers which were not reasonable to provide in a seminar form.
    – Cort Ammon
    Apr 15, 2015 at 4:02

In many mathematical proofs, you will have to consider two cases P or ~P, and for each of these two cases, you will have to consider Q or ~Q. (Sorry, an example eludes me at the moment.)

Consider a proof structured as follows:

Case 1: P

Sub-case 1: Q

(Here we consider P and Q)

Sub-case 2: ~Q

(Here we consider P and ~Q)

Case 2: ~P

Sub-case 1: Q

(Here we consider ~P and Q)

Sub-case 2: ~Q

(Here we consider ~P and ~Q)
  • Not sure I follow but I guess my question is why, for my three topics concerning liberty, equality and justice, Case 1, sub-case 1 and Case 2, sub-case 2, do not appear to be considered.
    – MikeP
    Apr 12, 2015 at 5:53

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