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Question of the day

I'm not entirely sure I follow, but perhaps this will be of some use.

Whether two propositions really have four possible combinations of truth values depends on the propositions. Non-philosophical examples make the point easier to follow.

Suppose P is "Paula is Canadian" and Q is "Quincy is Australian." In this case, the two propositions are logically independent, and all four combinations P&Q, P&~Q, ~P&Q and ~P&~Q represent genuine possibilities. But not all propositions are independent in this way; it depends on their content.

P and Q might be contradictories, that is, one might be the denial of the other. (If P means that Paul is Canadian and Q means that she is not Canadian, then we have this situation.) In that case, the only two possibilities are P&~Q and ~P&Q.

Or P and Q might be contraries, meaning that they can't both be true though they could both be false. For example: if P is "Paula is over 6 feet tall" and Q is "Paula is under 5 feet tall," then we only have three possibilities: P&~Q, ~P&Q, and ~P&~Q. The fourth case, P&Q, isn't possible.

Or P and Q might be subcontraries, meaning that they can both be true, but can't both be false. For example: if P is "Paula is under 6 feet tall" and Q is "Paula is over 5 feet tall," then the only possibilities are P&Q, P&~Q and ~P and Q. ~P&~Q isn't possible.

Or P might imply Q. If P is "Paula is over 6 feet tall" and Q is "Paula is over 5 feet tall," then the possibilities are P&Q, ~P&Q, and ~P&~Q. Here, P&~Q isn't possible.

Finally, P and Q might be equivalent. Suppose P is "The temperature is 32 degrees Fahrenheit" and Q is "The temperature is 0 degrees Celsius." In that case, P and Q are in effect the same proposition, expressed by different sentences. They are either both true or both false, leaving P&Q and ~P&~Q as the only possibilities.

All of this applies across the board, and in particular it applies in philosophy. Not all philosophical claims are independent, and so for some philosophical propositions, one or more of the four combinations won't represent possibilities. But at least some philosophical disputes are over the very question of what the logical relationship between two claims actually is. For example: consider "Paula's behavior is determined" and "Paula is responsible for her behavior." One important view is that these are contraries; they can't both be true. Other philosophers deny this, claiming, for example, that responsibility entails determinism, in which case "Paula is responsible, and her behavior is not determined" doesn't represent a genuine possibility. Other philosophers would claim that the two are independent, and so all four combinations represent genuine possibilities.

This kind of disagreement about the logical relations among philosophical claims is common in philosophy. But the larger point is that we can't simply assume in all cases that all four combinations represent genuine possibilities.