Philosophy 220. Truth-Functional Equivalence and Consistency
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1 Philosophy 220 Truth-Functional Equivalence and Consistency
2 Review Logical equivalency: The members of a pair of sentences [of natural language] are logically equivalent if and only if it is not [logically] possible for one of the sentences to be true while the other sentence is false. Logical consistency: A set of sentences is logically consistent if and only if it is [logically] possible for all the members of that set to be true [at the same time].
3 Truth-Functional Equivalence: Sentences and of SL are truth-functionally equivalent if and only if there is no truth value assignment [for the components of and ] on which and have different truth-values. This means that on a full truth table, the columns for any two truth-functionally equivalent sentences of SL will be identical.
4 Finding Equivalence on a shortened truth-table As with truth-functional truth and truth-functional falsity, we test for truth-functional equivalence by looking for a counterexample. If we assume that one of the sentences is true and the other false, then either we will or will not get a coherent truth-value assignment. If we do, then the two sentences are shown not to be truth-functionally equivalent. If we cannot get a coherent truth-value assignment assuming that one sentence is true while the other is false, then we must try it the other way before drawing any conclusions. (why?)
5 ~(B & ~A) and (A v B) A B ~ (B & ~A) A v B Which columns should be identical if these two sentences are truth-functionally equivalent?
6 ~(B & ~A) and (A v B) A B ~ (B & ~A) A v B Which columns should be identical if these two sentences are truth-functionally equivalent?
7 ~(B & ~A) and (A v B) A B ~ (B & ~A) A v B T F So assume that one is T and the other F.
8 ~(B & ~A) and (A v B) A B ~ (B & ~A) A v B T F Note that the only way for A v B to be false is for both A and B to be false.
9 ~(B & ~A) and (A v B) A B ~ (B & ~A) A v B F F T F Note that the only way for A v B to be false is for both A and B to be false.
10 ~(B & ~A) and (A v B) A B ~ (B & ~A) A v B F F T T F Now we see if this truth-assignment is coherent
11 ~(B & ~A) and (A v B) A B ~ (B & ~A) A v B F F T F T F It is coherent. If A and B are both false, then ~(B & ~A) must be true.
12 F & (J v H) and (F & J) v H F H J F & (J v H) (F & J) v H Which columns should be identical if these two sentences are truth-functionally equivalent?
13 F & (J v H) and (F & J) v H F H J F & (J v H) (F & J) v H Which columns should be identical if these two sentences are truth-functionally equivalent?
14 F & (J v H) and (F & J) v H F H J F & (J v H) (F & J) v H T F Assume one is T and the other F
15 F & (J v H) and (F & J) v H F H J F & (J v H) (F & J) v H F T F F If the disjunction is false, then both disjuncts must be false.
16 F & (J v H) and (F & J) v H F H J F & (J v H) (F & J) v H T F T T F F If the conjunction is true then both of its conjuncts must be true.
17 F & (J v H) and (F & J) v H F H J F & (J v H) (F & J) v H T F? T T F F Is there a truth-value assignment for J that is coherent?
18 F & (J v H) and (F & J) v H F H J F & (J v H) (F & J) v H T F T T T F F FAIL! Is there a truth-value assignment for J that is coherent? T is not coherent because it would make (F & J) true, which would in turn make ((F & J) v H) true.
19 F & (J v H) and (F & J) v H F H J F & (J v H) (F & J) v H T F F T T F F FAIL! Is there a truth-value assignment for J that is coherent? F is not coherent because it would make (J v H) false, which would in turn make (F & (J v H)) false.
20 F & (J v H) and (F & J) v H F H J F & (J v H) (F & J) v H T F X T T F F FAIL! Does this mean that F & (J v H) and (F & J) v H are equivalent?
21 F & (J v H) and (F & J) v H F H J F & (J v H) (F & J) v H T F X T T F F FAIL! Does this mean that F & (J v H) and (F & J) v H are equivalent? IT DOES NOT! We have shown that F & (J v H) cannot be true while (F & J) v H is false, but it is still possible that F & (J v H) can be false while (F & J) v H is true. So let s check:
22 F & (J v H) and (F & J) v H F H J F & (J v H) (F & J) v H F T So assume one is F and the other T
23 F & (J v H) and (F & J) v H F H J F & (J v H) (F & J) v H F T F T We have many ways to proceed here, so let us assume that F is false (to make the conjunction it is in false) and that H is true (to make the disjunction that it is in true). If this turns out to be incoherent, there are several other possibilities to try.
24 F & (J v H) and (F & J) v H F H J F & (J v H) (F & J) v H F T? F T Now we must see if any truth value of J would yield a coherent table
25 F & (J v H) and (F & J) v H F H J F & (J v H) (F & J) v H F T T F T Let s try T first.
26 F & (J v H) and (F & J) v H F H J F & (J v H) (F & J) v H F T T F T F T This is a coherent truth-value assignment for F, H, and J that reveals that these two sentences are not truthfunctionally equivalent.
27 The Full Truth-Table (for illustration) F H J F & (J v H) (F & J) v H T T T T T T T T T F T T F T T F T T T T T T F F F F F F F T T F T F T F T F F T F T F F T F T F F We proved with our first shortened truth-table that the first sentence is never true while the second sentence is false F F F F F F F
28 The Full Truth-Table (for illustration) F H J F & (J v H) (F & J) v H T T T T T T T T T F T T F T T F T T T T T T F F F F F F F T T F T F T F T F F T F T F F T F T F F F F F F F F F however, we showed with the second shortened truth-table that the two are not equivalent because the second sentence can be true while the first sentence is false.
29 Truth-Functional Consistency A set of sentences of SL is truth-functionally consistent if and only if there is at least one truth value assignment [of the constituents of the set of sentences] on which all the members of the set are true. That means that if each of the set of sentences of SL were done on a truth-table, there would be one row of the truth table on which all of the sentences of the set are true.
30 Shortened tables Since a single example of a case in which all of the sentences of a set can be true shows that the set is consistent, when we check for consistency with a shortened truth-table, we should assume that all of the sentences of the set are true. If we get a coherent truth-value assignment from this assumption, then the set is consistent. If we cannot, then the set is inconsistent.
31 A consistent set? H J K H J J K K ~H
32 A consistent set? H J K H J J K K ~H T T T Assume that all are true.
33 A consistent set? H J K H J J K K ~H T T T T H J being true is consistent with several outcomes, so let s assume that H is true to start with.
34 A consistent set? H J K H J J K K ~H T T T T T If H is true, then J must be true to preserve the truth of H J.
35 A consistent set? H J K H J J K K ~H T T T T T T If J is true, then K must be true to preserve the truth of J K.
36 A consistent set? H J K H J J K K ~H T T T T T T T If K is true, then ~H must be true to preserve the truth of K ~H.
37 A consistent set? H J K H J J K K ~H T T T T T T F FAIL! Failure, H and ~H cannot be true at the same time.
38 A consistent set? H J K H J J K K ~H F T T T Assume that all are true. Then, since assuming H was true brought us an inconsistent set, let s assume H is false instead.
39 A consistent set? H J K H J J K K ~H F T T T T If H is false, then ~H is true.
40 A consistent set? H J K H J J K K ~H F F F T T T T The truth-value of H J is assured by H having a truth-value of false, whatever J s truth-value. Also, the truth of K ~H is assured by ~H being true, whatever K s truth-value. So we can select any truth values for J and K so long as they don t make J K false. (F for both will do)
41 A consistent set? H J K H J J K K ~H F F F T T T T This is one of several examples of truth-value assignments on which all three sentences end up true, so we have proven that these sentences are consistent.
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