Lec 9 Short-cut truth-table method
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Answer: 1) The standard form of the argument is as follows: (1) If you study hard, you will have good grades. (2) If you play hard, you will have happy days. (3) Either you study hard or you play hard. Therefore, (4) either you will have good grades or you will have happy days.
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Example: Prove the validity of the following argument: “If you study hard, you will have good grades. If you play hard, you will have happy days. Either you study hard or you play hard. It follows that either you will have good grades or you will have happy days.”
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The Short-Cut Truth-Table Method
The short-cut truth-table method is this: Suppose the corresponding wff of a certain argument is {[((α1 &α2) &α3)) & … αn] →β 1) We assume that the wff is not a tautology.
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2) From this assumption, we deduce a self-contradiction. 3) We can then conclude that the wff is a tautology and that the argument in question is valid.
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2) P: You study hard. Q: you will have good grades. R: You play hard. S: You will have happy days. The above argument has becomes: (1) (P→Q) (2) (R→S) (3) (PVR) ∴(4) (QVS)
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Exercises
1. Prove the validity of the following argument form: (1) (P→Q) (2) (Q→R) (3) (R→S) (4) ~S (5) (PVP1) ∴ (6) P1
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2. Test the validity of the following argument by the short-cut truth-table method: “If China walks on the road of market economy, then China walks on the road of capitalism. If China walks on the road of capitalism, then China is not walking on the road of socialism. China walks on the road of market economy. Therefore, China is not walking on the road of socialism.”
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3) The corresponding wff of the above argument is this: {[(P → Q) & (R→S)] & (P V R) → (Q V S)}
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ห้องสมุดไป่ตู้
4) {[(P → Q) & (R→S)] & (P V R) → (Q V S)} FT F T FTF T FTF F
The Short-Cut Truth-Table Method
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Introduction
If an argument form has more than 3 statement variables, using the truthtable method to test the argument will become inefficient, for there will be too many rows in the truth-table. For these arguments, we should use the short-cut truth-table method(捷徑真值表法).
Self-contradiction
FFF
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Since the assumption that the wff is not a tautology leads to a self-contradiction, the assumption is false and the wff is a tautology. Therefore, the above argument is valid.
{[(((P→Q)&(Q→R))&(R→S))& (S→P1)] →(P→P1)} → → → → → TTF T FTF T FTF T FTF F TFF
Self-contradiction
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Since the assumption that the corresponding wff is not a tautology leads to a self-contradiction, this assumption is false and the above argument form is valid.
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Example: Prove the validity of the following argument form: (1) (P→Q) (2) (Q→R) (3) (R→S) (4) (S→P1) ∴ (5) (P→P1)
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The corresponding wff of the above argument form is:
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Answer: 1) The standard form of the argument is as follows: (1) If you study hard, you will have good grades. (2) If you play hard, you will have happy days. (3) Either you study hard or you play hard. Therefore, (4) either you will have good grades or you will have happy days.
7
Example: Prove the validity of the following argument: “If you study hard, you will have good grades. If you play hard, you will have happy days. Either you study hard or you play hard. It follows that either you will have good grades or you will have happy days.”
2
The Short-Cut Truth-Table Method
The short-cut truth-table method is this: Suppose the corresponding wff of a certain argument is {[((α1 &α2) &α3)) & … αn] →β 1) We assume that the wff is not a tautology.
3
2) From this assumption, we deduce a self-contradiction. 3) We can then conclude that the wff is a tautology and that the argument in question is valid.
9
2) P: You study hard. Q: you will have good grades. R: You play hard. S: You will have happy days. The above argument has becomes: (1) (P→Q) (2) (R→S) (3) (PVR) ∴(4) (QVS)
13
Exercises
1. Prove the validity of the following argument form: (1) (P→Q) (2) (Q→R) (3) (R→S) (4) ~S (5) (PVP1) ∴ (6) P1
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2. Test the validity of the following argument by the short-cut truth-table method: “If China walks on the road of market economy, then China walks on the road of capitalism. If China walks on the road of capitalism, then China is not walking on the road of socialism. China walks on the road of market economy. Therefore, China is not walking on the road of socialism.”
10
3) The corresponding wff of the above argument is this: {[(P → Q) & (R→S)] & (P V R) → (Q V S)}
11
ห้องสมุดไป่ตู้
4) {[(P → Q) & (R→S)] & (P V R) → (Q V S)} FT F T FTF T FTF F
The Short-Cut Truth-Table Method
1
Introduction
If an argument form has more than 3 statement variables, using the truthtable method to test the argument will become inefficient, for there will be too many rows in the truth-table. For these arguments, we should use the short-cut truth-table method(捷徑真值表法).
Self-contradiction
FFF
12
Since the assumption that the wff is not a tautology leads to a self-contradiction, the assumption is false and the wff is a tautology. Therefore, the above argument is valid.
{[(((P→Q)&(Q→R))&(R→S))& (S→P1)] →(P→P1)} → → → → → TTF T FTF T FTF T FTF F TFF
Self-contradiction
6
Since the assumption that the corresponding wff is not a tautology leads to a self-contradiction, this assumption is false and the above argument form is valid.
4
Example: Prove the validity of the following argument form: (1) (P→Q) (2) (Q→R) (3) (R→S) (4) (S→P1) ∴ (5) (P→P1)
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The corresponding wff of the above argument form is:
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