It is important in logic and mathematics to be able to distinguish between statements which are implications and statements which are universally quantified, because their truth value analysis can produce different results. I claim no special insight into the problem other than an observation that the otherwise flawless explanations that I often come across miss some key element which could help grasp the idea faster.
First, it is a lot easier to work with mathematical concepts in truth value analysis than with everyday English sentences. The reason: fish are aquatic animals, but suppose you were asked the question: ‘Are all fish aquatic animals?’ during a TV game show. Should you suspect a trap? Some people might. We live in an age of knowledge fragmentation where one never sees the full picture, and so can’t be sure of objective truths. The ancients and the medievals, by contrast, may have dwelled on fiction but they sought to consolidate knowledge and, conceivably, were less prone to relativist dilemmas.
Let our universe of discourse, UD, be integers then. Let Fx stand for ‘x is even’ and Gx for ‘x is evenly divisible by 2’. Our two sentences are:
(1) If a number is an even integer, then every integer is evenly divisible by 2.
(x)Fx ⊃ (x)Gx
(2) If a number is an even integer, then it is evenly divisible by 2.
(x)(Fx ⊃ Gx)
Sentence (1) is an example of material implication. Material implication is an example of a compound statement, with its truth value dependent on the logical operator ‘⊃’. In plain English, the two halves of the sentence are independent, or each is capable of taking on its own truth value. We know that sentences like this are false if and only if the consequent, here (x)Gx, is false while the antecedent, here (x)Fx, is true. Sentence (1) is patently false, because not every integer is evenly divisible by 2.
Sentence (2) can be translated to look like a conditional sentence, and in the ordinary sense of the English grammar it certainly is a conditional: For all x, if x is an even integer, then x is evenly divisible by 2. But the scope of the universal quantifier extends over the whole sentence, which means that the ‘x’ in the second half of the sentence has the same reference as the ‘x’ in the first half of the sentence, thus the sentence says in fact: All even integers are evenly divisible by 2. For this to be false, we would need to find only one counterexample, that is, some even integer that is not evenly divisible by 2. I can’t find such a counterexample. The sentence is true.
Sentences (1) and (2) are not equivalent then. However, where a conditional statement and a universally quantified statement are equivalent for some A and B, as in:
(3) (∃x)Ax ⊃ B
(4) (x)(Ax ⊃ B)
the truth values will of course be the same. The difference is in how we choose to express ourselves: (3) is false when B is false and (∃x) Ax is true, and (4) is false when for at least one A, B does not hold, but the fact remains that when (3) is false (4) will be false, and when (3) is true (4) will be true too.
First, it is a lot easier to work with mathematical concepts in truth value analysis than with everyday English sentences. The reason: fish are aquatic animals, but suppose you were asked the question: ‘Are all fish aquatic animals?’ during a TV game show. Should you suspect a trap? Some people might. We live in an age of knowledge fragmentation where one never sees the full picture, and so can’t be sure of objective truths. The ancients and the medievals, by contrast, may have dwelled on fiction but they sought to consolidate knowledge and, conceivably, were less prone to relativist dilemmas.
Let our universe of discourse, UD, be integers then. Let Fx stand for ‘x is even’ and Gx for ‘x is evenly divisible by 2’. Our two sentences are:
(1) If a number is an even integer, then every integer is evenly divisible by 2.
(x)Fx ⊃ (x)Gx
(2) If a number is an even integer, then it is evenly divisible by 2.
(x)(Fx ⊃ Gx)
Sentence (1) is an example of material implication. Material implication is an example of a compound statement, with its truth value dependent on the logical operator ‘⊃’. In plain English, the two halves of the sentence are independent, or each is capable of taking on its own truth value. We know that sentences like this are false if and only if the consequent, here (x)Gx, is false while the antecedent, here (x)Fx, is true. Sentence (1) is patently false, because not every integer is evenly divisible by 2.
Sentence (2) can be translated to look like a conditional sentence, and in the ordinary sense of the English grammar it certainly is a conditional: For all x, if x is an even integer, then x is evenly divisible by 2. But the scope of the universal quantifier extends over the whole sentence, which means that the ‘x’ in the second half of the sentence has the same reference as the ‘x’ in the first half of the sentence, thus the sentence says in fact: All even integers are evenly divisible by 2. For this to be false, we would need to find only one counterexample, that is, some even integer that is not evenly divisible by 2. I can’t find such a counterexample. The sentence is true.
Sentences (1) and (2) are not equivalent then. However, where a conditional statement and a universally quantified statement are equivalent for some A and B, as in:
(3) (∃x)Ax ⊃ B
(4) (x)(Ax ⊃ B)
the truth values will of course be the same. The difference is in how we choose to express ourselves: (3) is false when B is false and (∃x) Ax is true, and (4) is false when for at least one A, B does not hold, but the fact remains that when (3) is false (4) will be false, and when (3) is true (4) will be true too.
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