Suppose the following material conditional "If 1+1=2 then Moon is made of cheese". The antecedent refers to the context of mathematics. In that context "1+1=2" is true. Whereas "Moon is made of cheese" is false. But the truth value of the consequent depends on a different context (real world). So shouldn't we limit our material conditionals only for antecedents and consequents which are part of the same context? Does this contradict the notion of truth tables etc?
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2The point of the material conditional is that it depends on truth values only, this is why it does not always match semantic intuitions, but that is by design. Truth values may depend on contexts, paradigms or what have you, but that is not material conditional's concern. Of course, one can declare some sentences not well formed and/or lacking truth value depending on their context, but then material conditional simply does not apply to them.– ConifoldAug 5, 2020 at 11:23
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1Relevance Logic is one attempt to address issues of context, however the conditional statement of relevance logic is not the material conditional of classical logic.– nwrAug 6, 2020 at 3:48
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1 Answer
Not quite.
It is correct that "1 + 1 = 2" is neither true nor false if you haven't some context in mind. Same thing for "the Moon is made of green cheese".
If you write the truth table for "1 + 1 ≠ 2" ⊃ "The Moon is made of green cheese", the result is as follows:
A B C
T T T
T F F
F T T
F F T
KEY
A = "1 + 1 ≠ 2"
B = "The Moon is made of green cheese"
C = "1 + 1 ≠ 2" ⊃ "The Moon is made of green cheese"
As you can see, the implication is not valid, contrary to what most books on mathematical logic would have said only a few years ago on the basis that "1 + 1 ≠ 2" is usually considered false, and is indeed false in standard arithmetic.
To make it valid, you need to specify the context.
So, for example, you could make the standard assumptions that "1 + 1 ≠ 2" is false, and that "The Moon is made of green cheese" is false.
This still doesn't affect the truth value of the implication "1 + 1 ≠ 2" ⊃ "The Moon is made of green cheese".
However, you can write a new relation:
"1 + 1 ≠ 2" is false and "The Moon is made of green cheese" is false, therefore, the implication "1 + 1 ≠ 2" ⊃ "The Moon is made of green cheese" is true.
This only because "1 + 1 ≠ 2" is here assumed false.
This gives the following truth table:
A B C
F F T
And now, the implication is valid ... according to the definition of the material implication, which is arbitrary and doesn't provably refer to anything the human mind does.
answered Aug 6, 2020 at 18:41
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1Doesn't provably refer to anything the human mind does is very different from arbitrary. Material conditional is selected for desirable technical properties it has and its usefulness in mathematics, for example. Vector spaces also do not refer to what human mind does, but are not arbitrary.– ConifoldAug 6, 2020 at 19:56
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@Conifold Fallacious comparison. Vector spaces are concepts conceived to refer to some aspect of the material world. The non-fallacious parallel would be a concept to refer to some aspect of what the human mind does, in this case, the logical implication. This is something the material implication doesn't do. The material implication is arbitrary in the sense that it happens to be only the best approximation to the logical implication mathematicians were able to find in 170 years of mathematical logic. It is essentially an accident in the history of mathematics. Happenstance. Aug 6, 2020 at 20:55
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Vector spaces are used to model all sorts of things, including the human mind, and the use of conditionals is a part of the material world, along with the rest of human behavior, even if human mind were not. Not to mention that uses terms are put to today do not answer to what their predecessors were conceived for anyway. And if historically contingent makes arbitrary then all of mathematics is "arbitrary", so the word would be of little use.– ConifoldAug 6, 2020 at 21:35
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@Conifold This is also fallacious. You are trying to argue a point using fluffy notions like "arbitrary". I explained in what sense the material implication is arbitrary. You then implicitly extend my definition and apply this extended definition to mathematics and get an absurd result. Well done but nothing to do with me. If we all did what you just did, we would all be better off shutting our mouths. And the rest of your comment is irrelevant to my last comment and to my answer. Maybe you could try to understand my point? Aug 7, 2020 at 9:17
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I am not arguing a point, only pointing out your mistakes. Take it as you will.– ConifoldAug 7, 2020 at 9:58