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In mathematics, "P implies Q" means "P is false and/or Q is true." Or equivalently, that "it is not the case that P is true and Q is false."

What errors or confusion might arise in a natural language in which "P implies Q" meant "P is false and/or Q is true?" What, if anything, would be lost in terms of expressive power?


EDIT: As in classical propositional logic, I assume here that P and Q are unambiguously true or false logical propositions in the PRESENT. I am interested here only is what is NOW true or false, not what might be true or false in the future, or what hypothetically might have been in the pass.

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  • Um... How is what you describe different from what we find ourselves in at present?
    – Boba Fit
    Commented Dec 7, 2022 at 13:36
  • Some do not accept that an implication can be true, regardless of the consequent, if the antecedent is false, etc. Commented Dec 7, 2022 at 16:48
  • Does not really change the comment.
    – Boba Fit
    Commented Dec 7, 2022 at 17:21

3 Answers 3

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The literature on this subject is enormous. In the last 50 years there have been thousands of papers and scores of books on the subject of conditionals and the logic of how they work. It would be impossible to cover all the issues in the scope of a single question like this.

In very broad brush terms, there are many reasons why the material conditional does not cope with what we might call real-world conditionals. The real world is full of intensionalities, modalities, speech acts, counterfactual possibilities and uncertainty. Mathematicians have the luxury of ignoring these things, because they work in an artificial environment where such considerations do not apply.

Real-world conditionals are typically not truth-functional and hence are not material conditionals. Kripke was of the view that all conditionals are modal. I would prefer the more modest claim that most are. Some modal conditionals can be handled as strict conditionals within a C I Lewis style modal logic, i.e. as a material conditional within the scope of a box operator, but many cannot. Causal claims are a particularly common kind of modal conditional that are not truth-functional.

Most real-world conditionals are uncertain. Practically everything out here in the real world is uncertain. An uncertain conditional is systematically different from an uncertain material implication.

For a great many conditionals, "if A then B" is contrary to "if A then not B", and in a fairly large number of cases they are actually contradictory, whereas for material conditionals these are consistent.

For a great many conditionals, the denial of "if A then B" does not entail commitment to "A and not B".

For a great many conditionals, ¬A does not entail "if A then B". There have been some attempts to explain this away as a conventional implicature. Frank Jackson in his earlier work adopted this approach. Without going into detail, it really doesn't hold up. And even if it did, it would not go very far in accounting for all the other differences.

Some conditionals function as quantificational restrictors. Indeed, on Angelika Kratzer's account, this is their main function.

Most real-world conditionals are context sensitive. Many, if not most, conditionals must be evaluated against a backdrop of the pragmatics of the circumstances in which they are used.

In the real world, we have not just conditional statements but conditional commands, conditionals questions, conditional promises, conditional bets, conditional threats, conditional offers, conditional obligations, conditional all kinds of stuff. The logic of these in most cases does not follow the logic of material implication.

There are counterfactual conditionals, which cannot be understood as material. Most accounts are forced to treat counterfactuals as a completely different kind of construction, but the difference is exaggerated. There are unified accounts of conditionals. The word 'if' is not ambiguous.

Incidentally, it is rather oversimplying to say that in mathematics implication is always material. Probability theory has its own conditional, which has a logic quite different from that of the material conditional. Also, some mathematicians use non-classical logics, such as intuitionistic logic.

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  • Perhaps I should have narrowed it down a bit to the case of P and Q being logical propositions that are unambiguously either true or false in the present. Does that make a difference? Commented Dec 7, 2022 at 1:58
  • Ambiguity is not really an issue. Requiring P and Q to be propositions would eliminate the cases where the consequent is a speech act. Limiting cases to the present tense would eliminate counterfactuals and future contingents. Nothing else changes. Most real world conditionals are modal and are not truth functions. To clarify, I'm not claiming that material implication is not a conditional, or that there are no cases where it is useful. It has its uses, but it is at best a very narrow and specialised kind of conditional.
    – Bumble
    Commented Dec 7, 2022 at 3:47
  • So, if we are talking about logical propositions P and Q that are either true or false in the present, can we can assume that "P implies Q" means "P is false and/or Q is true" without any errors or confusion arising, and without any loss of expressive power? Commented Dec 7, 2022 at 3:59
  • No. Even with those restrictions, most real-world conditionals are not material implications.
    – Bumble
    Commented Dec 7, 2022 at 5:16
  • Can you give even one example of those "real-world conditionals" involving a pair of logical propositions that are unambiguously either true or false in the present? Commented Dec 7, 2022 at 5:52
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If "P implies Q" meant "P is false and/or Q is true" in a natural language, it could be confusing because it does not match the standard logical meaning of "implies." In logic, "P implies Q" means that if P is true, then Q must also be true. This is different from the meaning given in the question, which says that if P is false or Q is true, then the implication holds.

This confusion could lead to errors in reasoning and misunderstandings in communication. For example, if someone said "P implies Q" to mean that P is false or Q is true, another person might interpret it as meaning that P must be true in order for Q to be true, which is not the intended meaning.

In terms of expressive power, using this alternate meaning of "implies" would also be less powerful because it would not allow for expressing the standard logical implication. This would limit the ability to reason and make logical statements using this natural language. For example, it would not be possible to express the statement "If it is raining, then the ground is wet" using this alternate meaning of "implies," because it does not allow for the possibility that P is true and Q is false.

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  • +1 but your last sentence is wrong. If p is true and q is false, then ¬p ∨ q is false, which is what we want since p implies q is also false in this case, so there is no problem here. Commented Dec 11, 2022 at 10:49
  • I assume here that P and Q are unambiguously true or false logical propositions in the PRESENT. (See my edit just now.) Therefore, if "P is true," and "P is false or Q is true," then Q must also be true. Thus "P is false or Q" gives the same result as "if P then Q." in such cases. Commented Dec 13, 2022 at 23:18
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What disadvantages in a natural language if "P implies Q" meant "P is false and/or Q is true?"

The horseshoe ¬p ∨ q does not mean "If p, then q" and so does not mean "p implies q".

The consequence is inescapable. Any reasoning using the expression ¬p ∨ q to tell us the truth value of the conditional "If p, then q" or of the implication p → q is pie in the sky. Any reasoning using the expression ¬p ∨ q to tell us the truth value of the conditional "If p, then q" or of the implication p → q is just wrong reasoning. It is a logical fallacy.

Sometimes, many times, a potential countable infinity of times, ¬p ∨ q will give the correct answer. This is why mathematicians feel good enough about using the horseshoe.

Sometimes, however, indeed many times, and in a potential countable infinity of times, ¬p ∨ q will inevitably give the wrong answer.

This applies irrespective of the subject matter: philosophy, metaphysic, everyday life, engineering, science and indeed mathematics itself. This much is in plain view and has been for more than a century at least. In fact, we know that the horseshoe is the wrong model since the critique Diodorus made 2,300 years ago of Philo's view of the truth conditions of the conditional (as reported by Sextus Empiricus). No logician could possibly reasonably pretend to be ignorant of Diodorus' point.

Still, apparently, mathematicians don't mind. They have somehow been able to accommodate themselves to the fact. However, in the real world, we know that erroneous logical reasoning can produce dire consequences. More to the point, systematic erroneous logical reasoning will inevitably produce dire consequences sooner or later and using the horseshoe as mathematical model of the conditional introduces a systematic error.

We are perhaps not quite there yet but as complex systems become more integrated and artificial intelligence come to be used in their development, the riskier it will become, both in terms of probability of the consequences and in terms of their seriousness. Think for example of systems and procedures necessary to ensure the safety of nuclear plants, of the use nuclear warheads. Think also of trains, airplanes and mass transit. Think also of the security of the systems used in the financial world or to ensure national security.

So the disadvantages are potentially enormous and will inevitably be realised sooner or later.

We also need to look at this edit of the question:

EDIT: As in classical propositional logic, I assume here that P and Q are unambiguously true or false logical propositions in the PRESENT. I am interested here only is what is NOW true or false, not what might be true or false in the future, or what hypothetically might have been in the pass.

First, I don't remember any mathematical logic author insisting that propositional logic only applies to propositions which are true or false in the present.

Second, logic is useless in situations where we already know which propositions are true and which are false. When we do, we don't need logic to help us decide which are true and which are false.

Further, the question is explicitly about the effectiveness of mathematical logic for natural language statements:

What errors or confusion might arise in a natural language in which "P implies Q" meant "P is false and/or Q is true?" What, if anything, would be lost in terms of expressive power?

If you start by narrowing the scope of the question to pure mathematics, then the question becomes meaningless metaphysics. The only value of mathematics is in its application to the real world.

Also, mathematicians are presumably human beings and there is no reason whatsoever to believe that they rely on a different logic than other human beings. If they follow a particular method of proof, they remain logical in the ordinary sense, in the same way that a chess player remains logical even when he or she follows chess rules. However, whatever is the result of his or her reasoning only applies to the game at hand, not to the wider world. Whether mathematics applies to the wider world is an empirical matter which as such cannot be decided solely by looking at the mathematics. This applies to mathematical logic. Mathematicians doing mathematical logic proofs do, presumably, follow the rules they accepted. They try to be logical in their application of the rules. However, just as in the case of the game of chess, whether the result tells us something true about the real world, and in particular about the logic of human reasoning, cannot be decided just by considering the rules of mathematical logic. To see if the rules are any good, you have to consider the empirical data about actual human reasoning, including the actual reasoning of mathematicians. As it turns out, even in mathematics, actual human reasoning contradicts mathematical logic.

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  • Re: "So the disadvantages are potentially enormous and will inevitably be realised sooner or later." How about sooner? An example would be nice. Bear in mind that your example should deal only with logical propositions that are unambiguously either true or false in the present. Commented Dec 11, 2022 at 16:15
  • In mainstream mathematics (the basis of most science, engineering and commerce) "A implies B" is usually defined as either "not A OR B" or "not (A AND not B)". Commented Dec 11, 2022 at 16:22
  • @DanChristensen "An example would be nice." There are plenty of such examples on the Internet. Some of them date back to the beginning of the 20th century. One goes back to Diodorus at the time of the Stoic, 2,300 years ago. People with a real expertise in formal logic cannot possibly ignore these examples. 2. "In mainstream mathematics" Defined? Who cares. What matters is what people actually do, including mathematicians, and no mathematician relies on the horseshoe. Commented Dec 11, 2022 at 17:10
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    So, no example is forthcoming. Thanks anyway. If you are interested, the above "definitions" can also be derived from first principles. See dcproof.wordpress.com/2017/12/28/if-pigs-could-fly Commented Dec 11, 2022 at 18:08
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    If you cannot present not even a single example of any supposed inconsistencies resulting from material implication (i.e. a theorem of the form A & ~A), there is no point in continuing this discussion. Thanks anyway for your time and consideration. Commented Dec 12, 2022 at 16:48

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