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Suppose m is a dry match. Under most circumstances, if m is struck, it will light. But if somebody were to wet m, then we might stipulate that m would not light even if it were struck.

  1. Suppose S denotes the sentence "m is struck".

  2. Suppose L denotes the sentence "m will light".

  3. Suppose W denotes the sentence "m is wet".

Question: Is the conditional S > L true?

My intuition is that S > L is not true. This is because S is not enough for us to guarantee that L is true. For example, suppose S & W is true. Then S is true. Yet L would be false. Hence S > L is false.

The problem with this is argument is that if it is true, then almost every conditional we utter is false. When we assert conditionals like S > L, we are often conversationally implying S & ~W > L. And in fact this isn't even quite right. There could be other things besides just W that could prevent L from being true. So really, the antecedent of S > L would have to contain an infinite number of further conditions (for example, that the match isn't deprived of oxygen at the moment it is struck) to truly guarantee that L is the case.

I'm temporarily calling this argument the "Underspecificiation of Conditional Antecedents Argument (UCAA)" for why many conditionals, as uttered, could be false. Is this a common thought, and has there been much written about this argument?

  • If S & ~W => L (the two sufficient conditions for L),then you could not prove S => L. If, however, L => S & ~W (two of the necessary conditions for L), then you could, of course, prove that L => S. – Dan Christensen Jul 3 '14 at 5:53
  • I would suggest reading Carl Hempel's "Empiricist Criteria of Cognitive Significance: Problems and Changes." Although it is not directly related to your question, there is a discussion in the paper about Carnap's theory that terms are significant when their meaning can be given partially by a conditional. For example, Carnap might say, "If an object x is struck at t, x is flammable iff x bursts into flames at t." Roughly, the idea is that for objects that do not happen to light at t, the meaning of the word "flammable" is left undetermined... – leibnewtz Jul 12 '14 at 2:25
  • ...and the sentence it is embedded in fails to gain a truth value of T or F. – leibnewtz Jul 12 '14 at 2:29
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I haven't seen anything written on this, but I think this is strongly related to lexical semantics. The reason is that you have picked only one of a profusion of problems with "if the match is struck, it will light".

The match will not light if it is wet, at absolute zero, struck too gently, struck in an atmosphere without oxygen, struck at relativistic speeds, struck the way you strike a drum, struck on a smooth surface, etc. etc..

That you are supposed to import all the relevant context (standard temperature and pressure, appropriate strength, motion, and contact surface for a match, and so on) is a typical part of natural language--and indeed, if you're not pretty savvy in this regard anyway, you will misunderstand the word "light" to be the same thing as what a lightbulb does.

But just as lexical semantics opens up all sorts of wiggle-room for comprehension, so does context-aware analysis of truth statements. If one wants to know why formal logic is not always as useful as one might hope for everyday situations, this is a big part of it.

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As you've already stated, S→L cannot be proven (in the formal sense of the word) from any assumptions we would normally make. Therefore, because propositional logic is sound, we conclude that S→L must be false.

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I would take the position that they are "conditionally" true, unless/until some underlying or assumed condition is false. Using your example, most people familiar with matches, would not strike a wet match. They would wait until it dries, before striking it. Therefore, the "reality" statement should be S & (dry & O2 & T & etc.) > L. This statement is true, as long as all underlying conditions are true.

In "normal" conversations, where neither side has anything to gain (or loose), the underlying conditions have the highest provability of being true. The more a side has to gain(or loose), the more likely one or more of the conditionals is false. So, the truth or falsity of utterances, depends on the "degree of manipulation" of the conditionals.

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I would say that your issue here is conflating the everyday usage of conditional statements with the notion of logical implication. One way in which the conditional is used in everyday conversation is as a sort of model that uses past experiences and makes generalized statements about them. "If you strike a match, then it will light" is an example. It's generally assumed unless otherwise stated that the match is not wet and that all other conditions are such that the match will light.

On the other hand, the material implication of logic is not a model in the aforementioned sense. It's defined such that the implication is false just in case the antecedent is true and consequent false.

The solution is that we need to make our premises explicit such that there is no room for ambiguities that could make a premise both true and false so-to-speak. That's not to say that it could be both true in false, it's to say that we could interpret it in one sense that would make it true and in another sense make it false.

And example: "Mary likes John." If we interpreted this as referring to today, it could be true, yet it could be false 5 years from now. So we need to be more precise. "Mary likes John at time t," where Mary and John have specific referents (i.e. we're not speaking about two different Marys or two different Johns.) He we removed all ambiguities.

So in your example, just be more precise. "If you strike match x at time t, then match x does light at time t." Notice that it doesn't matter what conditions hold for the match and the match box here, since the truth-value of the implication will be determined by what actually happens. It will be false just in case you strike it and it doesn't light.

One last tidbit: Notice how I changed the verb phrase "will light" to "does light". This is to remind us that the implication is determined by what happens. There is no element of prediction involved. If we let time t be something in the future, we wait until that time to see if the premise is true or not.

Your issue has been a problem for me in the past; I now make sure that my premises are all precise and verifiable, and thus the truth-value of the premise can be determined without any confusion.

p.s. we could have been even more explicit by making sure that "you" and "match x" and "time t" all have specifically determined referents, but for convenience I figured it would suffice to assume that we already had assumed that.

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