# Is the use of this conditional proposition correct, or does it violate the rules of logic?

Is the reasoning answering this logic puzzle correct? One of the answers was by Hexomino on question 1:

I think that the lawyer's statement does not help the client. Consider the following statement "If 0=1 then all elephants are yellow". This assertion is true simply because the initial predicate is false. We know that not all elephants are yellow but that doesn't matter because 0 is not equal to 1. Similarly if the accused man is innocent then the statement "if the accused man committed the murder, he had an accomplice" is true as is "if the accused man committed the murder, he did not have an accomplice". Therefore, for the defence lawyer to say that the assertion is false implies that the initial predicate must be true, i.e, the accused man committed the murder (and did not have an accomplice).

• It very much depends on whether the lawyer was using the material conditional of logic or the indicative conditional or ordinary reasoning, whose meaning is context-dependent. It the material conditional, the statement would be true if the accused didn't commit the murder, so saying it's false implies he did; but w/ the indicative conditional, saying it's false could mean "I'm not making any claims about whether he did or didn't, but even if he did he might not have had an accomplice". – Hypnosifl Sep 24 '20 at 6:48
• Thank you, I learned something new today 😀 – Prince Deepthinker Sep 24 '20 at 7:22
• This question certainly attacks an important topic regarding the nature of the conditional. If you're under the impression that argumentation is clear cut and mathematical, check out SEP's articles on informal logic and defeasible reasoning. Logic ain't math! – J D Sep 24 '20 at 15:50
• All of the Mathematical logic people should have been all over this one. The answer is YES the reasoning is correct because in both math & philosophy A TRUTH TABLE proves that the only way for a conditional statement to be false is if the antecedent (wording before the THEN PART) while the consequent is false. The consequent is the wording after the THEN PART. If P THEN Q. Where P is the antecedent & Q is the consequent. Google truth tables to find out why. Outside of math & philosophy people don't mean the same thing usually. It can be used Rhetorically. So a false part could be sarcastic. – Logikal Sep 24 '20 at 18:20
• BTW, one way to think about the indicative conditional in this case would be in terms of modal logic--the statement "if the accused man committed the murder, he had an accomplice" could be understood as "in every possible world where the accused man committed the murder, he had an accomplice". Or perhaps a probabilistic analogue, like a conditional probability statement described in Bumble's answer. In either case, one can judge the statement to be false even if one believes that in this world, the accused didn't do it, or that it's very unlikely he did. – Hypnosifl Sep 25 '20 at 0:14

In addition to Hypnosifl's comment, there is another way to think about what is happening here. When making statements about the real world, as opposed to mathematical statements, we take for granted that such statements are typically merely highly probable, not certain. When the prosecutor says, "if the accused man committed the murder, he had an accomplice," there is an unspoken "it is highly probable that..." After all, what the prosecutor says is based presumably on evidence of some kind, and evidence may be misinterpreted or falsified. So the prosecutor can be understood as saying P(accused had an accomplice | accused committed the murder) is high. The defending attorney expressing disagreement is then committed to P(accused had an accomplice | accused committed the murder) is low, or at least not high. Now commonly (though not always) the probability P(B|A) is not high just because the probability of A is low. So there is nothing inconsistent about the defending attorney believing that it is highly improbable that her client committed the murder, but also highly improbable that her client had an accomplice if he did. Arguably, the attorney is not helping her client by saying what she did, but she is not incriminating him.

Puzzles like this are often invented by mathematicians who spend all their time using material conditionals and appear to be oblivious to the fact that there are lots of other kinds.

• The problem here is transparency. There should be no unspoken ANYTHING. You directly stayed all the RELEVANT FACTS. What humans do instead is do what is CONVENIENT. They don't want to work too hard in certain wealth classes. You the lower people are supposed to understand or take something for granted. No it is your job to explain all facts relevant to the conclusion. This how pure deductive reasoning WORKS. None of this short cut stuff. We can't & shouldn't do so on proofs and leaders nor authorities should be doing so either. Pure deductive reasoning is checks & balances. Some don't like it – Logikal Sep 25 '20 at 6:27

I'm going attempt rolling Bumble's and Hypnosofil's responses into something and add a dash of background

NB: This has been challenged as a mischaracterization. See the comments below.

Yes and no. One way logic can be dichotomized is between the formal and informal, and the answer to your question depends on the types of logic you choose. Perhaps the most salient example to show the difference is between the material and indicative conditional which highlights the difference between natural and artificial languages.

The traditional disputes of philosophers are, for the most part, as unwarranted as they are unfruitful. The surest way to end them is to establish beyond question what should be the purpose and method of a philosophical enquiry. - A. J. Ayer

You have highlighted an important and often unnoticed phenomenon in logic, and that is the distinction between how the human reason actually works, and how we can model and mechanize human reason. Since the 1950's when AI researchers attempted to begin attacking the problem of how can we solve problems using reason and common sense, they brushed up a scientific fact that human reason is defeasible and does not use formal languages. Let's look at the difference:

Technically, introductory formal logic uses the material conditional which is actually a mathematical mapping, technically a 2-variable relation whose domain (T,F) is logically equivalent to the codomain value of T. Since each distinct ordered pair in the domain maps to an element, it's a function in the technical sense. It would have to be one-to-one to be injective and is not. See the mapping:

p q p → q
T T T
T F F
F T T
F F T

WHAT THE HECK DID YOU SAY? Odds are if you're not a mathematician, something similar ran through your mind. There's a strange middle ground where some thinkers know enough math to not know this isn't actually how people think, and this is a classic motif in the criticism of philosophy, that philosophers are a little too detached from reality even when they study ontology. Let's try that again for the non-formalists among us using natural language.

"If the accused man committed the murder, he had an accomplice." This is the natural language or indicative use of conditional. Note maybe the statement 'He committed the murder' is false and is counterfactual. What does the law say about the difference between murder and homicide? If we don't have persuasive evidence such as a videotape of the act and are forced to rely on (legal) testimony, how should we decide what to decide? Even if we saw the act, is mens rea present?

Notice the difference in the approach to problem-solving? The first is formal logic and deductively valid, and the second is informal logic and cogent. In philosophy, this distinction was drawn on and the overuse of technical definitions and abstractions such as formal logic was criticized by the ordinary language philosophers who appeal to intuition much more strongly.

Perhaps the most intriguing biography is that of Ludwig Wittgenstein who is characterized as having an early and later phase and embodying both. In fact, his thoughts about family resemblance were extremely influential in meta-ontological thinking. Another important figure is Stephen Toulmin whose model of argumentation in Uses of Argument seems to be practically ignored by those in computational fields like mathematics and computer science. (I speak from direct experience on both fronts).

• I disagree with your characterisation of the issue. Classical logic is not the only formal logic, and the material conditional is not the only formal conditional connective. Conditional probability is itself part of a formal calculus, and many other conditionals are too, including intuitionistic implication, relevant implication, strict implication, Lewis conditionals, Stalnaker conditionals, etc. The important distinction here is not between formal and informal logic, but between different different kinds of formal logic. Even defeasible reasoning can be formalised using non-monotonic logic. – Bumble Sep 24 '20 at 17:18
• @Bumble Thanks! I'll see where we disagree. I may have engaged in some oversimplification out of laziness or for clarity. – J D Sep 24 '20 at 21:13
• @Bumble It does indeed look like my two examples strongly suggest that dimensions of formality and defeasibility of logics aren't independent. I chose two extreme examples and didn't clarify. Thank you. Certainly, AGI is nothing more than trying to perfect the form of defeasibility in a non-biological system, so defeasibility and formalization are certainly not mutually exclusive. I also reread the original problem and question, and you are right to call out that this question, given the context of being a logical puzzle, the formalism is presumed in the speech act itself. – J D Sep 25 '20 at 14:41