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I wanted to understand the concept of non-truth function (which I found when reading about conditionals in logic). The definition of non-truth function that I have is (from reddit):

If a connective is not truth functional, you need something more than the truth value of the parts in order to know whether the compound is true.

Or as the absence of truth function:

So if a connective is truth functional, you can calculate the truth value of the compound sentence from the truth values of the parts.

In other words, if we have a functional that receives boolean values (caused by some propositions I assume) then we cannot unambiguously assign it a truth (boolean) value.

The abstract definitions sort of make sense to me but when I try to come up with a concrete single example that elucidates the concept transparently I can't find one with an explanation that makes sense to me.

For example, one that looks promising is an example provided in the reddit thread:

Sentence 2) John is happy because Mary is home.

It seems that the use of "because" is the "typical" example of non-functionals (and "before" or "it is possible that"), however, I find the ones using "because" the most troubling.

In the sentence above, it explains that John is happy because Mary is home. If it were for any other reason why would the sentence use the word "because"? It nearly feels that the "because" is equivalent to an implication:

If (Mary is home) then (John is happy)

If that is the case, then the "because" can be used in the same way as a conditional of propositional logic. If that were not the case, how is it that we need extra information to discover if that functional is true or not? It seems to unambiguously assign truth values. What "extra" info do we need?

Can someone show me a clear example of a function that could be assigned the value true and false, or unknown even, for example, F(A,B) = F(True,True) where A and B are propositions that might be true or false?


Related:

  • See Indicative Conditionals for discussion. – Mauro ALLEGRANZA Jul 3 '17 at 5:52
  • "because" conveys the expectation of a cause-effect relationship. If Mary is not at home, how we evaluate the "because-conditional" ? If truth-functional, we must evaluate it with the same truth value of: "Jhonny is happy because the Moon is red". – Mauro ALLEGRANZA Jul 3 '17 at 8:46
  • Connectives of classical logic are only approximations to their use in natural languages. The ones that are most commonly considered to deviate are implication (material conditional) and disjunction. For example, implication should not always be true when its premise is false, and some disjunctions should have truth values even if their terms do not, see supervaluationism. So functional dependence on truth values has to be relaxed. – Conifold Jul 3 '17 at 18:14
  • @Conifold you can add conjunction to that list since non-commutative conjunction is crucial to many dynamic logics and natural language. "Mary fell and was injured" vs "Mary was injured and fell". – Dennis Jul 6 '17 at 4:39
  • @MauroALLEGRANZA I think your onto something. Though I am not sure I appreciate examples, do you mind expanding on them a bit please? – Pinocchio Aug 27 '17 at 20:17
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Conditionals in English are used for a lot more than just expressing simple truth functions. Here are some general cases where the truth functional material conditional doesn't fit.

  1. Claims about causal relations. Often when we say "if A then B" we are making a causal claim. But "A causes B" is not a truth function. It is not true just in the event that A is false. "If I drink this poison it will kill me" is a causal claim, but it is not true just because I never drink it. Causal claims are not analyzable in any straightforward way into some combination of truth functions. This is why there is much discussion over how to analyze the concept of causation. A truth function expresses nothing more than a correlation, and it is a commonplace observation that correlation is not causation.

  2. We sometimes use conditionals to describe general relationships. Suppose Alice is in London. "If Alice is in Berlin then Alice is in Germany" is a reasonable thing to say, but not "If Alice is in Berlin then Alice is in Mexico." We wouldn't say that both of these are true just because they have a false antecedent and so the truth functional material conditional is true.

  3. Often we wish to express conditionals with an antecendent that is known to be false, just for the purposes of a hypothetical speculation. Two canonical examples discussed by Ernest Adams are the sentences, "If Oswald didn't kill Kennedy someone else did," and "If Oswald hadn't killed Kennedy someone else would have." Assuming we accept the standard account of the assassination of Kennedy, both of these conditionals have a false antecedent. But neither are trivially true and they are also significantly different in meaning from each other. The first is asking us to consider the epistemic possibility that Kennedy was actually killed but not by Oswald, the second is asking us to entertain the hypothetical metaphysical possibility that in some similar world to ours Oswald doesn't kill Kennedy but someone else does. These latter kind are often called counterfactuals and are difficult to assess because they involve making assumptions about which similarities are relevant and which are not. In another pair of examples given by Quine, "If Caesar had been in command in the Korean War he would have used nuclear weapons" versus "If Caesar had been in command in the Korean War he would have used catapults" we are stuck between deciding whether it is reasonable to assume that transplanting Caesar into the 20th century involves updating his knowledge of the weapons of warfare.

  4. Sometimes we use conditionals as quantificational restrictors. Indeed it is common in quantifier logic to write "all A's are B's" in a conditional form as (∀x)(Ax → Bx). The conditional here can be called a quantificational restrictor because it is restricting the range of x to only those x's that are A. But the truth functional material implication only works in one special case: that of the universal quantifier. English has lots of quantifiers, e.g. 'most', 'many', 'a few', etc. If we use the letter Q as a placeholder for a general quantifier and try to represent the sentence "Q A's are B's" as (Qx)(Ax → Bx) this gives the wrong meaning in all cases except for Q being 'all'. For example, "most penguins can fly" would come out true if we represented it as (most x)(Penguin(x) → Flies(x)) simply because most things are not penguins. English conditionals are not limited in this way. It is not unreasonable to say something like, "For most things it would be true to say if it's a bird it can fly."

  5. The truth functional material implication gives the wrong result if we are uncertain about whether it is true and are willing to express our uncertainty using probabilities. To say "it is probable that if A then B" does not mean that it is probable that A is false or B is true. The probability of a conditional is given by the conditional probability. If I roll a regular 6-sided die, the probability of "if it comes up even then it will be a 6" is one third, but the probability of the material conditional is two thirds. The same observation carries over to bets. If I offer to bet you that "if Brazil get to the final of the World Cup they will win," I am not betting that they either not get to the final or win.

  6. In English we can conditionalize lots of things other than statements. We can conditionalize almost any speech act, e.g. a question, a command, an obligation, a threat, a promise, a bet, etc. None of these conditionalized speech acts behave like the material conditional and they cannot be truth functions because their consequent part does not have a truth value. But it would be odd to believe that the word "if" is fundamentally ambiguous and means something completely different depending on whether it is conditionalizing a statement or some other speech act. Here are some examples... A conditional command: "if the patient is still alive tomorrow, change the dressing". This does not mean "make it the case that either the patient is not alive or the dressing is changed". A conditional promise: "I promise if I return safely I'll marry you," does not mean "I promise either not to return or to marry you".

  • why is it that "Causal claims are not analyzable in any straightforward way into some combination of truth functions"? (if you have a reference I would love to read more about how causation are analyzed). – Pinocchio Aug 27 '17 at 20:07
  • One account is given in John Mackie's book "The Cement of the Universe" in which he analyzes causation as an INUS condition, i.e. an insufficient but non-redundant component of an unnecessary but sufficient condition. This is a step in the direction of describing causation, but it is still quite weak. It doesn't capture the fact that a causal claim provides an explanation of an event, usually in the form of some kind of mechanism. Another good book on causation is James Woodward's "Making Things Happen". – Bumble Aug 29 '17 at 3:24

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