# Deductive reasoning & conditionals

What would be a good example of explicit deductive reasoning that doesn't seem to be possibly interpreted correctly as a conditional (If A, then B)?

• – Conifold Jun 27 '20 at 8:30
• @CanBeSaidClearly "Doesn't seem to be possibly" is an English phrase meaning "doesn't seem to be possibly". Don't think it could be expressed differently, I suspect it cannot, but prove me wrong if I am. If you did, you would prove that it is understandable. And if you don't, you confort my belief that it cannot be expressed differently. But no doubt it means something. I don't see how you could prove that this is not the case, since failing to understand something which is perfectly understandable is such a common occurrence. – Speakpigeon Jun 27 '20 at 9:14
• You can help by articulating yours in more than one sentence. – Conifold Jun 27 '20 at 9:35
• @Fizz 1. The question isn't about the material implication. 2. I don't expect anyone to come up with any good example; I just wanted to make sure. – Speakpigeon Apr 8 at 10:28
• Depending how much you want to stretch your argument, it would be interesting to hear from you how you argue that a bi-conditional (P if and only if Q) is expressible as a (single?) conditional (or I'm guessing you mean as a sequence of conditionals?). You need the "and" at the same meta-language level as the "if" for that to sound naturally. – Fizz Apr 8 at 18:16

It would be worthwhile distinguishing between a conditional sentence in the object language and a conditional in the metalanguage. Some deductive arguments have a conditional in the object language, e.g. those of the form modus ponens or modus tollens. Some arguments do not, e.g. those of the form conjunction elimination, disjunction elimination, etc.

But what is always possible (and I think this is what your question is asking) is that a deductive argument can be expressed as a kind of modal conditional at the meta level. A valid argument is one such that necessarily if all the premises are true then the conclusion is true.

The modal term 'necessarily' can be replaced by some other modality or generality, depending on what account of validity you find most congenial. Some common ones are: under all interpretations, under all substitutions of the non-logical constants, under all permutations of the domain of quantification, in all possible worlds, it is a priori knowable that, it is provable that, it is conceptually certain that, or whatever.

• 1. "distinguishing between" I don't see the need for this distinction. You would need to give an example proving I am wrong. Rules such as "conjunction elimination" are perfectly understandable in everyday English. As I see it, mathematical logic is not logic but it is logical.2. "modal conditional at the meta level" No, no extraneous notion of modality is implied in my question. Just the ordinary conditional. – Speakpigeon Apr 8 at 10:35
• @Speakpigeon: it's necessary (to make the distinction) if you want to reason about proofs as mathematical objects themselves. See sequent and proof theory. Also, the whole programme of abstract algebraic logic is built upon abstracting the notion of consequence relation to its properties. – Fizz Apr 8 at 16:51
• @Speakpigeon: Also, in logics besides the classical one are often not structurally complete. So one needs to distinguish derived rules (i.e. rules provable in the logic itself) and admissible rules which can be added to the logic whiteout changing provable tautologies . Again, you cannot make this distinction unless you have a notion of sequent. Since you tagged this modal logic: in that context, an example of admissible but non-derivable rule is "the rule of necessitation". If this is unclear, as a Q separately... although some like have been asked here before. – Fizz Apr 8 at 17:03
• @Fizz "it's necessary (to make the distinction) if you want to reason about proofs as mathematical objects themselves" Not at all. My question is about deductive reasoning and the natural language conditional. The logic of deductive reasoning is definitely sturdy enough to accommodate reasoning with metalevels. – Speakpigeon Apr 9 at 10:35
• @Speakpigeon: ok, see the 2nd part of my answer then which exemplifies the difficulties of expressing both logical and meta-logical "ifs" by the same word. – Fizz Apr 9 at 16:01

Perhaps you're looking for objections to characterizing natural language conditionals with the material conditional ("->"). If so, two examples come to mind.

First, from Van McGee:

(1) If that creature is a ﬁsh, then if it has lungs, it is a lungﬁsh
(2) That creature is a ﬁsh
(3) Hence, if it has lungs, it is a lungﬁsh

If we treat "if...then" in natural language as the material conditional, then we have:

(1*) A -> (B -> C)
(2*) A
(3*) B -> C

Van McGee claimed, however, that while (1*) and (2*) might be stipulated to entail (3*), one can construct scenarios in which (1) and (2) are true, but (3) false.

Second, natural language counterfactuals raise similar problems for material conditionals:

(4) If Hoover had been born in Russia, he would've been a communist
(5) If Hoover were a communist, he would've been a traitor
(6) Hence, if Hoover had been born in Russia, he would've been a traitor

Which - if using the material conditional to model - results in:

(4*) A -> B
(5*) B -> C
(6*) A -> C

But while (4*) and (5*) entail (6*), many believe (4) and (5) can be true while (6) false.

• 1. My question isn't about the material conditional. 2. In your "McGee" example, the implication (A → (B → C)) ∧ A ⊢ B → C is a simple modus ponens and we don't need the material implication to see that it is a logical truth. McGee's counterexamples are fallacious. 3. "counterfactuals raise similar problems" Counterfactuals do falsify the material implication, but the two arguments have the same form and are true. If the conclusion is false, then at least one of the premises is false, irrespective of what "many believe". McGee's claims are fallacious. – Speakpigeon Apr 8 at 10:51
• I suspect Van McGee's argument would be that in natural language the statement "if it has lungs, it is a lungfish" can be interpreted as a general statement about all things with lungs (including non-fish things), as opposed to some specific named object in your domain of discourse that the previous propositions were also specifically referring to (and which was established to be a fish by proposition 2). So it'd be an illustration of semantic ambiguity in if-then statements of natural language. – Hypnosifl Apr 8 at 20:02
• @Hypnosifl No, no semantic ambiguity here. and we don't need to call on the notion of domain. Logic is 100% about form. McGee's natural language argument is perfectly valid irrespective of domain. Replace by A's and B's, and this is still valid. Aristotle did that 2,500 years ago, why are we now going backward? McGee's fallacy is precisely not to respect the form of the argument. One can always equivocate and there is nothing to stop McGee doing that. – Speakpigeon Apr 9 at 10:23
• There is semantic ambiguity in what logical form you assign each statement, how you "translate" informal English into a formal logic statement. If "that creature" is referred to with the individual constant t, then statement #1 could be translated as Fish(t) -> (Lungs(t) -> Lungfish(t)) and statement #2 could be translated as Fish(t). But statement #3 does not refer to "that creature" so it could be translated as "For all x, Lungs(x) -> Lungfish(x)". And in this translation, the third statement would not follow deductively from the first two in first-order logic. – Hypnosifl Apr 9 at 13:08
• @Hypnosifl "There is semantic ambiguity in what logical form you assign each statement" There is no semantic in the operation of translating. There may be semantic ambiguity in the original and in the result. A translation is not ambiguous by itself. It may be acceptable or not acceptable, and possibly consistent or not consistent. The problem with McGee's example is perfectly obvious. This is a gross fallacy. He is essentially arguing on the basis of the ambiguities of informal arguments that the logic of formal arguments doesn't work. He is a waste of time. – Speakpigeon Apr 11 at 10:40

Given your other comments on this page, you identify the conditional with the consequence relation (at least in this context of deductive reasoning). This is fine in classical logic because it has a/the deduction theorem and is also structurally complete (meaning every admissible rule is also derivable in classical logic). So, sure in classical logic

A, B |- C iff
A |- B->C iff
|- (A /\ B) -> C (and even) iff
|- A -> (B->C)

And likewise

A->B, A |- B iff
|- ((A->B) /\ A) -> B iff
|- (A->B)->(A->B)

Because classical logic can be axiomatized just based on the material conditional (and modus ponens), one can claim there's no need for any connective besides the [material] conditional (and either negation or a false element) in classical logic. (More precisely the implication-negation fragment is functionally complete in propositional classical logic.)

In general though, in other logics, it's not safe to make this identification of the conditional with the consequence relation. Even intuitionistic logic admits rules that are not derivable in the logic itself. So if your "explicit deductive reasoning" allows any logic besides the classical one, you cannot make this claim that the conditional and the consequence relation are one and the same notion.

If (on the other hand) your "explicit deductive reasoning" refers to some natural language statement encoding a modus ponens (for instance), I don't see how it's possible to intelligibly express it just using consequential conditionals and not sound gibberish in natural language. For instance, what is your proposed re-encoding of this explicit reasoning:

If it rains, [then] the soil gets wet. It rains. Therefore the soil gets wet.

using just if-thens? Are you claiming that one can be easily understood if instead of the above one says:

If if it rains then the soil gets wet and it rains, then the soil gets wet.

I'm fairly sure you don't mean that. You're perhaps thinking that just "If it rains, then the soil gets wet" is the whole act of "explicit deductive reasoning", but the latter shortened version is actually not explicit about the fact that it rains.

Sure you can sound less gibberish by using something else than "if-then" for the inner conditional here

Assuming (or knowing) that whenever it rains the soil gets wet, then if it rains, then [surely] the soil gets wet.

While more comprehensible (because I've avoided repeating "ifs" by substituting a "whenever" for one of them), this latter form sounds like we're just repeating a hypothetical than concluding anything about what's going on right now. But in the "double if" sentence, this doesn't help much with getting a comprehensible form:

If whenever it rains the soil gets wet and it rains [right now], then the soil gets wet [right now].

... unless you add some helper words (like "right now") to hint at the distinction between stating a consequence (as a conditional) and applying it to some facts. (And even then it's barely comprehensible.) Which gets exactly to the point that there's a need to disambiguate the level/kinds of these two "ifs" in an "explicit deductive reasoning" like MP.

To make this last part a bit more precise (and perhaps more clear/intuitive), if you consider the Curry Howard correspondence, in which MP corresponds to function application, you're basically confusing a function with an application of it. (Before you protest that the Curry Howard correspondence only deals with intuitionistic logic--that is true, but not relevant for your confusion. In order to deal with classical logic under the Curry Howard correspondence, you need to additionally consider multi-conclusion sequents like A |- B, C which are intuitionistically impermissible, but in classical logic they are interpreted as A |- B \/ C. This requires an extended lambda calculus (e.g. the lambda-mu-mu calculus) in which a "co-term" can non-deterministically output "whatever it wants" from a disjunctive sequent.)

Given that in your comments you reject "nesting issues" like those discussed in two of the above answers as being relevant to your question (although you didn't not mention this in your initial question), I suppose your whole point is what is sometimes observed in intro classes to logic, e.g. by Dona Warren:

Conditionals are important to logic because logic doesn’t give us any particular facts about the world. It doesn’t tell us whether or not a certain animal is a poodle, for example, or whether or not a certain person is a bachelor. What logic tells us is the relationships that would obtain between these facts if they were true. It tells us, for example (and very roughly), that if an animals is a poodle then that animal is a dog, and that if a person is a bachelor then that person is unmarried. We might go so far as to say that conditionals are the very heart of logic.

However, the same kind of intro-level material would also note that:

because there are many different ways to express conditionals besides the straightforward “If...then...,” we need to know how to recognize and symbolize conditionals when they’re communicated through a variety of sentences.

The point to this is that when you say that any deduction can be expressed as a conditional it is true in this sense that logic roughly deals with the notion of logical consequence.

However, the "standard objection" to your approach of expressing logic in terms of language features like conditionals is that language is more vague. E.g., van Dijk:

'From a logical point of view' one of the notorious properties of natural language is its vagueness and ambiguity. Connectives are no exception. That is, we may in `surface structure' express a certain connection, e.g. some type of implication, with a connective, e.g. and, normally used to express another connection:

(1) John was not well prepared and failed his exam.

Similarly, an "underlying" connection need not be expressed by a connective at all:

(2) Peter won't come; he is angry.

From such examples it may be concluded that natural connection should be studied at a sufficiently abstract level, viz. at the level of "deep structure" or "logical form" of sentences.

Also

not all conditionals have implicational strength. Weak conditionals have semantic properties analogous to those of natural conjunction. [E.g.]

(40) If you go to the store, please buy me some cigarettes.

So, in summary, simply saying that logical consequence can be expressed by (natural language) conditional isn't a great opener about insights into logic or logical consequence because (i) there are other ways to phrase it in natural language besides if-then and (ii) 'ifs' serve other purposes in natural language.

Although in your comments you reject mathematical logic, it's not too clear to me what you're rejecting... because any attempt to reason about logic in a less nebulous fashion than resorting to analysis of natural language sentence (all the time) requires some formalization, be it merely the fixing of the meanings of words like "if" and "and" to narrower purposes than generally permissible in natural language. And once you do that, you are just doing math/symbolic logic in disguise, if you're not explicitly using formulas.

To explain this point a bit better, it's worth going back to (40) and explain it a little more, although I won't be citing van Dijk's own account (which uses "possible worlds") but rather resort to one in the style of Rescher in speaking (more broadly) of an enthymematic basis for a conditional, i.e. one with unstated but implied (extra) premises:

Sentence (40) is not merely expressing the entailment "Assume you go to the store. Therefore I want [you to buy me] some cigarettes." But rather the sentence (40) means "I want some cigarettes. (And) if you go to the store, you can fullfill my need by buying some cigarettes." Fully "conditionalized" that would be "If I want some cigarettes and you go to the store, then you can fullfill my need [by buying some]." But there's clearly a semantic difference between the latter which is just expressing a hypothetical and (40)--the latter also expresses an actual need of the speaker. But while (40) is perfectly comprehensible, one clearly can't formalize "Assume x=1 and y=1. Therefore x+y=2." as "If x=1, then x+y=2." as the latter is not true except for specific values of y. (For those who like the "possible worlds" style of explanation, it means that's true in the world/model where y=1.) So, the conclusion of a natural language "if" sometimes carries additional hints/implicature as to what the (full) antecedent actually is. This is sometimes called a "weak conditional" (van Dijk) or more generally an "enthymematic [basis for a] conditional" (by Rescher). Before moving on, I should note that from a pragmatics perspective the main point of (40) seems to be to assert the speaker's need/want of cigarettes, rather than the more obvious fact that they can be bought at a/the store if one happens to go there.

Rescher's point (in his book Conditionals) is actually somewhat like your in that he opens (chapter 3) by saying:

Deducibility is clearly a route to conditionals. Whenever q is derivable from p by logico-conceptual means—that is, when p |- q obtains—then the claim ‘‘If p, then q’’ is clearly in order.

However after discussing that one needs to settle for a specific logical consequence (e.g. the classical one) before such an account is well defined, and after discussing some examples of enthymematic [basis for] conditionals, some more obvious like:

If that spoon is made of silver [and we don’t know this one way or the other], then it will conduct electricity

is validated by the fact that the conjunction of the antecedent (‘‘That spoon is made of silver’’) together with some suitable item of background information (‘‘Silver conducts electricity’’) makes it possible to deduce the conclusion by logic alone.

Rescher goes on to note that

Seen from [a certain] point of view, a conditional effectively summarizes an enthymematic argument. On this approach an if-then statement of the format:

if antecedent then consequent

is, in effect, the abbreviated report of a deductive argument of the form:

(antecedent + enthymematic premisses) |- consequent

[endnote:] Mackie (1973) calls this the ‘‘condensed argument account’’ of conditionals. It goes back to F. P. Ramsey’s studies of the 1920s, but only gradually found exponents later on.

[...]

Factual, speculative, and counterfactual conditionals can all rest on the same enthymematic basis. [...]

It is worth noting that our enthymematic-deductive account of conditionals does not constitute an explicatory analysis of conditionals that ‘‘reduces’’ them to nonconditional statements. After all, |- itself represents a conditional of sorts. Instead, what it accomplishes from an explanatory point of view is to ‘‘reduce’’ conditionals in general—a category that includes many obscure and problematic cases—to a particular form of conditionality that is (at least comparatively) clear and well understood.

So if this is your point that a specific kind of "clear" conditional is what "explicit deductive reasoning" is all about, you likely won't get any counter-arguments (modulo trying to thus reason about conditionals themselves, without some form of disambiguation, which you think it's a non-issue). But even then, there's the issue that you're either trying to explain/reduce a clearer notion of "explicit deductive reasoning" to a more nebulous one (conditionals in natural language in general), or that you're being idiosyncratic by assuming "conditional" refers only to the "clear" ones. The vast majority of published philosophical efforts around conditionals try to explain [natural language] conditionals in terms of some kind of logic, rather than the other way around. In fact, I can't recall a work that tries to do the opposite.

A more striking every-day example of "if" not used to express the logical entailment one might judge from form alone is given by Elder & Jaszczolt

If you’re thirsty, there’s some beer in the fridge.

(This is based on "Austin’s (1961) famous ‘biscuit conditional’"; the original was: "There are biscuits on the sideboard if you want them".) One reading here is "There's beer in the fridge. [And] If you’re thirsty, you may drink it." Unlike the (enthymematic) case of missing/incomplete premises, "biscuit conditionals" are actually missing their true conclusion from the natural language statement... and the conclusion seemingly presented is actually a premise (and it's not even a hypothetical one, but also a statement of fact). This seems to be a fairly favored way of reading such statements Elder (p. 72 in their 2019 book) and also Fogelin (1972); however Sweetser (1990) though that this reading isn't exactly satisfactory either, because the offer [of beer] isn't really contingent on the hearer being thirsty, i.e. the actual meaning is more like "There's beer in the fridge that you may drink anytime [you want]" so there's no conditional in the offer whatsoever. In this view, the only "conditionality" is for the hearer to accept of reject the offer of beer in the sense of actually drinking it or not. But Elder disagrees: "If, however, it transpires that you weren’t really [thirsty] (and that you just really wanted a [beer]), I won’t be happy: my permission for you to have a [beer] was conditional on your [thirst]." I suppose the crux of the dispute is that such conditionality like "if you want it" and perhaps even "if you're thirsty" are difficult for the speaker to assess if they really hold true of the hearer. In any case, suffice to say that while such "biscuit conditionals" have somewhat disputed interpretations, they clearly are not to be read at face value in terms their actual conclusion (and possibly their premises shouldn't be interpreted at face value either).

You again might argue that none of this is relevant to your point. (As you like to say it--"[the] problems of language are [not the] problems of logic".) But your point seems rather trivial to me: that one can express a (more) precise thing ("explicit deductive reasoning") by a more vague one--a conditional (in the natural language meaning of that term). The only way for this to be a deeper observation is [for you] to narrow what you mean by conditional.

• I looked up the Dona Warren paper you link here. Two fallacies already p14. First, a fallacious claim: "Philosophers and logicians treat all conditionals as material conditionals". Second, the fallacious argument of conditionals as promises: "Material conditionals are like promises. The conditional is false when (and only when) the promise is broken". – Speakpigeon Apr 11 at 10:26
• @Speakpigeon: well, that's an introductory course like I mentioned... I wasn't holding it up as some kind of example of depth or rigor; just pointing out that your general argument is usually made, even at that level. – Fizz Apr 11 at 10:37
• 1. "your general argument is usually made" What argument? I didn't argue it. Further, Warren does not argue that deductive reasonings are conditionals. 2. Your answer is interesting but I hope you understand you cannot derive from it anything relevant to my question. In other words, it is itself not logically valid. People arguing in support of the material implication invariably resort to fluffy arguments, such as "conditionals as promises". In other words, you infer too much from the data you exhibit and this is essentially what mathematical logic does... – Speakpigeon Apr 11 at 11:08
• "(i) there are other ways to phrase it in natural language besides if-then" This is irrelevant unless you can show that the logic of the statement cannot be expressed using a conditional. - "*(ii) 'ifs' serve other purposes in natural language" And how is that relevant to whether deductive reasoning can always be expressed by a conditional? – Speakpigeon Apr 11 at 17:10
• "Fully ''conditionalized'' that would be ''If I want some cigarettes and you go to the store''" No. The clause "I want some cigarette" is obviously not a part of the condition here. and it is in fact completely irrelevant to the logic of the conditional. If you go to the store, then you can by me some cigarettes. Why wouldn't that be enough? – Speakpigeon Apr 11 at 17:22