# What is the relation between a conditional sentence and the corresponding universal statement?

I had a technical question about conditionals. To use an example, consider the following conditional statement,

(1) If X is a man, then X is a father

I consider (1) to be false because I consider the following universal statement to be false,

(2) All men are fathers

My question then is,

a) what is the technical relationship between (1) and (2)? I don't think (2) is derived from (1) however I intuitively feel like the truth of (1) necessitates the truth of (2).

b) Does someone who asserts the truth of (1) also implicitly assert the truth of (2)?

c) Is it possible for someone to claim (1) is true while acknowledging (2) is false? (for example, by using the following reasoning, X is a unique and special man for whom manhood entails fatherhood. Wouldn't such a justification be a special pleading fallacy? The person also claims that since (1) does not specifically mention "all men", therefore (2) is not relevant to the truth of (1))

d) If indeed the above justification is special pleading, what is the best way to demonstrate that (1) is a false conditional statement to that person (who denies (1) implies the truth of (2) )

Thank you

• From a logical perspective, they're equivalent: P(x) -> Q(x). From a linguistic perspective, (1) is a compound sentence whereas (2) is a simple sentence with a generalized quantifier, that is, their logical forms are different. From a pragmatic perspective, their contextually dependent interpretation bottoms out in a defeasible implication. Jul 7, 2015 at 6:50
• "All men are fathers" must be formalized as : "for all x, if man(x), then father(x)"; thus, the only difference is that in (1) there is no leading quantifiers. If you read it as implicitly universally quantified, the two have exactly the same "logical form". Jul 7, 2015 at 8:00
• @MauroALLEGRANZA This is not true, logical forms are linguistic constructs that reflect the structure of sentences. The two sentences are structurally different. Jul 8, 2015 at 7:09
• From comments on an answer by the original poster, it seems like his(?) 1 might better be rendered as "If John is a man then John is a father." This clarification might lead to quite different answers, but I hesitate to make the change without confirmation... Jul 8, 2015 at 16:10
• @ChrisSunami I had the same feeling and I actually interpreted (1) in this sense. Jul 10, 2015 at 11:18

Let's first formalise the two sentences:

All men are fathers  –  ∀x∈M F(x)

If X is a man, then X is a father  –  M(x) → F(x)
This latter one would normally be considered to have an implicit universal quantifier over all mankind (or all objects, ...), i.e. it's actually ∀x∈H M(x) → F(x).

Here, I'm using M(x) for the predicate "x is a man" and M for the set of all men. F(x) is the predicate "x is a father". H is the set of human beings, so M ⊆ H.

## A

The technical relationship is that the two are equivalent, i.e.

x∈M F(x)  ≡  ∀x∈H M(x) → F(x)

This follows from basic set theory and the notion that ∀!x∈M M(x)  (that M(x) holds precisely for all elements in M).

## B

If he doesn't, he is not being logical.

## C

The argument you gave doesn't allow formalising the first sentence using a universal quantifier over all mankind as above. This is possible if X was mentioned before. So, it depends on the context. For example, if we were to say

Let's look at this man, let's call him X. If X is a man, ...

Then of course we cannot formalise "If X is a man" with "For every element X of the set of human beings, if X is a man ...". This is because X is a free variable (i.e., not bound by a quantifier).

If X isn't specified however, we should formalise the sentence with a universal quantifier in order to introduce X ourselves.

## D

As said above, the equivalence of the above two statements follows from basic set theory.

• This maybe weird. What if the conditional was (1) "If the Book of Mormon (BOM) has no errors, then BOM is divine". I consider it wrong because I think it is not true that (2) "a text without errors entails that it is divine". The opponent then claims BOM is special (for arbitrary reasons) and that (1) is true specifically for BOM and not for all or most texts. He also denies (2) is implicit in or relevant to the truth of (1). Intuitively, I feel there is some relation b/w the two. Is there any sensible way to approach his argument? Jul 7, 2015 at 8:23
• @Hesperus (1) and (2) in your comments are in fact different claims. What might be lost in your original question and clarified in the comment you make here is that you meant X to be a particular person rather than a stand-in variable. The claims For any x, x is P is different than (If A is X, then A is P). Jul 7, 2015 at 9:12
• @Hesperus it would be correct to say that (1) doesn't have to be true if you don't think (2) is true, however, that does not mean (1) cannot be true. (2) would imply (1), but if (2) is false that leaves all options open for (1). That's material implication.
– user2953
Jul 7, 2015 at 9:12

The relation depends on how the statements are formalized and "meant". There is however a straightforward formalization that does not require set theory, only predicate logic. The first statement is M(X) → F(X), the second one is ∀x(M(x) → F(x)).

Assuming X is meant as a particular individual the second infers the first by the universal instantiation rule. Intuitively, if something is true for all instances then it is true for any one in particular. However, even if the second sentence is false the first may still be true, one can assert (1) and deny (2). X does not have to be "unique and special man", all it takes is for him to be a father, then (1) holds because material conditional is always true if its conclusion is. In fact, X does not even have to be a father or a man, if X=Eve then "if Eve is a man then Eve is a father" is true. Because the premise is false, and material conditional is always true if its premise is false.

On the other hand, if X is meant as a variable then the first sentence infers the second by the universal generalization rule, in fact they are equivalent. Intuitively, because "each is" and "all are" are two different ways so say the same thing. So one can not assert (1) and deny (2). But if a person insists that (1) does not specifically mention "all" and talks about X being "unique and special" then they probably have the first meaning in mind.

In fact, the phrasing you quote suggests that they also have in mind something different from the material conditional M(X) → F(X), which only depends on truth values. Perhaps, the indicative conditional M(X) ⇒ F(X), which requires something like a causal link between M and F to hold. This conditional is context dependent, so the reference to X being "unique and special" is then relevant as indicating that for X specifically the required link does exist. This would then be the case not of special pleading but of hidden assumptions. It is not just being a man that compels X to be a father, but also unnamed assumptions that make him unique and special. The premise M(X) by itself does not justify the entailment, so I would say that even M(X) ⇒ F(X) is a fallacy, but this is informal, there is no formal definition of indicative conditional.

• When using quantifiers you're always using set theory. "For all x, ..." implies a set X from which x is taken.
– user2953
Jul 8, 2015 at 8:31
• Set theory is an extension of predicate, a.k.a pure quantificational, logic with a special predicate ϵ and a list of axioms involving it, usually ZFC. You may use set theory to build models of predicate logic, but it is defined independently of any models, and is much simpler than set theory. Indeed, Aristotle used "all" and "some" in his syllogistic millenia before set theory was even introduced. See plato.stanford.edu/entries/quantification/#PurQuaLog Jul 8, 2015 at 23:14
• Yes, you're right with that. But it is not "more straightforward" than set theory, like you seem to suggest. The explanation with set theory doesn't use elements from set theory that are not in predicate logic.
– user2953
Jul 8, 2015 at 23:59
• I guess it's a matter of perspective, but you do use ∈ and say "follows from basic set theory". Which presumably means you assume at least some of its axioms that specify the usage of ∈, e.g. extensionality, where just inference rules for quantifiers suffice. Jul 9, 2015 at 0:26
• You need to look at the context. The 'in' operator is used in a straightforward manner.
– user2953
Jul 9, 2015 at 7:45

We consider the two following statements:

(1) If x is a man, then x is a father

(2) All men are fathers

First question:

a) what is the technical relationship between (1) and (2)?

You said it yourself, sentence (1) is a conditional, while (2) is a universal affirmative, to use Aristotelian lingo.

I don't think (2) is derived from (1)

This would depend on what you mean by "derived". The two are logically equivalent. That is, if one is true, then the other is true. Of course, given the semantic of the vocabulary involved, both are false of the real world, and also false of all men, but if we restrict what we are talking about to fathers, or to a subset of fathers, the two become true. Clearly, they will be true or false together. They are logically equivalent.

The difference seems to be only that the conditional refers to one thing, x, while the universal refers to a whole set of things. However, if x is a man, then x belongs to the set of all men, and vice versa. This is certainly an interesting fact but the logic remains the same.

This is enshrined in mathematics by the logical equivalence of the two following mathematical expressions:

x ∈ M → x ∈ F

M ⊂ F

The implication x ∈ M → x ∈ F is the translation of the conditional "If is is a man, then x is a father", while the expression M ⊂ F is a translation of "All men are fathers".

however I intuitively feel like the truth of (1) necessitates the truth of (2).

And vice versa. (1) implies (2) and (2) implies (1). They are logically equivalent.

b) Does someone who asserts the truth of (1) also implicitly assert the truth of (2)?

If they are logical, presumably yes, but they may have a secret reason for denying the implication.

c) Is it possible for someone to claim (1) is true while acknowledging (2) is false?

Obviously, yes, if people are free to make whatever claim they want, but not without being illogical, unless they make some secret assumption about the semantic of the words involved.

(for example, by using the following reasoning, X is a unique and special man for whom manhood entails fatherhood. Wouldn't such a justification be a special pleading fallacy?

Prior assumptions may change the truth value of a statement. It is false that all men are fathers but if you assume first that "all men" refers to some group of fathers, then it becomes trivially true that all (those) men are fathers. Which is why you cannot do good logic without identifying all relevant assumptions.

The person also claims that since (1) does not specifically mention "all men", therefore (2) is not relevant to the truth of (1))

Both are true on their own merits, independently of the other, yet they are logically equivalent.

They are de facto relevant to each other because of the semantic: if x is a man, then x belongs to all men, and vice versa.

a) (2) necessitates (1) [ because (1) is false if and only if X is a man and X is not a father, but since all men are fathers, that can never be the case. ]

b) not necessarily - for ex. what if by X we're talking about parents? Then (1) is true and (2) is not true.

c) yes possibly ( see b ). It's not a special pleading fallacy, we just don't know what the domain of X is explicitly and are speaking about what the truth relationships of (1) and (2) would be under different assumptions.

d) make explicit the domain of X.

For example:

(1) For any given human, if X is a man, then X is a father

(2) All men are fathers

now the the 2 statements are equivalent and (b) = yes and (c) = no

First things first, is there no LaTex support?

S: If x is a man then, x is a father C: All men are fathers M: If x is a member of the set of men then, x is a member of the set of fathers.

S = M = C [all three statements are logically equivalent]