31.3Hitcockedited


© David Hitchcock. Informal Logic, Vol. 31, No. 3 (2011), pp. 
191-228. 

Inference Claims 
 
DAVID HITCHCOCK 
 
Department of Philosophy 
McMaster University 
Hamilton, ON  L8S 4K1 
hitchckd@mcmaster.ca 
 
 
Abstract: A conclusion follows 
from given premisses if and only if 
an acceptable counterfactual-
supporting covering generalization 
of the argument rules out, either 
definitively or with some modal 
qualification, simultaneous 
acceptability of the premisses and 
non-accepta-bility of the conclusion, 
even though it does not rule out 
acceptability of the premisses and 
does not require acceptability of the 
conclusion independently of the 
premisses. Hence the reiterative 
associated conditional of an 
argument is true if and only it has 
such a covering generalization, and a 
supposed unexpressed premiss 
supplied to make an argument 
formally valid should be a covering 
generalization. 
 

Résumé:  Une conclusion d’un 
argument s’ensuit de ses prémisses 
si et seulement si une proposition 
vérifonctionnelle acceptable dont 
l’antécédant est une conjonction de 
toutes les prémisses de l’argument et 
dont le conséquent est la conclusion 
(a) appuie une proposition 
contrafactuelle et (b) exclut, soit 
avec certitude ou  soit avec réserve 
par l’usage d’une expression 
modale, simultanémment 
l’acceptabilité des prémisses et la 
non-acceptabilité de la conclusion, 
même si elle n’exclut pas 
l’acceptabilité des prémisses et ne 
requiert pas l’acceptabilité de la 
conclusion indépendamment des 
prémisses. Donc, une prémisse 
implicite avancée pour rendre un 
argument formellement valide doit 
être une telle proposition 
vérifonctionnelle. 

 
 
Keywords: argument, associated conditional, consequence, counterfactual-
supporting generalization, covering generalization, inference, inference 
claim, material conditional, truth-preservation, truth-transmission 
 
 
 
  
1. Introduction 
 
As individuals and as communities, we increase our knowledge 
by making inferences from things we already know. 
Argumentation involves such inferences, and invites its 
addressees to accept them. The arguer implicitly claims that the 
conclusion of each constituent argument follows from the reason 
or reasons from which it is drawn.What is the general form of 



David Hitchcock 192 

such inference claims? What does it mean to say that a conclu-
sion follows from a reason or reasons? 
 
2.  Following as logically necessary truth-preservation: two 

objections 
 
Introductory logic texts nowadays identify following with ne-
cessary truth-preservation: if the reasons are true, then the con-
clusion must be true. See for example Tarski (2002/1936, pp. 
178, 183-184), Salmon (1963, p. 18), Etchemendy (1990, pp. 
81-82), Forbes (1994, p. 3), Copi and Cohen (2001, p. 43), 
Hurley (2006, p. 41), and Jeffrey (2006, p. 1). The textbooks go 
on to explain this necessity as due to a logical form of the argu-
ment: if the conclusion follows, it does so because the argument 
has a contentless form that cannot have an instance with true 
reasons and an untrue conclusion. (I use the term ‘untrue’ rather 
than the usual ‘false’ so as to accommodate the possibility of a 
conclusion that is neither true nor false.) 
This conception of following, I maintain, is in one respect too 
broad and in another respect much too narrow. 
 
2.1 Too broad 
 
It is too broad in counting a conclusion as following merely be-
cause the conclusion must be true or merely because the reasons 
cannot be true. On the contrary, I hold, there must be a connec-
tion between the reasons and the conclusion. Consider the fol-
lowing medieval example: 
 

(1) You are sitting and you are not sitting; therefore Tom 
is in the corner. 

 
Intuitively, the conclusion does not follow. The principle ex 
falso quodlibet (from a falsehood anything follows) is at best 
dubious. Similarly, consider the following parallel example: 
 

(2) Tom is in the corner; therefore, you are not both sitting 
and not sitting. 

 
Here too, it seems implausible to hold that the conclusion fol-
lows. So the principle ex quolibet verum (from anything a truth 
follows) is also dubious. Rejection of the principles ex falso 
quodlibet and ex quolibet verum requires revising standard logic, 
whether classical or intuitionist. Tennant (1979, 1980, 1984) has 
shown one way of doing so, although he modifies the relations 
of deducibility and entailment rather than the relation of 
following logically. He treats entailment as the converse of 



Inference Claims 193 

deducibility, and then puts restrictions on deducibility that 
simultaneously restrict the extension of the entailment relation. 
Thus, in example (1) above, Tennant would say that Tom is in 
the corner follows logically from the premiss you are sitting and 
you are not sitting but is not entailed by it, because it is not 
deducible from it once restrictions are imposed on what can 
count as a proof. 
 If (unlike Tennant) we identify being entailed with 
following logically, then we can capture the force of Tennant’s 
restrictions by requiring for a conclusion to follow logically that 
the argument has a form that not only cannot have an instance 
with both true reasons and an untrue conclusion, but also can 
have an instance with true reasons and can have an instance with 
an untrue conclusion. We can label the additional requirement a 
requirement of non-triviality: the ruling out of true reasons and 
an untrue conclusion must be non-trivial, in the sense that it 
must not be due merely to the ruling out of true reasons or 
merely to the ruling out of an untrue conclusion. Following 
standard usage in logic textbooks, let us call an argument in 
which the conclusion follows logically in this sense a formally 
valid argument. 
 
2.2 Too narrow 
 
The received conception of following is much too narrow in 
forbidding the form that rules out true premisses and an untrue 
conclusion to have any content. The restriction to logical or con-
tentless forms seems to be an unwarranted prejudice. If an ar-
gument has a form that rules out true reasons and an untrue con-
clusion in a non-trivial way, then why not admit that the conclu-
sion follows, even if the form has some content? 
 Consider the hackneyed argument from Socrates’ hu-
manity to his mortality: 
 

(3) Socrates is human, so Socrates is mortal. 
 
This argument is an instance of many forms. One of them is the 
form: x is human, so x is mortal. Let us suppose that, as a matter 
of physiological necessity, every human being will eventually 
die. Then the form cannot have an instance with a true premiss 
and an untrue conclusion. But it can certainly have an instance 
with a true premiss, as in the case of our example, where the 
name ‘Socrates’ replaces the variable x. And it can have an in-
stance with an untrue conclusion, as when we replace the vari-
able x with the name ‘seven’: seven is not mortal, because num-
bers are not living organisms and so are not subject to dying. 
The conclusion that Socrates is mortal thus seems to follow ne-



David Hitchcock 194 

cessarily from the premiss that Socrates is human, even though 
the necessity in question is physiological rather than formal or 
even (I would maintain) semantic. 
 The usual response in the western logical tradition to ar-
guments like that from Socrates’ humanity to his mortality is to 
hold that they have an unexpressed premiss, variously described 
as ‘unstated’, ‘hidden’, ‘tacit’, ‘suppressed’ or ‘missing’. In the 
Socrates argument of example (3), an argument analyst would 
attribute to the argument the unexpressed premiss that every 
human is mortal. With the addition of this premiss, the argu-
ment’s conclusion follows purely formally, so the received con-
ception of consequence is vindicated. But of course it is vindi-
cated only because the postulation of an unexpressed premiss 
presupposes that a conclusion that follows from an argument’s 
premisses must follow purely formally. To use the existence of 
an unexpressed premiss that every human is mortal as a reason 
for holding that the conclusion of the Socrates argument follows 
formally would be to reason in a circle, assuming what is to be 
proved. Further, it is odd to hold that an argument has a premiss 
that it does not have. Like the emperor in the fairy tale of Hans 
Christian Andersen who had no clothes, the Socrates argument 
does not in fact have as a premiss that every human is mortal. If 
we look at it carefully, with the eyes of a child uncorrupted by 
logical indoctrination, we will see that it has just one premiss, 
that Socrates is human. Further, why would a person omit a 
premiss of their argument? The usual explanation, going back to 
Aristotle (Rhetoric I.2.1357a16-21) and repeated for example by 
Quine (1972, p. 169), is that arguers omit a premiss for economy 
of expression, when the addressees can supply the premiss for 
themselves, as a matter of common knowledge. Much human 
linguistic communication is indeed elliptical, relying on context, 
both linguistic and extra-linguistic, to provide the addressee with 
what is required to understand what is said or written. But the 
difficulty with supposing that arguers routinely suppress a pre-
miss that they conceive their argument as having is that we have 
no awareness of such a supposedly suppressed premiss, even 
when we are reasoning things out for ourselves (Hitchcock 
1985). Readers can check this phenomenological fact directly by 
reflecting on inferences they make for themselves, immediately 
after making them. It will readily be discovered both that the 
inference is not formally valid and that there is no awareness of 
having omitted a premiss. 
 Both the reasoning that people use to draw their own con-
clusions and the arguments that they make to others to support 
their claims are typically not formally valid. In two collections 
of arguments selected by random sampling methods, one from 
books in a university library and the other from calls to phone-in 



Inference Claims 195 

radio and television talk shows, fewer than 10 percent of the ar-
guments were formally valid, or candidates for being formally 
valid (Hitchcock 2002, 2010a). In the remaining arguments, the 
conclusion would follow, if at all, in virtue of a form with con-
tent that ruled out true premisses and an untrue conclusion in a 
non-trivial way. The Socrates argument of example (3), though 
artificial, is typical in that respect of how we humans reason and 
argue. 
 
 
3.  First reformulation: following as necessary truth-

transmission 
 
If we adjust the received conception of following as formally 
grounded necessary truth-preservation so as to accommodate the 
two objections just mentioned, we get an alternative conception 
of following as what we might call necessary truth-
transmission, where the necessity need not be purely formally 
grounded. On this alternative conception, a conclusion follows 
from one or more premisses offered in its support if and only if 
the argument has a form that non-trivially rules out true pre-
misses and an untrue conclusion: no argument of that form can 
have true premisses and an untrue conclusion, even though an 
argument of that form can have true premisses and an argument 
of that form can have an untrue conclusion. The three conditions 
in this alternative conception can be expressed as conditions on 
a covering generalization of the argument: there is a generaliza-
tion of the argument’s associated material conditional (the truth-
functional conditional whose antecedent is the conjunction of 
the argument’s premisses and whose consequent is the argu-
ment’s conclusion) that is necessarily true, even though it can 
have an instance with a true antecedent and can have an instance 
with an untrue consequent. 
 The received conception of consequence as formally 
grounded necessary truth-preservation can be articulated in sub-
stitutional, model-theoretic or schematic ways. To articulate the 
alternative conception of consequence as necessary truth-
transmission in these ways, one permits treatment of extra-
logical constants like the name ‘Socrates’ in the same way as 
logical constants like the conjunction operator ‘and’ are treated 
and one rules out trivial consequences (Hitchcock 1998). 
 The substitutional, model-theoretic and schematic articula-
tions of the two conceptions each replace the component of ne-
cessity in the original conception with a component of univer-
sality. The necessity with which truth is preserved or transmitted 
disappears, to be replaced by the universality with which a sub-
stitution or interpretation or instantiation produces an argument 



David Hitchcock 196 

with a true conclusion when it has true premisses—against a 
background in which the universe is as it is, with no consider-
ation of ways it might otherwise be. Equivalently, the impossi-
bility of true premisses and an untrue conclusion disappears, to 
be replaced by the mere factual absence of a counter-example, 
where a counter-example is either a parallel argument with true 
premisses and an untrue conclusion or an interpretation on 
which the premisses are true and the conclusion untrue or an in-
stance of a scheme of the argument that has true premisses and 
an untrue conclusion. The modal requirement in the truth-
transmission conception that the argument can have true pre-
misses and can have an untrue conclusion is replaced by the 
factual requirement that as a matter of fact at least one substitu-
tion or interpretation or instantiation produces an argument with 
true premisses and at least one produces an argument with an 
untrue conclusion. 
 John Etchemendy (1990) objected to the replacement of 
modality by universality. He argued that the model-theoretic 
conception of logical consequence, which is generally accepted 
by contemporary logicians as the gold standard against which 
other conceptions are to be measured, is an incorrect precization 
of the modal conception, one that both under-generates and 
over-generates consequences, missing consequences that obtain 
and inventing consequences where there are none. Gila Sher 
(1996) has replied that, because all the extra-logical constants in 
an argument’s premisses and conclusion are subject to 
reinterpretation and variation of the domain is possible, the 
absence of a counter-interpretation is not just a matter of fact, 
but is a matter of logical necessity. 
 Sher’s reply is however not open to defenders of the 
model-theoretic articulation of the truth-transmission conception 
of consequence, since on that conception not all the extra-logical 
constants in the premisses and conclusion of an argument need 
be subject to interpretation (or equivalently subject to re-
interpretation if the extra-logical constants are already 
interpreted, as in arguments in a natural language). Consider for 
example the argument: 
 

(4) Napoleon ruled France; Napoleon was exiled to Elba; 
so Napoleon was short. 

 
Intuitively, the conclusion of this argument does not follow from 
its premisses: although Napoleon was in fact short, this fact does 
not follow from the biographical facts mentioned in the 
premisses. Yet the model-theoretic articulation of the truth-
transmission conception implies that it does follow, since there 
is no re-interpretation of the name ‘Napoleon’ on which the 



Inference Claims 197 

premisses are true and the conclusion untrue (because nobody 
other than Napoleon ruled France and was exiled to Elba, and 
Napoleon was in fact short), even though the “re-interpretation” 
of ‘Napoleon’ as the name of Napoleon produces an argument 
with true premisses and the re-interpretation of ‘Napoleon’ as 
the name of Giscard d’Estaing produces an argument with an 
untrue conclusion. The model-theoretic articulation of the truth-
transmission conception of consequence thus over-generates 
consequences. Furthermore, the substitutional and schematic 
articulations fare no better, since no substitution for the word 
‘Napoleon’ will produce an argument with true premisses and an 
untrue conclusion and no instance of the schema ‘person x ruled 
France; person x was exiled to Elba; so person x was short’ has 
true premisses and an untrue conclusion, even though in each 
articulation the non-triviality requirement is met. In a previous 
article (Hitchcock 1998, p. 32), I raised but did not answer the 
question how closely the model-theoretic, substitutional and 
schematic specifications of the truth-transmission conception of 
consequence fit our intuitive judgments of when a conclusion 
follows from stated premisses. The Napoleon argument in 
example (4) makes the answer clear. All three articulations are 
too loose a fit: they count conclusions as following when 
intuitively they do not follow. And the articulations are not on 
solid enough ground to over-rule our intuitive judgments about 
arguments like the Napoleon argument.  
 The crucial question for theoretical purposes is to figure 
out what is wrong with the Napoleon argument. Its premisses 
are impeccable: both true and known (independently of 
knowledge of the conclusion) to be true. Its conclusion is also 
true. There is also topical overlap, so the premisses cannot be 
stigmatized as lacking relevance in the sense in which relevance 
logicians make relevance a necessary condition for entailment. 
The premisses are evidently irrelevant to the conclusion in some 
broader sense that still needs theoretical articulation. It does not 
seem very illuminating, for example, to say that the premisses 
are not germane to the conclusion and have no bearing on it 
(Johnson and Blair 1993, p. 324) or that the truth of the 
premisses provides no basis for supposing that the conclusion is 
true, or indeed for supposing that it is false (p. 55). In the 
context of a conception of following as broader than following 
logically, we need to know in virtue of what feature of the 
Napoleon argument its premisses are non-germane to its 
conclusion, have no bearing on it, and provide no basis if they 
are true for supposing that the conclusion is true. 
 Unfortunately, the account of premissary relevance in 
(Hitchcock 1992) is not much help. According to that account, a 
premiss is irrelevant to a conclusion for which it is offered as 



David Hitchcock 198 

support if it cannot ineliminably be put together with other at 
least potentially accurate information to provide a set of 
premisses that is sufficient to justify the conclusion (p. 260). But 
the premisses of the Napoleon argument can be put together 
ineliminably with the argument’s associated material conditional 
to produce a formally valid argument. And the associated 
material conditional is not just potentially true; it is actually true 
and known to be true: a quick check of reputable sources will 
tell us that the associated material conditional has a true 
consequent (Napoleon was short), and hence is true. If the 
account of relevance in (Hitchcock 1992) is to be rescued, we 
need some account of why this formally valid expanded 
argument with premisses known to be true is not sufficient to 
justify its conclusion. On reflection, the problem appears to be 
that the added premiss (the material conditional associated with 
the original Napoleon argument) cannot be known to be true 
independently of knowing that the conclusion is true, so that the 
argument cannot produce knowledge of the truth of its 
conclusion. In view of this problem, the account of relevance in 
(Hitchcock 1992) needs to be modified by requiring for 
relevance that one can discover that the other potentially 
accurate information is actually correct without assuming the 
truth of the argument’s conclusion. On the modified account, 
then, a premiss is relevant to a conclusion for which it is offered 
as support if and only if there is a set of premisses that (a) when 
combined with the relevant premiss are sufficient to justify the 
conclusion, (b) are not jointly sufficient by themselves to justify 
the conclusion, (c) are at least potentially accurate, and (d) if 
accurate can be discovered to be accurate without assuming the 
truth of the conclusion. 
 The question is how to modify or replace the 
substitutional, model-theoretic and schematic articulations of the 
truth-transmission account of consequence so as to capture these 
partly epistemological constraints on relevance. On the surface, 
the Napoleon argument seems to meet the conditions under 
which a conclusion follows from given premisses. It has a 
covering generalization that is not only true, but known to be 
true: Every ruler of France who was exiled to Elba was short. 
Further, this covering generalization is non-trivially true: there is 
at least one ruler of France who was exiled to Elba (namely, 
Napoleon) and there is at least one person who was not short—
for example, Giscard d’Estaing. But the non-trivial truth of this 
covering generalization does not license those who know of it to 
draw the conclusion from the premisses. Why not? One salient 
fact about the argument, already mentioned, is that, if you did 
not already know that Napoleon was short, the argument would 
give you no reason to believe that he was. This fact corresponds 



Inference Claims 199 

to the fact that our only basis for knowing that the argument’s 
associated material conditional is true is that we already know 
that the conclusion is true. A fortiori, the only way to establish 
the truth of its generalization is to show for each instance where 
the antecedent is true that in this case the consequent is also true. 
One way to test this supposition is to consider a hypothetical 
case, e.g. another ruler of France who was not short, such as 
Jacques Chirac. If Chirac were exiled to Elba, he would not 
thereby become short. Nor is there any reason for restricting ex-
rulers of France eligible for exile to Elba to those who were 
short. It seems then that a crucial requirement for a conclusion 
to follow from given premisses is that it has a covering 
generalization that is not only non-trivially true, but also can be 
known to be true independently of knowledge of truth of the 
conclusion. And it appears that we know that an argument has a 
true covering generalization independently of knowledge of the 
truth of the argument’s conclusion if and only if we know that 
the covering generalization holds not just for actual cases that 
satisfy its antecedent but also for hypothetical cases that might 
satisfy it. 
 Consider another example: 
 

(5) Abraham Lincoln was a president of the United States. 
So Abraham Lincoln was a man. 

 
As a matter of fact, every previous president of the United States 
has been a man. But if we require an inference-licensing 
covering generalization to hold for hypothetical cases that might 
satisfy its antecedent, the fact that every previous president of 
the United States has been a man does not license us to infer 
from Abraham Lincoln’s presidency that he was a man. For 
there is no rule against a woman being elected president of the 
United States. And indeed, counterfactually, if Walter Mondale 
had defeated Ronald Reagan in the 1984 US presidential 
election, and had then died in office, his running mate Geraldine 
Ferraro would not have become a man simply by succeeding 
him in the office of US president. The generalization that all 
previous US presidents have been men does not hold for all 
possible previous US presidents, even though it holds for the 
actual ones. This limitation corresponds, it seems, to the fact that 
the only way we can determine that every previous president of 
the United States has been a man is by discovering, directly or 
indirectly, about each of the 39 previous presidents that he was a 
man. The generalization cannot license an inference to the 
maleness of a particular previous president because our 
knowledge of its truth rests on already knowing that the 
particular previous president was a man. 



David Hitchcock 200 

 It is tempting to identify the requirement that an 
argument’s true covering generalization supports counterfactual 
instances with a requirement that the argument has a true law-
like covering generalization. The covering generalization of the 
Socrates argument of example (3), that every human is mortal, is 
law-like if it is true. And such philosophers of science as Ernest 
Nagel have held (1961, pp. 71-72) that a singular counterfactual 
conditional holds if and only if the indicative form of its 
consequent follows logically from the indicative form of its 
antecedent in combination with a law and the requisite initial 
conditions for the law. The law would thus be a generalization 
of the indicative counterpart to the singular counterfactual 
conditional, with its antecedent supplemented by a statement of 
initial conditions. But requiring a law-like covering 
generalization may be too demanding. Consider the argument: 
 

(6) Obama lives in the White House, so he lives in 
Washington. 

 
Intuitively, the conclusion of this argument follows from the 
premiss: from the fact that someone lives in the White House, 
we are entitled to conclude that this person lives in Washington. 
The reason, of course, is that the White House is located in 
Washington. However, neither this fact nor the generalization 
based on it is law-like. The White House, i.e. the residence of 
the president of the United States, could have been built 
somewhere else. Or the political boundaries might be different, 
with Washington and the District of Columbia divided up 
among the surrounding states, with each of them giving their 
portion of present-day Washington its own name. Nevertheless, 
given that at present the White House is in Washington, the 
generalization that whoever lives in the White House lives in 
Washington supports counterfactuals. If John McCain lived in 
the White House, he would live in Washington. If Vladimir 
Putin lived in the (US) White House, he would live in 
Washington. And so on. It should be noted in making these 
counterfactual judgments that the generalization is being taken 
to hold only for the present, conceived as a short indefinitely 
bounded stretch of time surrounding the time of its inscription; it 
could not license for example an inference from the claim that 
the 50th president of the United States will live in the White 
House to the conclusion that he or she will live at that time in 
Washington, DC, since either the location of the president’s 
residence or the boundaries of Washington, DC might have 
changed between now and then in such a way as to falsify the 
generalization that whoever lives in the White House lives in 
Washington, DC. The time-boundedness of the covering 



Inference Claims 201 

generalization of the Obama argument immediately shows that it 
is not a natural law. Nevertheless, it supports counterfactual 
instances. Hence, since one could not know that counterfactual 
instances of a universal generalization were true if one could 
discover the truth of the generalization only by discovering the 
truth separately of each of its instances, the covering 
generalization of the Obama argument is known to be true 
independently of knowledge of the truth of the conclusion that 
Obama lives in Washington.  
 If one insists that any covering generalization that licenses 
an inference must be law-like, then one would be obliged to 
treat the Obama argument as having an unexpressed premiss, 
assumed to be mutually believed by both arguer and intended 
audience, that the White House is located in Washington, DC. 
With the addition of this premiss to the stated premiss, the 
conclusion would follow in virtue of the law-like generalization 
that whoever lives in a building that is located in a certain 
municipality lives in that municipality. But there seems to be no 
theoretical advantage to this approach over the approach of 
treating non-law-like covering generalizations that support 
counterfactual instances as licensing inferences, even if the 
license is temporally constrained. And, practically speaking, it is 
easier to ask directly whether an argument has a true covering 
generalization that supports counterfactual instances than to ask 
whether it can be supplemented with additional correct 
information in such a way that the expanded argument has a true 
law-like covering generalization. With this latter approach, for 
example, one must make sure that the conclusion does not 
follow from the additional correct information independently of 
the argument’s stated premisses. Further, treating the conclusion 
of the Obama argument in example 6 as following just from the 
stated premiss corresponds better to the phenomenological fact 
that one would not use any other premiss than the stated premiss 
in reasoning for oneself from it to the stated conclusion. 
 
 
4.  Second reformulation: following as counterfactual-

supporting truth-transmission 
 
Thus it appears that, in the case of non-logical consequence, we 
cannot capture the consequence relation through a merely 
universal condition, whether expressed substitutionally, model-
theoretically or schematically. But, contrary to the position 
adopted in (Hitchcock 2009), the non-logical consequence 
relation need not hold as a matter of nomic necessity. It is 
enough if the universal condition holds counterfactually, of any 
individual supposed to satisfy its antecedent. Necessarily true 



David Hitchcock 202 

universal covering generalizations do hold counterfactually, and 
as we have seen the necessity of the generalization need not be 
semantic but might be for example physiological or 
constitutional. But, as the Obama argument of example (6) 
shows, some universal covering generalizations hold 
counterfactually even if they are not nomically necessary. 
 Substitutional and model-theoretic articulations of the 
consequence relation cannot be modified to express the 
requirement that a universal condition holds counterfactually. 
But schematic articulations can. We can revise the schematic 
articulation of the truth-transmission conception of consequence 
to read as follows: A conclusion is a consequence of given 
premisses if and only if the argument is an instance of an argu-
ment scheme, which may or may not be purely formal, that has 
no actual or counterfactual instances with true premisses and an 
untrue conclusion, even though it has an instance with true pre-
misses and an instance with an untrue conclusion. 
 The counterfactual-supporting version of the schematic 
articulation of the truth-transmission conception of consequence 
automatically addresses a limitation of standard substitutional 
and schematic articulations—namely, the limitation that an ar-
gument may lack a counter-example because of the limitations 
of the language in which it is expressed. In opening up a space 
for instances that are hypothetical rather than actual, we allow 
that the language may have no name for such merely hypotheti-
cal instances. We can accommodate such presently nameless 
possible instances by allowing the addition of new names to the 
language in which the argument is expressed. 
 Counterfactual-supporting truth-transmission in virtue of 
an argument scheme depends on the truth of a covering gener-
alization of the argument that will support counterfactual instan-
ces. The generalization might be physically contingent, like the 
generalization that whoever lives in the White House lives in 
Washington. But it must support counterfactual instances. A true 
contingent generalization that does not support counterfactuals 
will not license an inference from its antecedent to its conse-
quent. Thus the fact that every president of the United States has 
been a man does not license an inference from Abraham Lin-
coln’s having been president of the United States to his having 
been a man, for the generalization about the sex of American 
presidents has false counterfactual instances: if Walter Mondale 
had been elected U.S. president in 1984 and had died in office, 
to be succeeded by his running mate Geraldine Ferraro, she 
would not have been a man.  
 It would make things theoretically neat if in general know-
ledge of the truth of true universal generalizations that do not 
support counterfactual instances had to rest ultimately on know-



Inference Claims 203 

ledge of the actual instances that fall under them. But consider 
the universal generalization that all spheres of gold are less than 
one mile in diameter, constrasted to the law-like generalization 
that all spheres of uranium are less than one mile in diameter 
(Carroll 2011). Here our belief in the universal generalization 
does not rest on knowledge of the diameter of all the actual 
spheres of gold, past, present and future, but on knowledge of 
facts about the cost, scarcity and uselessness (for all but orna-
mental and fetishistic purposes) of gold. Thus an argument like: 
 

(7) This sphere is pure gold, so its diameter is less than a 
mile. 

 
though its conclusion does not follow just from its stated 
premiss, could be rescued by attributing to it an unexpressed 
premiss that every sphere of gold has a diameter less than a 
mile. 
 The requirement that a generalization licensing an infer-
ence support counterfactual instances implies an asymmetry in 
the treatment of real-life arguments. If someone were to argue 
that Abraham Lincoln was a man, because he was president of 
the United States, one could as far as I can see get the conclu-
sion to follow from the premiss only by attributing to the argu-
ment an unstated premiss that all previous presidents of the 
United States were men. But, in contrast to the gold argument in 
example (7), this maneuver would not make the argument re-
spectable, since any epistemic justification for the added premiss 
would have to appeal ultimately to the information in the con-
clusion. 
 The theoretical asymmetry in the treatment of formally 
invalid arguments is however not as extensive as one might im-
agine. Occasionally people advance arguments whose conclu-
sion follows, if at all, in virtue of a merely contingent generali-
zation. But, rather surprisingly, it seems that these merely con-
tingent generalizations support counterfactuals. Consider the 
following remarks by a caller to a radio phone-in show soon af-
ter the invasion of Iraq in March 2003, at a time when there 
were reports of Iraqi families being told they will be murdered if 
they don’t fight: 
 

(8) I think you will make a very poor soldier if you put a 
gun on his head or on his family and say, “Go and 
fight”. We have to acknowledge that the Iraqis are 
fighting an aggression whether rightly or wrongly. 
They think that they are going to be occupied. And 
even the US army generals are acknowledging that 
they are having a stiff resistance. You do not get stiff 



David Hitchcock 204 

resistance from soldiers who are under duress. 
(Hitchcock 2010a, p. 41.) 

 
The context makes clear that the caller is arguing that the Iraqi 
soldiers are not fighting under duress, a conclusion that follows 
almost logically from his last two statements. The preceding 
statements appear to give two additional independent arguments 
for the same conclusion. If we take the first statement as the 
premiss of one of those arguments, and supply the implicit 
conclusion, we get the following argument: 
 

(9) …you will make a very poor soldier if you put a gun on 
his head or on his family and say, “Go and fight”; so 
the Iraqi soldiers are not fighting under duress. 

 
If we acknowledge the semantic connection between fighting 
under duress and being told to go and fight with a gun to one’s 
head or on one’s family, we can see that this argument is an in-
stance of the schema ‘People who are F are very poor soldiers, 
so the Iraqi soldiers are not F’. The universally generalized ma-
terial conditional associated with this schema turns out to be 
logically equivalent to the statement that the Iraqi soldiers are 
not very poor soldiers; for the proof, see the appendix. This 
statement is what a skilled argument analyst would intuitively 
attribute to the caller as an assumption involved in using his 
general point about very poor soldiers to support a claim about 
the Iraqi soldiers in particular. But, even though the covering 
generalization is not law-like, it does support counterfactuals. If 
the Iraqi soldiers are not very poor soldiers, then, if short people 
were very poor soldiers, then the Iraqi soldiers would not be 
short. And so on. 
 Thus, contrary to the approach recommended in (Freeman 
2011a, pp. 186-189), the fact that an argument lacks a true 
nomically necessary covering generalization does not require 
adoption of the unexpressed premiss approach in order to under-
stand and then evaluate the argument. A true nomically contin-
gent covering generalization will license the inference in the 
stated argument if it supports counterfactual instances. Even 
singular contingent statements can license inferences, as in the 
Iraqi soldiers argument of example (9). In particular, there is no 
need to revert to an unexpressed premiss in order to understand 
and evaluate the following artificial argument discussed by 
Freeman (2011a, p. 183): 
 

(10) All humans are mortal, so Socrates is mortal. 
 



Inference Claims 205 

The argument is an instance of the schema: All humans are F, so 
Socrates is F. The corresponding universal covering generaliza-
tion is that Socrates has every property that all humans have. By 
parallel reasoning to that used for the Iraqi soldiers argument of 
example (9), this generalization is logically equivalent to the 
contingent singular statement that Socrates is human. Assuming 
that this statement is true, it will license the inference in the Soc-
rates argument of example (10), provided that the universal 
generalization to which it is logically equivalent supports 
counterfactual instances. And in this case it does. If all humans 
had green skin, Socrates would have green skin. If all humans 
had blue eyes, Socrates would have blue eyes. And so on. (The 
Socrates argument of example (10) is of course artificial; it is 
hard to imagine someone putting it forward seriously in an at-
tempt to establish its conclusion. But one can readily imagine a 
parallel argument with the name ‘Jesus of Nazareth’ replacing 
‘Socrates’ being seriously advanced by a critic of a certain posi-
tion in Christian theology.) 
 As far as I can see, people do not reason and argue in ac-
cordance with argument schemes whose corresponding univer-
sal generalization does not support counterfactual instances. To 
put the point another way, for all but one of the arguments I 
have collected in two rounds of random sampling of arguments 
(Hitchcock 2002, 2010a), resulting in more than 100 inferences 
for evaluation, I have been able to construct a covering generali-
zation that, if true or otherwise acceptable, would apparently 
hold for counterfactual instances. To bolster this claim, it would 
be necessary to get independent and unbiased judgments from at 
least two people as to whether a given universal generalization if 
true would support counterfactual instances. Such judgments 
may require real-world knowledge, as in the case of a sphere of 
gold versus a sphere of uranium. But, despite ongoing disputes 
about the truth conditions for counterfactual singular condition-
als (Lewis 1973, Pearl 2000, Arregui 2009), there seems little 
disagreement about the truth value of counterfactual instances of 
a true universal generalization in a world otherwise much like 
ours or with the same invariant causal relationships as ours. For 
example, the reader should readily agree that Jacques Chirac 
would not be short if he not only was a former ruler of France 
but also had been exiled to Elba; that Socrates would have had 
green skin if all humans had green skin; that a sphere of gold 
would not have a diameter less than a mile if it had a circumfer-
ence of four miles; and so on for the other examples in this arti-
cle.  
  
 



David Hitchcock 206 

5. Elaboration and extensions of counterfactual-supporting 
truth-transmission 

 
On the counterfactual-supporting truth-transmission account, 
then, a conclusion follows from given premisses if and only if 
the argument is an instance of an argument scheme, which may 
or may not be purely formal, that has no actual or possible in-
stances with true premisses and an untrue conclusion, even 
though it has an instance with true premisses and an instance 
with an untrue conclusion. 
 This account, it turns out, needs elaboration and extension, 
in four respects. 
 
5.1. Restrictions on the range of the variables in an argument’s 
schema: First, in testing to see whether an argument has a coun-
terfactual-supporting covering generalization that is non-trivially 
true, we may need to restrict the range of the variables in a gen-
eralization under consideration. Such restrictions are implicit in 
the examples of covering generalizations already given, where 
the range of variables is restricted to persons, countries, places 
and so on. To take another example: if someone argues that 
marijuana should be legalized on the dual ground that it is less 
harmful than alcohol and that alcohol is justifiably legal, it is 
reasonable to restrict the variables in the covering generalization 
to psychotropic drugs (Hitchcock 1985). Such restrictions pre-
suppose the background information that the name or other term 
over which one generalizes falls within the specified range: Na-
poleon is a person, France is a country, marijuana and alcohol 
are psychotropic drugs, and so forth. Background information of 
this sort can be treated as an unstated premiss if one wishes, or 
more accurately as an unstated presupposition: in a debate about 
legalizing marijuana, one does not need to state that marijuana is 
a mind-altering drug. 
 
5.2. Generalization from truth to acceptability: Second, some 
may balk at assigning truth-values to deontic generalizations. To 
accommodate such sensitivities, we can replace the word ‘true’ 
in the articulation of the truth-transmission conception of conse-
quence with a word like ‘acceptable’ or ‘justifiable’ (in their 
normative senses), treating truth as a property that confers ac-
ceptability or justifiability in the intended sense. 
 
5.3. Allowance for conclusions that are not assertives: Third, 
allowance needs to be made for conclusions that are not asser-
tives. One can argue for any of the kinds of speech acts distin-
guished by Searle (1979), including commissives, directives, 
expressives, declaratives and suppositives (Hitchcock 2006). For 



Inference Claims 207 

example, the following exchange on a radio phone-in show 
about the epidemic of Sudden Acute Respiratory Syndrome 
(SARS)1 in Toronto in the spring of 2003 concludes with an ar-
gument for a question: 
 

(11) Caller (Gina from Toronto): ―Hi. I wanted to speak 
about the SARS. 

Roy (program host): ―Yes. Go ahead, please. 
Gina: ―OK. On the weekend they had... were looking for 

court orders for two people who had not obeyed the 
quarantine. They now have a student who cut short 
her quarantine after ten days to write an exam. She’s 
now ill, and they have now 25 students and a teacher, 
I believe. So, considering how the government mis-
handled Walkerton2 and the East Nile virus, where... 
it‘s worked out OK with SARS, I realize they‘re not 
saying an epidemic, but last week before they really, 
really knew and the TTC [Toronto Transit Commis-
sion–DH] driver wanted to wear a mask, where was 
their union? They said their hands were tied, that they 
can‘t do anything when the Board of Health says this, 
but why do people so blindly believe government of-
ficials? 

Roy: ―Well, you know, I think when it comes to an issue 
like SARS, you do look to official explanations and 
official sources...  

Gina: ―Yes. Just like Walkerton and the East Nile virus, 
which they mishandled 

Roy: ―West Nile. 
Gina: ―West Nile virus,3 I‘m sorry. Which they 

mishandled. So why do you want to believe them 
now? (Hitchcock 2010a, p. 55) 

 

                                                 
1 SARS was a new, highly contagious respiratory viral infection that turned 
out to have a fatality rate of about 15%. Public health authorities reacted by 
imposing a quarantine on anybody suspected of being at risk for contracting 
the disease. The disease was eventually eradicated as the result of an aggres-
sive world-wide campaign to stamp it out. 
2 Walkerton is a small town in Ontario whose water supply became 
contaminated in May 2000. Almost half the population became ill, and seven 
people died. A study put the cost of dealing with the tragedy at $64.5 million. 
(http://www.cbc.ca/news/background/walkerton/; accessed 2011 03 14)  
3 In the summer of 2002, 19 people in the southern part of Oakville, Ontario 
were hospitalized for diseases caused by the West Nile virus, which is 
transmitted to humans by mosquito bites. 
(http://www.health.gov.on.ca/english/public/pub/ministry_reports/wnv_rep_2
003/wnv_rep03.html; accessed 2011 03 14). 
 



David Hitchcock 208 

Gina’s concluding sub-argument, supporting her question, “Why 
do people so blindly believe government officials?” runs as 
follows: 
 

(12) Premiss: The government mishandled Walkerton and 
the West Nile virus. 

Intermediate conclusion: Why do people so blindly believe 
government officials <on this public health crisis–
DH>? 

 
We can construct a covering generalization in the usual way, 
recognizing the shared background information that Walkerton 
and the West Nile virus were public health crises: If a 
government mishandled two previous public health crises, why 
do people so blindly believe what officials of the government say 
on another public health crisis? In evaluating whether this covering 
generalization is acceptable, we need to judge the force of the 
‘why’ question. Taken literally, it is a request for an explanation of 
people’s blind belief in the statements of government officials. 
What motivates such a request is the presupposition that such blind 
belief makes no sense. The acceptability of the question is thus a 
function of the prima facie foolishness of blindly believing 
statements about a public health crisis by officials of a government 
that recently mishandled other public health crises. 
 As with this example, one can develop criteria for the 
acceptability of speech acts of all types, criteria that can be used in 
judging whether a commissive or directive or expressive or 
declarative follows from the reasons given in its support. 
 
5.4. Allowance for rebuttable inferences: Fourth, allowance 
needs to be made for rebuttable inferences, where the conclusion 
does not follow definitely from the premisses but is merely 
made probable or possible or presumptively acceptable by them. 
Such inferences are rebuttable in the sense that further 
information compatible with the premisses can make the 
conclusion false or otherwise unacceptable. A conclusion that 
follows definitely from the premisses, on the other hand, is not 
rebuttable in this sense; if it is unacceptable, there must be 
something wrong with at least one premiss. It is however 
underminable, given the non-triviality requirement for 
transmission of acceptability. The information that Arthur is 
either a farmer or a grocer but is not a farmer necessitates as a 
consequence that Arthur is a grocer, but additional information 
that Arthur is a farmer undermines the inference. It does not 
follow necessarily from the premiss set {Arthur is a farmer, 
Arthur is not a farmer, Arthur is either a farmer or a grocer} that 
Arthur is a grocer. 



Inference Claims 209 

To signal a rebuttable inference, arguers sometimes use 
modal auxiliaries or adverbs, as in the following argument from 
a caller to a phone-in radio show: 
 

(13) … when someone is purchasing a puppy, many times 
they are going to a breeder that‘s just in it for profit, so 
the puppy‘s already starting out with maybe not a good 
head start. (Hitchcock 2010a, p. 18) 

 
The caller acknowledges that not every puppy bought from a 
breeder just in it for profit lacks a good head start. The qualifier 
‘maybe’ reduces the strength of the inference claim, which thus 
needs for its support only a counterfactual-supporting existential 
generalization that some animals bought from breeders just in it for 
profit do not get a good head start. 
 With these four complications, the truth-transmission 
account, which should now be called an acceptability-transmission 
account, appears capable of handling all inferences. We can sum it 
up as follows. A conclusion follows from given premisses if and 
only if an acceptable counterfactual-supporting generalization rules 
out, either definitively or with some modal qualification, 
simultaneous acceptability of the premisses and non-acceptability 
of the conclusion, even though it does not rule out acceptability of 
the premisses and does not require acceptability of the conclusion 
independently of the premisses. (The account is framed in terms of 
the non-acceptability of the conclusion rather than its 
unacceptability, to allow for the possibility that a conclusion is 
neither acceptable nor unacceptable, because it is neither worthy or 
acceptance nor worthy of rejection.) An inference claim is thus the 
claim that a counterfactual-supporting covering generalization is 
non-trivially acceptable. 
 This consequence relation has structural properties that 
deserve investigation. For example, it is transitive only under 
certain conditions, and obeys the cut rule only under certain 
conditions. Hence it implies restrictions on the chaining of 
arguments together to support a final conclusion by means of 
one or more intermediate conclusions drawn along the way. The 
present paper prescinds from investigation of these restrictions, 
because of limitations of space and time. In a recent 
investigation of rebuttable inference, Verheij (2010) shows that 
a non-monotonic consequence relation in “reason-based 
argumentation” has seven properties, which he calls logical 
equivalence, restricted reflexivity, antecedence, right 
weakening, conjunctive cautious monotony, mutual attack and 
conjunctive cumulative transitivity. It needs to be investigated 
which of these properties belong to consequence conceived as 
acceptability-transmission licensed by a non-trivially acceptable 
counterfactual-supporting covering generalization. 



David Hitchcock 210 

 If one insists that a conclusion follows from given premisses 
only if it follows formally from them, then one can treat the 
inference claim of an argument that is not formally valid as the 
claim that the argument has as an unstated premiss a 
counterfactual-supporting covering generalization that is non-
trivially acceptable. Hence, if one thinks that formally invalid 
arguments have a gap between premisses and conclusion that needs 
to be filled by a gap-filling implicit assumption (Ennis 1982), then 
one will supply as the gap-filler the most non-trivially plausible 
counterfactual-supporting covering generalization of the stated 
argument. 
 
 
6. A sceptical rejoinder 
 
Sceptics can counter that this account of inference claims is 
unnecessarily complicated, that the inference claim of an 
argument is just its so-called “associated conditional” 
(Hitchcock 1985), i.e. the singular conditional whose antecedent 
is the conjunction of the premisses and whose consequent is the 
conclusion. Doesn’t the connective ‘if’ simply mean that the 
consequent follows from the antecedent, as Stoic logicians long 
ago maintained (Diogenes Laertius 7.71)? And don’t theorists of 
argument reconstruction object that supplying an argument’s 
associated conditional as its unstated gap-filling premiss is 
merely reiterative, not really informing us of the substantive 
assumption used or needed to infer the conclusion? 
 Let us grant that in at least one of its uses a singular 
indicative conditional sentence means that its consequent 
follows from the antecedent. In fact, in the ancient dispute about 
the truth-conditions for singular indicative conditionals, it was 
common ground that a conditional is true when its consequent 
follows from its antecedent (Sextus Empiricus 2.112; cf. Barnes 
2007, pp. 125-126). The dispute was about what it took for this 
condition of following to be met. The earliest proposal was that 
of the logician Diodorus Cronus, who proposed that a true 
conditional was one that neither could nor can begin from a truth 
and end in a falsity (Sextus Empiricus 2.115). This modal 
conception would imply that a conclusion follows from given 
premisses if and only if it was and is impossible for the 
premisses to be true and the conclusion false. The present 
conception of the consequence relation is similar, with the 
additional requirement that the impossibility be non-trivial (i.e. 
not due to the impossibility that the premisses are true and not 
due to the impossibility that the conclusion is false), the 
clarification that the impossibility need not be logical or 
semantic (and indeed can be established by any counterfactual-



Inference Claims 211 

supporting covering generalization), allowance for the ruling out 
of true premisses and a false conclusion to hold only for a 
specious present rather than omni-temporally, the expansion of 
the property of truth to other kinds of acceptability, and the 
recognition of modally qualified rebuttable consequences. I 
propose then, following the ancient tradition, to take the present 
account of the consequence relation to be also an account of the 
truth conditions of a singular indicative conditional, in at least 
one of its senses. In this sense, a singular indicative conditional 
is definitively acceptable if and only if it has a counterfactual-
supporting generalization that is non-trivially acceptable. It is 
acceptable in some qualified way if and only if it has a 
counterfactual-supporting generalization that is non-trivially 
acceptable in the same qualified way. 
 Of course, some philosophers of language, starting with 
Diodorus’ own pupil Philo of Megara (Sextus Empiricus 2.113-
114) and continuing as recently as in the work of Paul Grice 
(1989, pp. 58-85), maintain that the singular indicative 
conditional is true if and only if it does not have a true 
antecedent and a false consequent. But, although Philo thought 
that the consequent of a conditional with such truth conditions 
follows from its antecedent, contemporary logicians and 
philosophers of language generally reject this claim. They take 
the medieval conception of a consequentia materialis as a 
consequence that holds unless the premisses are true and the 
conclusion false (Kneale and Kneale 1962, pp. 274-297) to 
reflect confusion between the truth of a conditional and the 
validity of an argument from its antecedent to its consequent. 
Likewise, they regard as unfortunate the use by Whitehead and 
Russell in Principia Mathematica (1910-1913) of the phrase 
‘material implication’ for the Philonian truth-functional 
conditional, since they generally follow C. I. Lewis (1912) in 
regarding the truth-functional conditional as not capturing the 
sense in standard English of the word ‘implies’, which signifies 
the converse of the relation of following. The problem with 
construing the Philonian conditional as expressing the 
implication of its consequent by its antecedent, it is generally 
thought, is the so-called “paradoxes of material implication”: a 
falsehood would imply anything and anything would imply a 
truth. 
 
 
7. Anti-generalist alternatives 
 
Some contemporary theorists of argumentation have, however, 
defended the view that the inference claim of an argument is its 
associated singular conditional, taken by some of them to be 



David Hitchcock 212 

truth-functional (i.e. Philonian or “material”) and by others to be 
sui generis. Others have argued, within the unexpressed premiss 
approach, that either always or sometimes the unexpressed 
premiss of an argument that is not formally valid is just the 
argument’s associated singular conditional, whether truth-
functional or sui generis. I shall consider each of the four views. 
 
8.1 Bermejo Luque: Lilian Bermejo Luque (2006, 2011a) 
interprets the inference claim made in the complex speech act of 
arguing as the claim that the argument’s associated material 
conditional is true. Her interpretation seems incorrect on its face, 
for the mere absence of the circumstance that the argument’s 
reasons are true and its conclusion untrue seems weaker than the 
circumstance that the conclusion follows from those reasons. 
Suppose, for example, that someone were to argue: 
 

(14) 8 is divisible by 2, so 8 is divisible by 4. 
 
This is clearly a bad argument, even though it has a true premiss 
and a true conclusion. You cannot legitimately infer from the 
fact that 8 is divisible by 2 that 8 is also divisible by 4, even 
though as a matter of fact it is true that 8 is divisible by 4. Thus 
the inference claim in the divisibility argument of example (14) 
is false. But the associated material conditional is true, because 
it has a true consequent. 
 In this counter-example, the conclusion is known to be 
true independently of the argument put forward in its support. 
Bermejo-Luque tries to ward off this sort of counter-example by 
remarking that an argument’s associated material conditional 
 

is to be valued under the argumentative conditions in 
which it arises, namely, that the reason alleged in the 
argument (whose content is the antecedent of the 
conditional) is supposed to be true or highly acceptable, 
and also that we have not already independently 
determined the real value of the claim for which we argue 
(whose content is the consequent of the conditional). 
These conditions suffice to free us from the paradoxes of 
material implication... (Bermejo-Luque 2006, p. 79; cf. her 
2011a, p. 79.) 

 
So we are to construe the supposed counter-example as an 
argument presented in a situation where it is supposed to be true 
that 8 is divisible by 2, but we have not already determined 
independently whether 8 is divisible by 4. To make such a 
situation plausible, let’s vary the example slightly to one where 
the arithmetical premiss has been determined to be true, the 



Inference Claims 213 

arithmetical conclusion not yet determined to be true, and the 
inference is apparently incorrect. An example might be the 
following argument: 
 

(15) 79,974 is divisible by 3, so 79,974 is divisible by 9. 
 
According to Bermejo-Luque, the fact that the premiss of this 
argument is supposed to be true (and can in fact easily be 
checked to be true) but that we have not yet determined whether 
the conclusion is true (because we have not yet tried to divide 
79,974 by 9) frees us from the paradoxes of material implication 
when we come to appraise the argument’s associated material 
conditional. To avoid confusion with other senses of the 
conditional, let us consider the situation in terms of the logically 
equivalent negajunction ‘Not both 79,974 is divisible by 3 and 
79,974 is not divisible by 9.’ Given that we know that the first of 
the two conjuncts in this negated conjunction is true, the easiest 
and most direct way to determine whether the negajunction is 
true is to check whether the second conjunct is also true. If the 
second conjunct is also true, then the whole conjunction is true, 
the associated negajunction is false and the conclusion (on 
Bermejo Luque’s account) does not follow. If the second 
conjunct is false, then the whole conjunction is false, the 
associated negajunction is true and the conclusion (on Bermejo 
Luque’s account) does follow. The fact that we have not already 
determined the value of our conclusion does not bar us, when it 
comes to appraising the argument’s associated material 
conditional, from making such a determination independently of 
the argument offered in its support. If we have a way of making 
such an independent determination, and that determination 
yields the result that the conclusion is true, we will rightly 
conclude that the associated negajunction is true. Thus, on 
Bermejo-Luque’s account, every act of arguing for a conclusion 
whose propositional content can be determined to be true 
independently of the propositional content of its premisses 
makes a true inference-claim: its conclusion does in fact follow 
from the premisses offered in its support. 
 This is a clearly unacceptable consequence, as we can 
readily see by constructing simple examples of arguments with 
an obviously true conclusion and an obviously irrelevant 
premiss. Consider: 
 

(16) Snow is white, so grass is green. 
 
This is not a good argument, even though both its premiss and 
its associated negajunction are both known to be true. The 
conclusion that grass is green simply does not follow from the 



David Hitchcock 214 

premiss that snow is white, which is obviously totally irrelevant 
to the conclusion. The word ‘so’ when used inferentially 
implies, as part of its meaning and not as some pragmatic 
implicature of its ordinary use, that the statement preceding it is 
relevant to the statement following it, in the sense explicated in 
(Hitchcock, 1992) and qualified earlier in the present article, that 
it helps to establish the truth of the conclusion. The truth of an 
argument’s associated negajunction is not sufficient to secure 
such relevance. 
 In a symposium on Bermejo-Luque’s Giving Reasons, 
Freeman (2011b) and Pinto (2011) raise similar objections to 
Bermejo-Luque’s identification of the propositional content of 
an arguer’s inference claim with the argument’s associated 
material conditional. In reply, Bermejo-Luque reiterates and 
elaborates her position as follows: 
 

… following Grice’s account of conditionals, I take infer-
ence-claims of the form “if 79974 is divisible by 3, then 
79974 is divisible by 9” to be conversationally inappropri-
ate, but not false. In other words, I think that an act of ar-
guing such as “79974 is divisible by 3, therefore 79974 is 
divisible by 9” is semantically correct, but pragmatically 
flawed because the reason is irrelevant. To my mind, this 
kind of irrelevance is pragmatic: we put forward a reason 
in order to show a target claim to be correct, but the reason 
does not work for this end. On Grice’s account, the only 
circumstances in which it is appropriate to assert a condi-
tional is where the speaker is ignorant of the truth values 
of R [the reason—DH] and C [the conclusion—DH], but 
believes that if R happens to be true, C will as well. (Ber-
mejo-Luque 2011b, pp. 229-230.) 

 
This restatement helps to defuse the obvious objection to 
Bermejo-Luque’s interpretation of an arguer’s inference-claim 
as the claim that the argument’s associated material conditional 
is correct. On her account, a good act of arguing requires not 
just that the arguer’s inference-claim be true but that the arguer 
be in a position to assert it. That is, the arguer must have a basis 
for asserting it other than the denial of the reason or the 
assertion of the conclusion. Bermejo-Luque herself 
acknowledges that this basis can include (2011a, p. 65), and 
indeed will consist in (p. 198), general rules or general facts 
from which the material conditional follows. Thus in the end her 
position is not far from that of the present article, except that she 
rests the claim that an argument has a covering generalization on 
the pragmatics of advancing the argument rather than on the 
semantics of inferential particles and phrases. Later in this 



Inference Claims 215 

article, in the discussion in section 8.3 of Janne Maaike Gerlofs’ 
position, I will return to the question whether justification of a 
material conditional without using a paradox of material 
implication requires appeal to some generalization of it. 
 
8.2. Verheij: Bart Verheij (2006, p. 186) also identifies the 
inference-claim of an argument with its singular, ungeneralized 
associated conditional. Verheij takes an argument to express that 
its premisses collectively support its conclusion, and takes this 
support relation to be expressed by the conditional sentence ‘if 
<the premisses>, then <the conclusion>’. For example, in the 
following argument: 
 

(17) Harry was born in Bermuda, so Harry is a British 
subject. 

 
the claim that the premiss ‘Harry was born in Bermuda’ 
supports the conclusion ‘Harry is a British subject’ is expressed 
by the sentence: 
 

(18) If Harry was born in Bermuda, he is a British subject. 
 
Unlike Bermejo-Luque, however, Verheij refuses to identify an 
argument’s inference claim with the material conditional, 
because the material conditional is truth-functional: 
 

...a material conditional is truth-functional: its truth value 
is determined by the truth values of the conditional’s 
antecedent and consequent... The conditional ‘If D [data—
DH], then C [conclusion—DH]’ implied by an argument 
‘D. So C’ should however intuitively reflect some relation 
between D and C that is not captured by the truth values of 
D and C alone. (Verheij 2006, p. 187.) 

 
Let us call Verheij’s non-material, non-truth-functional singular 
conditional an ‘inferential conditional,’ since it expresses the 
condition that an argument’s premisses support its conclusion, 
i.e. that it is legitimate to infer the conclusion from the 
premisses. Verheij does not give a complete account of the 
semantics of the inferential conditional. He tells us that it 
validates modus ponendo ponens; in other words, an argument 
from an inferential conditional and its antecedent to the 
consequent of the inferential conditional is formally valid. He 
tells us that one cannot derive an inferential conditional on the 
basis of logic alone (e.g. from a deduction of its consequent 
from its antecedent), but must always base its derivation on 
premisses. But he says very little about what premisses would 



David Hitchcock 216 

entitle us to infer the truth of an inferential conditional. In fact, 
he claims that, if the logic of the inferential conditional were 
expressed in a natural deduction system, it would have the 
standard elimination rule for conditionals, namely modus 
ponendo ponens, but would have no introduction rule. The only 
derivation of an inferential conditional that he recognizes as 
legitimate is its derivation using modus ponendo ponens from a 
more complex inferential conditional of which the inferential 
conditional is the consequent. In particular, the inferential 
conditional associated with a particular argument follows from 
what he calls (following Toulmin (1958)) the argument’s 
warrant, which is a conditional scheme expressed in ordinary 
language as a rule statement. For example, the inferential 
conditional (18) that Harry is a British subject if he was born in 
Bermuda follows from the following statement: 
 

(19) A person born in Bermuda is a British subject. 
 
The inferential conditional expressing that this statement (19) 
supports the original argument’s associated inferential 
conditional (18) is the following statement: 
 

(20) If a person born in Bermuda is a British subject, then 
Harry is a British subject if he was born in Bermuda.  

 
The consequent of this second-order inferential conditional is 
according to Verheij an instance of its antecedent. 
 Verheij needs to complete his account of the semantics of 
the inferential conditional. In particular, since a rule statement 
like “a person born in Bermuda is a British subject” sounds 
awfully like a generalized material conditional, and the 
inferential conditional associated with an argument is supposed 
to be an instance of such a rule statement but not to be a mere 
material conditional, Verheij needs to tell us how a warrant 
differs from a generalized material conditional. The answer, I 
suspect, will be an account like the one I have been developing 
in this paper, that it must be a counterfactual-supporting 
generalization of the argument’s associated material conditional. 
In that case, Verheij’s singular inferential conditional will have 
just the meaning that I am attributing to singular indicative 
conditionals when they signify that their consequent follows 
from their antecedent. 
 
8.3. Gerlofs: Janne Maaike Gerlofs works within the mainstream 
logical tradition according to which an argument that is not ex-
plicitly formally valid is to be reconstructed by supplying an un-
expressed premiss whose addition will make the argument for-



Inference Claims 217 

mally valid. She argues (2009, 2011) that in general the argu-
ment analyst should supply as the unexpressed “connecting” 
premiss the argument’s ungeneralized associated material condi-
tional. (In her (2009), she makes an exception for cases where 
what she calls the “connecting premise” is explicit. If the stated 
premiss is a singular indicative conditional and the conclusion is 
the consequent of that conditional, one needs to supply its 
antecedent as the unexpressed premiss; an example is the 
argument, “There is no reason to inform the Child Protection 
Office. If there is no evidence the father started the fire 
deliberately to hurt his children, there is no need to do so”, 
where one should supply as the unexpressed premiss, “There is 
no evidence the father started the fire deliberately to hurt his 
children” (p. 86). If the stated premiss is a counterfactual 
conditional offered in support of the denial of its antecedent, one 
needs to supply the denial of its consequent as the unexpressed 
premiss; an example is the argument, “Daniel is no athlete. If 
Daniel were an athlete, he would have stamina”, where one 
should supply as the unexpressed premiss, “Daniel does not 
have stamina.” (p. 103).) 
 In defence of her position, Gerlofs points out that the as-
sociated material conditional does do the job of transferring ac-
ceptability from the expressed premiss to the conclusion, by 
means of the valid form of argument modus ponendo ponens. 
Given this role, she argues, the associated material conditional is 
what Davies (1979) calls a “knowledge conditional”, defined as 
a conditional in which the antecedent contains accepted know-
ledge and the consequent a conclusion “deduced” (i.e. inferred) 
from this knowledge: in argumentation the antecedent is put 
forward as already accepted and the connecting premiss 
(whether expressed or unexpressed) is put forward as a means of 
transferring acceptability from the non-connecting premiss to 
the conclusion. Further, material conditionals derived using the 
paradoxes of material implication cannot be used as a connect-
ing premiss. If the author of an argument defends its connecting 
premiss by appealing to the truth of its consequent, the reason-
ing is circular if the premiss is being used to transfer accept-
ability from its antecedent to its consequent and involves a 
contradiction if the connecting premiss is a counterfactual con-
ditional being used to transfer unacceptability from its conse-
quent to its antecedent (Gerlofs (2009), p. 107). Similarly, if the 
author defends the connecting premiss by appealing to the false-
hood of its antecedent, the reasoning is circular if the premiss is 
being used to transfer unacceptability from its consequent to its 
antecedent and involves a contradiction if it is being used to 
transfer acceptability from its antecedent to its consequent. Thus 
the paradoxes of material implication are deprived of their sting. 



David Hitchcock 218 

 An argument’s associated material conditional is the 
logical minimum among the statements that can be added to 
make an argument formally valid, in the sense that it follows 
logically from any statement whose addition as a premiss will 
make the argument formally valid without making a stated 
premiss redundant. In her (2011), Gerlofs argues that treating 
the connecting premiss of an argument as this logical minimum 
has heuristic advantages, in that it enables one to classify critical 
questions that one can raise about an argument and to determine 
whether critical questions posed for a given argumentation 
scheme cover all the points where a particular argument fitting 
the scheme can be questioned. She points out that the critical 
reactions to an argument, which correspond to the critical 
questions belonging to its argument scheme, can concern the 
acceptability of the argument’s premisses, the ground for the 
logical minimum or the connection between the ground and the 
logical minimum. 
 Gerlofs’ proposal is carefully articulated and defended, 
and deserves serious consideration. My suspicion is that, if one 
investigates the ways in which a singular material conditional 
can be defended, other than by denying its antecedent or 
affirming its consequent, one will discover that one needs to 
appeal to a counterfactual-supporting covering generalization, or 
to something logically stronger that implies a counterfactual-
supporting covering generalization. If so, then, given the 
pragmatic constraints that Gerlofs imposes on the justification of 
an unexpressed connecting premiss, her position would turn out 
to be equivalent to mine: the argument as stated implies that it 
has a counterfactual-supporting covering generalization that is 
non-trivially acceptable. 
 Gerlofs herself proposes (2009, pp. 111-117) to evaluate 
connecting premisses by means of argument schemes, whose 
expression as a generalized conditional would provide what she 
calls the “ground” for the connecting premiss. Such schemes are 
in fact generalizations, at some level of abstraction, of the 
particular argument taken to fit them, and are taken to hold 
counterfactually and non-trivially, given satisfactory answers to 
the “critical questions” associated with them that pertain to the 
transfer of acceptability from premiss to conclusion (Hitchcock 
2010b). Thus her position turns out to be equivalent to mine. But 
her only argument for restricting the justification of connecting 
premisses to an appeal to argument schemes is that an arguer 
cannot justify a connecting premiss by appealing directly to the 
truth values of its antecedent and consequent. 
 How can one justify a singular material conditional, other 
than by denying its antecedent or affirming its consequent? The 
typical strategy for proving a conditional is to assume its 



Inference Claims 219 

antecedent and derive its consequent. One can then discharge 
the assumption and assert the conditional on the basis of the 
assumptions other than the antecedent that were used in deriving 
the consequent from the antecedent. In a situation where we are 
given an argument from the antecedent to the consequent and 
are treating the material conditional as the logically minimal 
gap-filler, the assumptions enabling us to derive the consequent 
from the antecedent would serve as what the pragma-dialectical 
approach calls the “pragmatic optimum” (Van Eemeren and 
Grootendorst 1992, pp. 63-64). In general, an argument for some 
conclusion would transfer acceptability from the stated 
premisses to the conclusion if and only if the associated singular 
material conditional follows from acceptable assumptions none 
of which either are identical to or rely for their support on the 
conclusion. Thus, the inference claim of any argument is the 
claim that its associated singular material conditional follows 
from acceptable assumptions that are epistemically independent 
of the conclusion. It is clear that a generalization of the 
associated conditional that supports counterfactuals must be 
based on more than the truth of the consequent or the falsehood 
of the antecedent. What is not clear is that any assumptions 
epistemically independent of the conclusion that are sufficient 
for it to follow logically from the stated premisses must be at 
least as strong as some counterfactual-supporting generalization. 
However, two suggestions of epistemically independent 
assumptions other than a counterfactual-supporting 
generalization turn out to entail a counterfactual-supporting 
covering generalization. Bermejo-Luque suggested (personal 
correspondence) that a reason for “if you promised, you have to 
do it” (construed as a material conditional, i.e. a negajunction) 
may be something like the very definition of “promising”. She 
also suggested that a reason for “if the litmus paper turned red, 
then the liquid in which it was dipped is an acid” may be 
something like a chemical explanation. Definitions and chemical 
explanations, however, although they are not covering 
generalizations, entail covering generalizations, which support 
counterfactual instances if the definition or chemical explanation 
is correct. So reasons of this kind imply that the author of the 
argument is committed to a covering generalization of its 
associated negajunction. These two attempts thus failed to find a 
way of deriving a conclusion from premisses that is 
epistemically independent of the truth of the conclusion but does 
not entail a counterfactual-supporting covering generalization. 
Their failure, however, at best justifies a presumption that the 
approach of Gerlofs and the similar approach of Bermejo-Luque 
are de facto equivalent to the position of the present article. 
 



David Hitchcock 220 

8.4. Ennis: The present account of inference claims implies that 
all inference is general. If a conclusion follows from given 
premisses, then a parallel conclusion follows also from 
relevantly parallel premisses, even ones that are purely 
hypothetical. The generality of all inference undergirds the 
strategy sometimes called “refutation by logical analogy” (e.g. 
by Copi and Cohen (2001)), of challenging an inference by 
supplying a parallel argument with true (or otherwise 
acceptable) premisses and an untrue (or otherwise not 
acceptable) conclusion, perhaps prefixing one’s challenge with 
the frame, “you might as well say that ...” On the broader truth-
transmission conception of consequence that includes non-
logical consequences, a conclusion that follows from stated 
premisses does so in accordance with a counterfactual-
supporting covering generalization that is non-trivially 
acceptable. If one restricts consequence to logical consequence 
or logical/semantic consequence, then one should look for such 
a generalization to add as an unexpressed premiss when one 
reconstructs the argument. 
 Proponents of the unexpressed premiss approach, 
however, sometimes object to the insistence that a gap-filling 
unexpressed premiss must be general by citing examples of what 
I will call “occasional arguments”, which they claim have only a 
singular unexpressed premiss. Woods (2004, pp. 249-250) cites 
the argument: 
 

(21) It’s raining, so Eveline won’t be driving to Calgary. 
 
Although he does not describe the context of utterance of this 
apparently actual argument, it is a reasonable assumption that no 
counterfactual supporting covering generalization (such as 
‘Eveline doesn’t drive when it’s raining’ or ‘Unless it’s an 
emergency, Eveline won’t drive long distances when it’s 
raining’) is acceptable. The conclusion follows, we may 
suppose, in virtue of particulars of the occasion of utterance of 
the argument rather than in virtue of some covering 
generalization. 
 Robert Ennis claimed about a similar example that the 
gap-filling unexpressed premiss was the argument’s associated 
singular conditional: 
 

(22) ... when Michael Scriven and I were trying to find our 
way to Detroit airport in the car he rented, I said at 
one point, “The sign says ‘Chicago’ [to the right], so 
we should turn right there.” (We were trying to get 
on I 94 going to the airport.) There was no 
generalization there (this is clear from the situation) 



Inference Claims 221 

and I would resist one. It was straight modus ponens, 
if anything. (e-mail communication, 2009 June 8) 

 
Ennis later clarified in personal correspondence that he would 
not interpret the singular associated conditional that he took to 
be an unexpressed premiss of his argument as a material 
conditional. Aside from assuming that it licenses modus ponens, 
he did not ascribe truth-conditions to it. Much like Verheij, he 
envisaged it as an inferential conditional asserting that the 
conclusion of the stated argument follows from its premiss. 
 The inference in an occasional argument like those in 
examples (21) and (22) applies, on its face, only to the particular 
situation that is the occasion of its utterance. The word 
‘occasional’ echoes Quine’s use of the phrase ‘occasion 
sentences’ for sentences whose truth-value is partly a function of 
the occasion of their utterance (Quine, 1960). In the same way, 
the inferential goodness of occasional arguments is partly a 
function of the occasion of their utterance. But only partly, I 
shall argue. Once the relevant particular features of the occasion 
are specified, the conclusion follows if and only if some 
counterfactual-supporting covering generalization is non-
trivially acceptable. 
 To understand an occasional argument, someone not 
present on the occasion of its utterance needs to know who 
uttered it, what particular individuals are being referred to by its 
constituent proper names and definite descriptions, and what 
background knowledge about those particular individuals is 
being taken for granted as shared between the arguer and the 
argument’s addressees. Consider for example the argument cited 
by Ennis in example (22): 
 

(23) The sign says ‘Chicago’ [to the right], so we should 
turn right there. 

 
Ennis himself evidently found it necessary (or at least useful) to 
give part of the relevant background information in his e-mail 
communication: “Michael Scriven and I were trying to find our 
way to Detroit airport in the car he rented... We were trying to 
get on I 94 going to the airport.” To complete the picture, we 
need to know that they were on I 96, having come across the 
Ambassador Bridge from Windsor. The sign on the right to 
which Ennis referred is located about half a mile before the exit 
to I 94. It reads: “I 94 West Chicago / I 94 East Port Huron”. To 
fully understand the argument of example (23), we should add 
this additional information as unexpressed premisses mutually 
believed by both arguer and addressee: 
 



David Hitchcock 222 

(24) We are trying to find our way to Detroit airport. We 
are trying to get on I 94 going to the airport. We are 
on I 96, having come across the Ambassador Bridge 
from Windsor. The sign says ‘Chicago’ [to the right]. 
So we should turn right there. 

 
Thus expanded with the information required for someone not 
present on the occasion to understand the argument, the 
argument fits very well the approach of the present article. The 
argument is an instance of the scheme: 
 

(25) x is trying to find x’s way to Detroit airport. x is 
trying to get on I 94 going to the airport. x is on I 96, 
having come across the Ambassador Bridge from 
Windsor. The sign says ‘Chicago’ [to the right]. So x 
should turn right there. 

 
This argument scheme has no actual or counterfactual instances 
with true premisses and an untrue conclusion, even though it has 
an actual instance with true premisses (namely, the argument of 
example 24) and an actual instance with an untrue conclusion 
(for example, the instance in which the variable x is replaced 
with the name of someone at the same spot on I 96 who intends 
to continue on that freeway past its intersection with I 94). Thus, 
the conclusion of the argument in example 24 follows from its 
premisses, since it is an instance of the scheme in example 25. 
The validity of that scheme corresponds to the truth of the 
covering generalization of the argument that anyone on I 96 who 
has come across the Ambassador Bridge from Windsor and is 
trying to get on I 94 going to the Detroit airport should turn right 
where the sign says “Chicago” to the right. And this covering 
generalization is non-trivially true, and supports counterfactual 
instances, as is required for the conclusion of the argument to 
follow from its premisses. It should be construed as temporally 
restricted to a specious present with indefinite boundaries, like 
the inference-licensing covering generalization of the Obama 
argument of example 6. 
 I suspect strongly that all occasional arguments will yield 
to a similar treatment. In other words, if the conclusion of an 
occasional argument intuitively follows from its premisses, it 
will follow from them in accordance with a non-trivially 
acceptable counterfactual-supporting covering generalization 
once the stated premisses are supplemented with the additional 
information about the occasion of the argument’s utterance that 
is necessary for someone not present on that occasion to 
understand the argument. I invite readers to test this claim by 



Inference Claims 223 

working through an occasional argument that they themselves 
have experienced as arguer or addressee. 
  Occasional arguments exhibit the kernel of truth in the 
unexpressed premiss approach. Such arguments do need gap-
filling supplementation by information about the topic of the 
argument (i.e. the individual person or thing referred to in both 
premisses and conclusion), information that is taken for granted 
as known (or at least believed) by both arguer and addressees. 
But this supplementation usually does not produce an argument 
that is formally valid. Rather, it produces an argument with an 
inference-claim that is at least arguably correct, in the sense that 
some non-trivially acceptable counterfactual-supporting 
covering generalization licenses the drawing of the conclusion 
from the supplemented premiss set. In general, the function of 
such context-available information is to narrow the scope of an 
obvious covering generalization so that it is a plausible 
candidate for being non-trivially true and supporting counter-
examples. In the argument of example 23 about turning right 
where the sign says “Chicago”, for example, the additional 
premisses narrow the scope of the generalization that one should 
turn right where the sign says “Chicago” from all persons to all 
persons in a specious present who are travelling from the 
Ambassador Bridge on I 96 and trying to get on I 94 going to 
Detroit airport 
 
 
8. Summary 
 
Contemporary logicians generally construe consequence as 
formally necessary truth-preservation: a purely formal feature 
rules out that the implicans is true while the implicatum is 
untrue. Two objections can be raised to this conception. First, it 
counts something as a consequence simply because a purely 
formal feature rules out that the implicans is true or simply 
because a purely formal feature rules out that the implicatum is 
untrue. Second, it rejects something as a consequence when a 
general but not purely formal feature rules out a true implicans 
and untrue implicatum. An alternative truth-transmission 
conception holds that a consequence relation obtains when and 
only when a general feature rules out that the implicans is true 
while the implicatum is untrue, even though it does not rule out 
that the implicans is true and does not rule out that the 
implicatum is untrue.  Both conceptions can be given 
substitutional, model-theoretic or schematic articulations. 
However, each of these three articulations of the truth-
transmission conception has counter-examples where the 
conclusion of an argument obviously does not follow from its 



David Hitchcock 224 

premisses even though the argument satisfies the articulation in 
question. The remedy is to abandon the substitutional and 
model-theoretic articulations and to modify the schematic 
articulation so as to require that it holds in virtue of a 
counterfactual-supporting covering generalization. This 
modification introduces an awkward asymmetry into the 
evaluation of arguments: covering generalizations that hold 
merely of actual cases have to be treated as unexpressed 
premisses rather than as principles licensing an inference. The 
suspicion that the asymmetry indicates something wrong-headed 
about the whole approach can be countered by noting that 
people rarely argue for conclusions that follow only in 
accordance with a covering generalization that if true would not 
support counterfactual instances; an explanation of their rarity is 
that to argue in this way is to beg the question at issue. 
 The modified truth-transmission conception of 
consequence can be elaborated and expanded in four ways: by 
recognizing the legitimacy of restricting the range of the 
variables in an argument’s schema, by generalizing from truth to 
acceptability to cover deontic conclusions, by allowing for 
conclusions that are not assertives, and by allowing for 
rebuttable inferences. On the elaborated and expanded account, 
a conclusion follows from given premisses if and only if an 
acceptable counterfactual-supporting covering generalization of 
the argument rules out, either definitively or with some modal 
qualification, simultaneous acceptability of the premisses and 
non-acceptability of the conclusion, even though it does not rule 
out acceptability of the premisses and does not require 
acceptability of the conclusion independently of the premisses. 
 This elaborated and expanded account is rather complex. 
A sceptic might find more attractive the simpler view already 
held by ancient logicians that the inference claim of an argument 
is just its associated conditional, i.e. the singular conditional 
whose antecedent is the conjunction of the argument’s premisses 
and whose consequent is the argument’s conclusion  In response 
to such a sceptic, we may grant that a singular indicative 
conditional in one of its senses signifies that its consequent 
follows from its antecedent, but take the expanded acceptability-
transmission account of the consequence relation to be an 
account of the truth-conditions of a singular indicative 
conditional in that sense. The alternative view that the inference 
claim of an argument is singular rather than general is either 
subject to objections or equivalent to the expanded 
acceptability-transmission account. The view that the 
unexpressed premiss of a formally invalid argument is at least 
sometimes the argument’s associated conditional is likewise 
either subject to objections or equivalent to the present account. 



Inference Claims 225 

 
 
Appendix 
 
The universally generalized material conditional associated with 
the schema ‘People who are F are very poor soldiers, so the 
Iraqi soldiers are not F’ is logically equivalent to the statement 
that the Iraqi soldiers are not very poor soldiers. 
 Proof: (L to R) Suppose that, for any F, if people who are 
F are very poor soldiers, then the Iraqi soldiers are not F. Then 
in particular, by universal instantiation, if people who are 
identical with the Iraqi soldiers are very poor soldiers, then the 
Iraqi soldiers are not identical with the Iraqi soldiers. But, by the 
meaning of identity, the Iraqi soldiers are identical with the Iraqi 
soldiers. Hence, by double negation and modus tollendo tollens, 
people who are identical with the Iraqi soldiers are not very poor 
soldiers. In other words, the Iraqi soldiers are not very poor 
soldiers. 
 (R to L) Suppose that the Iraqi soldiers are not very poor 
soldiers. Now suppose, for conditional proof, that for an 
arbitrarily chosen property F, people who are F are very poor 
soldiers. Suppose, for reductio, that the Iraqi soldiers are F. 
Since they are people, they are people who are F, and thus are 
very poor soldiers, contrary to our original supposition. Hence, 
by reductio ad absurdum, the Iraqi soldiers are not F. Hence, by 
conditional proof, if people who are F are very poor soldiers, 
then the Iraqi soldiers are not F. Hence, since we have 
discharged all assumptions about F, by universal generalization, 
for any F, if people who are F are very poor soldiers, then the 
Iraqi soldiers are not F. QED 
 The proof uses only rules of inference that conform to the 
truth-transmission sense of consequence. It can be imitated for 
any argument in which an additional singular premiss would 
make the argument formally valid. Any singular statement is 
thus equivalent to a second-order universal generalization, 
which if it supports counterfactual instances can license 
inferences. 
 
 
Acknowledgements: I wish to express particular thanks to 
Lilian Bermejo-Luque, Robert Ennis, James Freeman and 
Robert Pinto for sustained correspondence on issues discussed 
in this article. I thank as well for helpful comments on earlier 
versions of this article George Boger, Ralph Johnson, Ernest 
Lepore, Harvey Siegel and Mark Weinstein. They will not 
necessarily agree with the result, the responsibility for which is 
mine. 



David Hitchcock 226 

 The present article is to appear in slightly different form in 
the proceedings of a colloquium entitled “Inside arguments: 
logic vs. argumentation theory”, held at the University of 
Coimbra in Portugal from March 24 through 26, 2011. I am 
grateful to the editors of the conference proceedings and of 
Informal Logic for permitting the dual publication. 
 
 
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