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diff --git a/papers/4.tex b/papers/4.tex new file mode 100644 index 0000000..995a510 --- /dev/null +++ b/papers/4.tex @@ -0,0 +1,746 @@ + +\chapter{A Defence of the Interpretational Account of Validity} +\chaptermark{A Defence of the Interpretational Account of Validity} +\chapterauthor{Audrey Hammer, +\textit{University of Cambridge}} + +\begin{quote} +Both the interpretational account and the representational account +provide contrasting accounts of validity for natural-language arguments. +While the interpretational account captures formal validity, unlike the +representational account, it does not capture materially valid +arguments. Therefore, materially valid arguments are viewed as +counterexamples to the interpretational account. I motivate why we may +want to defend the interpretational account over the representational +account and then proceed to defend the interpretational account using +the suppressed premise strategy. The first objection to the suppressed +premise strategy is by Stephen Read, who argues that the supressed +premise is redundant. My contribution is to demonstrate how his +objection fails. I also discuss and defend the suppressed premise +strategy against other objections, which concern the nature of the +supressed premise and the problem of modus ponens. +\end{quote} + +\vspace{\credgap} + +\section*{Introduction} + +Validity, a key concept in logic, concerns whether an argument is +truth-preserving. The interpretational account of validity defends the +view that for an argument to be valid it must be formally valid. I turn +first to the importance of logical form, its role in logic, generally, +and validity, specifically. My discussion then moves to the +interpretational account alongside its rival, the representational +account. Both accounts face distinct issues. While I do not hold that +the representational account is incoherent, I do hold that its +formulation has weaknesses that are absent in the interpretational +account, giving a motivation for preferring the latter rather than the +former. Materially valid arguments, which are not formally valid, +present counterexamples to the interpretational account. The remainder +of the essay is devoted to showing how the suppressed premise strategy +can defend the interpretational account against this main objection. The +suppressed premise strategy will in turn be defended against pressing +objections, primarily Stephen Read's objection that the suppressed +premise is redundant. This objection to the supressed premise strategy +aims to prove that there is a contradiction in adding a suppressed +premise to an already materially valid argument, and my contribution is +to show how this objection fails. I then go on to defend the suppressed +premise strategy against a few other objections, including objections +concerning the nature of the suppressed premise and the argument, and +the problem of modus ponens. The result is a defence of the +interpretational account of validity, using the suppressed premise +strategy. + +\section*{Understanding the Relation Between Logical Form and Validity} + +Logic is considered the science of deduction: it deals with arguments +and their validity. In formal logical languages, like truth functional +logic and first order logic, we can capture validity using the standard +notion of logical consequence. A formal argument is valid if the +conclusion is a logical consequence of the premises. As Owen Griffiths +and Alexander Paseau put it, ``A formal sentence $\phi$ is a logical +consequence of a set of formal sentences $\gamma$ just if every model of $\gamma$ is a +model of $\phi$''.\footnote{Owen Griffiths and Alexander Paseau, \emph{One + True Logic} (Oxford: Oxford University Press, 2022), 8.} Thus, we can +describe the formal notion of validity for a logical language, using a +model-theoretic notion of logical consequence. + +Once we have captured the notion of validity for logical languages, we +can move on to understanding the concept of validity as applied to +natural language, as the accounts of validity that will be discussed are +accounts of validity for natural language. To understand validity as +applied to natural language, we must introduce the concept of logical +form. Logical form is generally considered to be a property of a +sentence of natural language. The logical form of a sentence is when, +keeping the logical constants fixed, the non-logical expressions get +replaced with variables of the appropriate sort. Thus, the logical form +of a sentence can be expressed using a schema. Given this schematic +representation of form, we can follow Alfred Tarski in the view that +logic is topic neutral, because a schema abstracts from the content of +the sentence, only retaining the form of the sentence. For example, take +the following sentence: + +\begin{enumerate}[leftmargin=42] +\def\labelenumi{(\arabic{enumi})} +\item + Pigeons wear vests and cats wear hats. +\end{enumerate} + +\noindent This sentence can be expressed using the following schema: + +\begin{enumerate}[leftmargin=42] +\def\labelenumi{(\arabic{enumi})} +\setcounter{enumi}{1} +\item + A $\land$ B. +\end{enumerate} + +\noindent This is because the logical expression in sentence (1) is ``and'' which +can be formalised using the symbol ``$\land$'', and the non-logical +expressions in the sentence are ``pigeons wear vests'' and ``cats wear +hats'', and thus these expressions are replaced with variables. + +One stipulation with this account of logical form, is that it requires +us to have an understanding of what a logical constant is. Thus far +formality has been captured by its topic neutrality, and since a +demarcation of logical notions is crucial to form, it makes sense to +construct this demarcation using this quality of topic neutrality. Here +we can invoke Tarski's account of isomorphism invariance. Tarski defines +logical notions using an analogy from geometry. Just as we may demarcate +particular geometrical objects by their invariance under +transformations, so too can we demarcate logical notions. Thus, ``we +call a notion 'logical' if it is invariant under all possible one-one +transformations of the world onto itself''.\footnote{Alfred Tarski, + ``What are Logical Notions?,'' \emph{History and Philosophy of Logic} + 7\emph{,} (1986), 149.} To explain this further, we can consider an +isomorphism to be a bijective function, so between two structures there +is a one-one mapping, which preserves all the relevant relations. This +isomorphism is the transformation that Tarski is speaking of. For a +relation to be isomorphic invariant it must remain unchanged over this +sort of transformation. A relation that is isomorphic invariant is thus +indifferent to individual objects. The only notions that do this are +logical notions, and this confirms neutrality. Thus, we can define a +logical notion as being isomorphically invariant and non-logical notions +as not being isomorphically invariant. This allows for the demarcation, +which is necessary to define logical form. + +This understanding of logical form can now aid us in capturing the +notion of formal validity for natural language. It is common in the +literature to equate an argument being formally valid with it being +valid in virtue of its form.\footnote{Mark Sainsbury, \emph{Logical + Forms: An Introduction to Philosophical Logic}, (Oxford: Blackwell, + 2001): 37.} However, using this as a definition for formal validity is +unsatisfactory, for we still need to define being valid in virtue of +form, which I find to be no more informative than formal validity. +Therefore, I define formal validity to be the following: an argument is +formally valid iff it has a form which has only valid instances. An +example of a formally valid argument is: + +\begin{enumerate}[leftmargin=42] +\def\labelenumi{(\arabic{enumi})} +\setcounter{enumi}{2} +\item + All men are mortal, Socrates is a man $\therefore$ \ Socrates is mortal. +\end{enumerate} + +\noindent The logical form of the argument can be captured using a schema, as +described above. Given the use of quantifiers in (3), the schema of the +argument is simply its first order formalisation (on the obvious +formalisation key): + +\begin{enumerate}[leftmargin=42] + +\def\labelenumi{(\arabic{enumi})} +\setcounter{enumi}{3} +\item + $ \forall x(Fx \rightarrow Gx), \ Fa \ \therefore \ Ga.$ +\end{enumerate} + +\noindent There are no invalid arguments with this form, therefore all the +instances of this form are valid, consequently the argument is formally +valid. It is clear from this explanation that this definition of +validity for natural languages coincides with the definition for formal +languages, meaning that a natural-language argument is formally valid +iff its formalisation is valid. + +\section*{Two Accounts of Validity} + +We can now examine two model-theoretic accounts of validity for +natural-language arguments. Generally, model-theoretic accounts of +logical consequence are now viewed as more successful compared to other +accounts of logical consequence, and the two accounts that are the focus +of this essay are model-theoretic. As such the central thesis of both +accounts understands logical consequence as concerning truth +preservation across models.\footnote{This contrasts with proof-theoretic + accounts which hold that the nature of logical consequence involves + there being a proof from the premises to the conclusion.} The first +account is the interpretational account of validity, which originates +from Bolzano but was promulgated by Tarski.\footnote{Jc Beall, Greg + Restall, and Gil Sagi, ``Logical Consequence'', \emph{The Stanford + Encyclopaedia of Philosophy} (Summer 2024 Edition); Stephen Read, + ``Formal and Material Consequences'', \emph{Journal of Philosophical + Logic} 23, no. 3, (1994): 249.} This account holds that an argument is +valid if there are no possible interpretations of the argument (except +for a reserved class of logical interpretations) where the premises are +true and the conclusion false. An interpretation of an argument is any +argument that has the same logical form as the initial argument. The +second account is the representational account of validity, which holds +that an argument is valid if it is impossible for the premises to be +true and the conclusion false.\footnote{Read, "Formal and Material + Consequences'', 250.} + +The interpretational account only accepts arguments that are formally +valid. The account achieves this by examining different logical +interpretations of the argument; if there is no interpretation that has +true premises and a false conclusion then the argument is considered +valid. On the other hand, the representational account allows for +arguments that are materially valid, alongside those that are formally +valid. Materially valid arguments are arguments in which the validity of +the argument is in part due to the meaning of the non-logical terms +involved. An example of a materially valid argument is: + +\begin{enumerate}[leftmargin=42] +\def\labelenumi{(\arabic{enumi})} +\setcounter{enumi}{4} +\item + Jill is a paediatrician $\therefore$ \ Jill is a doctor. +\end{enumerate} + +\noindent The representational account intends to capture a more ``intuitive'' +notion of validity. Defenders of this account hold that materially valid +arguments are contained within this intuitive notion of validity, and so +an account of validity must capture material as well as formal validity. +This belief is rooted in the idea that there is an analytic connection +between certain words or phrases, and these connections make the +argument valid, even though the argument is not formally valid. + +The main objection to the interpretational account is that it is subject +to counterexamples, which take the form of materially but not formally +valid arguments. To establish the success of the interpretational +account we must meet this objection. One example of a materially but not +formally valid argument is (5) above, and another is: + +\begin{enumerate}[leftmargin=42] +\def\labelenumi{(\arabic{enumi})} +\setcounter{enumi}{5} +\item + Adam is taller than Bill and Bill is taller than Cathy $\therefore$ \ Adam is +taller than Cathy. +\end{enumerate} + +\noindent Neither of these arguments is formally valid, since there are invalid +arguments with the same form as (5) and (6). The interpretational +account would not accept that they are valid arguments given there are +interpretations of (5) and (6) for which the premises are true and the +conclusion false. A formalisation of these arguments in first order +logic reveals their logical form: + +\begin{enumerate}[leftmargin=42] +\def\labelenumi{(\arabic{enumi})} +\setcounter{enumi}{6} +\item + $Fa \ \therefore \ Ga$ +\item + $ (Tab \land Tbc) \ \therefore \ Tac $ +\end{enumerate} + +\noindent Another interpretation of each of these arguments demonstrates the point further: + +\begin{enumerate}[leftmargin=42] +\def\labelenumi{(\arabic{enumi})} +\setcounter{enumi}{8} +\item + Pat is a postman \ $\therefore$ \ Pat is a father. +\item + Alice is friends with Bonnie and Bonnie is friends with Carl \ $\therefore$ \ Alice is friends with Carl. +\end{enumerate} + +\noindent These arguments are clearly invalid, yet they have the same logical form +as (5) and (6), respectively. It is due to these alternative +interpretations that (5) and (6) are not valid. + +However, the arguments (5) and (6) would be accepted under the +representational account due to this account's use of modality. The +representational account identifies logical consequence with +metaphysical consequence. The reference to ``impossible'' in the +representational account is a modal notion, whereas the interpretational +account does not include such modal notions. The reference to ``no +possible interpretations'' in the interpretational account may be made +actual using substitutional classes, and thus does not need to rely on +an analysis of modality.\footnote{Read, "Formal and Material + Consequences'', 252.} Yet, it is because of its use of modality that +the representational account can attribute validity to (5) and (6), for +there is no possible world where the premises of (5) and (6) are true +and the conclusion false. + +On the other hand, modality is an issue for the representational +account, for it requires that we have an analysis of +modality.\footnote{It should be noted that this conversation concerns + analyses of the metaphysical notion of modality, which is distinct + from a discussion of modal logic, which is considered to be well + understood. Metaphysical modality deals with the fundamental nature of + modal notions, whereas modal logic is a formal system which reasons + about sentences containing modal operators.} Commonly, modality is +cashed out in turns of possible worlds. This prompts the question of +what a possible world is. The answers to this question are +controversial. We have modal realists, like David Lewis, who endorse a +view that possible worlds exist, as real concrete entities.\footnote{David + Lewis, \emph{On the Plurality of Worlds}, (Basil Blackwell, 1986) 2-3, + 86.} Adopting this analysis for our account of validity would also +mean adopting the ontological commitments of this account. Other +analyses of modality include modal sceptics, who deny that modal +statements can be known. In adopting this approach, we could not know +whether our arguments are valid, which is entirely counterintuitive. +While there are some more modest approaches to modality, like those +taken by Stalnaker\footnote{Robert C. Stalnaker, ``Possible Worlds,'' + \emph{Noûs} 10, no. 1, (1976): 65-75.} and Adams\footnote{Robert + Merrihew Adams, ``Theories of Actuality,'' \emph{Noûs} 8, no. 3, + (1974): 211-231.}, there are still issues surrounding whether these +accounts can provide a reductive analysis. This is all to say that while +modality is often invoked in philosophical topics, the debate +surrounding modal notions is not uncontroversial, and thus any time it +is invoked in a theory, that theory faces the same controversies. This +is not to say that modal notions should never be used in philosophical +theories, but just that we should be aware of the commitment and, all +things being equal, adopt theories without modal notions. This gives us +a motivation to prefer the interpretational account over the +representational account. Indeed, Read, who accepts the representational +account over the interpretational account, admits that the lack of modal +notions in interpretational account is a possible motivation to prefer +this account rather than the representational account.\footnote{Read, + "Formal and Material Consequences'', 252.} + +While this general criticism concerning the use of modal notions is +important to note, there is a more specific problem with the +representational account; namely, the identification of logical +consequence with metaphysical consequence then provides no account of +the importance of formality in logical consequence.\footnote{Beall, + Restall, and Sagi, ``Logical Consequence.''} Similarly, the account +does not provide a basis for distinguishing between logical and +non-logical vocabulary. This is because the representational account +determines that all expressions used in the argument contribute to the +validity of the argument. Consequently, the representational account +undermines the topic neutrality of logic. + +Given that the representational account faces the above challenges, I +suggest that this should motivate us to adopt the interpretational +account instead. While I do not view these issues as being +insurmountable, I simply hold that if there is an alternative we should +favour it. If the problem of counterexamples to the interpretational +account can be overcome, then this account becomes a preferrable +alternative to the representational account of validity. I devote the +remainder of this essay to considering and defending a possible solution +the interpretational account can adopt to resolve the problem of +counterexamples. This solution is the suppressed premise strategy. + +\section*{The Suppressed Premise Strategy} + +The suppressed premise strategy (hereafter SPS) can be employed by the +interpretational account to overcome the problem of materially valid +arguments. SPS holds that materially valid arguments have suppressed +premises which when revealed make the argument formally valid, and thus +valid under the interpretational account. These suppressed premises are +true given they usually explicitly reveal true analytic connections +between words.\footnote{Read views these suppressed premises not just as + true but as logically true because he associates logical truth with + analytic truth (Read, ``Formal and Material Consequences'', 258). + Since I have not made this association, I will avoid understanding + suppressed premises as logically true.} Since they are true, the +addition of the suppressed premise is largely unproblematic, although +this claim will be defended further. + +SPS applied to the argument (5) gives: + +\begin{enumerate}[leftmargin=42] +\def\labelenumi{(\arabic{enumi})} +\setcounter{enumi}{10} +\item + Jill is a paediatrician, all paediatricians are doctors $\therefore$ \ Jill is a doctor. +\end{enumerate} + +\noindent This argument can be formalised as follows: + +\begin{enumerate}[leftmargin=42] +\def\labelenumi{(\arabic{enumi})} +\setcounter{enumi}{11} +\item + $Fa, \ \forall x(Fx \rightarrow Gx) \ \therefore \ Ga$ +\end{enumerate} + + +\noindent There are no possible interpretations of the argument (11) that will +have true premises and a false conclusion, thus under the +interpretational account (11) is valid, although (5) remains invalid. Of +course, this strategy applies to (6), where the suppressed premise is +that ``taller than'' is transitive. No suppressed premise can be added +to (9) or (10), since it is not true that all postmen are fathers, there +is no analytic connection between being a postman and being a father, +and the relation ``being friends with'' is not transitive. + +\subsection*{The Redundancy Objection} + +The first objection to SPS is put forward by Read and states that the +suppressed premise is either false or redundant, and since it cannot be +false it must be redundant. \footnote{Read, ``Formal and Material + Consequences," 257-9.} Read gives his argument as follows: + +\begin{quote} +The extra premise is strictly redundant. For if the original argument +were invalid, the added premise would not be logically true. Given that +it is logically true, it follows that the unexpanded argument was +already valid. Hence it was (logically) unnecessary to add the extra +premise.\footnote{Read, "Formal and Material Consequences", 259.} +\end{quote} + +\noindent This objection is best demonstrated using an example. Take argument (5), +which is considered invalid under the interpretational account. Read +says that because of its invalidity, it is possible for the premises of +(5) to be true and the conclusion of (5) to be false. This entails that +it is possible for Jill to be a paediatrician but not be a doctor. Yet +the suppressed premise for this argument is that ``all paediatricians +are doctors'', clearly contradicts the possibility Jill is a +paediatrician and not a doctor. It follows if we accept that (5) is +invalid, then we also accept that the suppressed premise is false. Yet +this suppressed premise is true, so the initial assumption that (5) is +invalid must be false, and therefore the addition of the suppressed +premise is made redundant for it is not necessary for the argument to be +considered valid. According to Read, the suppressed premise's redundancy +means we should reject the interpretational account in favour of the +representational account. + +Read's objection, while presented convincingly, lacks any actual force. +This is due to a key error it makes: it presupposes the representational +account, when it should presuppose the interpretational account. It is +not the case that (5) is invalid because the premise ``Jill is a +paediatrician'' is compatible with it being false that ``Jill is a +doctor'', which (if true) is what the representational account would +suppose, rather (5) is invalid because there is an interpretation of (5) +for which the truth of the premises is compatible with the falsity of +the conclusion. (9) is an interpretation of (5) for which it is +compatible that it is true that ``Pat is a postman'' and false that +``Pat is a father'', and therefore (5) is considered invalid under the +interpretational account. Under the interpretational account, nothing +specifically is said about the premises of (5), and so Read is wrong to +infer that attributing invalidity to (5) will make the suppressed +premise false. Since Read is wrong to assert that the invalidity of the +argument shows the suppressed premise's falsity, he cannot then infer +that since the suppressed premise is true, it must therefore be +redundant. Under the representational account, invalidity is saying +something about the specific premises of the argument under +consideration. Yet under the representational account a materially valid +argument, like (5), would not be considered invalid. + +Some may reply here that I am begging the question: why is it that we +should assume the interpretational account and not the representational +account? However, this line of thought is also mistaken. Read clearly +starts by assuming that materially valid arguments are invalid, which is +only the case under the interpretational account, not the +representational account. From this assumption of invalidity, he +attempts to prove a contradiction, but then uses the representational +account's understanding of validity in this contradiction, even though +the representational account would not attribute invalidity to something +that is materially valid. However, if the interpretational account is +used, then there is no contradiction in using SPS. In addition, this +strategy is only used by the interpretational account. Thus, Read must +assume the interpretational account if he is going to show a +contradiction; given he does not use the interpretational account in his +objection and that even if he did use the interpretational account there +would be no contradiction, this implies that his objection holds no +weight. + +\subsection*{Objections about the Nature of the Suppressed Premise and the +Argument} + +A second problem for SPS is that we have not been committed to the view +that the suppressed premise is logically true. This may lead to the +question: why is it acceptable to add to an argument an extra premise +that is not logically true? Surely only logically true propositions may +be added to the premises of an argument to retain the same argument. To +answer this question, an important point must be reiterated: I do not +agree that the argument prior to the addition of the suppressed premise +is the same argument as the argument after the addition of the +suppressed premise. To me this point is obvious, for the two arguments +have different properties: one argument is valid, the other invalid, and +they have a different number of premises. Since we are speaking of two +different arguments, I do not need to prove that the first argument is +``retained'' in the second. However, this does not mean SPS can be used +on any argument. If the premise ``all postmen are fathers'' is added to +(9) then we have a new argument: + +\begin{enumerate}[leftmargin=42] +\def\labelenumi{(\arabic{enumi})} +\setcounter{enumi}{12} +\item + Pat is a postman, all postmen are fathers $\therefore$ \ Pat is a father. +\end{enumerate} + +\noindent (13) is a valid argument, but we should not consider (13) to be using SPS. Therefore, we must identify what differentiates (11) from (13), and +why (11) is determined as using SPS and thereby linking it closely with +(5) in a way that (13) is not linked with (9). The difference is that +the suppressed premise revealed in (11) that ``all paediatricians are +doctors'' is true, but the premise ``all postmen are fathers'' is not +true. Indeed ``all paediatricians are doctors'' is an analytic truth. +However, it is not necessary that this be considered a logical truth. To +begin with, there seems to be no necessity to consider analytic truths +to be logical truths, particularly if we retain the commonly held view +that logic has no special content. And secondly, the goodness of an +argument can be characterised by whether it is sound, i.e., it is valid +and has true premises, which does not require the premises to be +logically true. So long as the suppressed premise is true, its addition +to the argument does not hinder the chances of the argument being sound +and should in fact improve this since the argument will now be formally +valid. Since one of the characteristics of a suppressed premise is that +it is true, there is no issue that it is not logically true. Considering +(13), the premise ``all postmen are fathers'' cannot be a suppressed +premise of the argument (9) for it is not true. Therefore, the +suppressed premise does not need to be logically true, but this does not +mean that SPS can be applied to any argument to make it valid. + +Moreover, we may consider that SPS might even allow us to consider +contingent truths as suppressed premises. Let us suppose that it were a +contingent fact that ``all postmen are fathers'', then it might make +sense to consider this to be a suppressed premise of argument (9). Say +Mr. Black presented argument (9) to Mr. White and both Mr. Black and Mr. +White were aware that ``all postmen were fathers'', then the argument +might be accepted as sound in the rhetoric (even though it is not +formally valid) because both understood that the argument has a +suppressed premise, and that Mr. Black in fact meant to make the +argument (13). Now suppose Mr. Smith questioned the validity of the +argument because he was not aware that it was a contingent fact that +``all postmen were fathers''. Yet, once this would be revealed to him, +Mr. Smith would certainly accept the validity of the argument. +Therefore, we may accept that a suppressed premise may be contingently +true, and it becomes clear that only truth, and not logical truth, is +necessary for the suppressed premise. + +A counterexample to this argument has been pointed out to me.\footnote{By + Owen Griffiths, in personal communication.} This is that if we take +the argument: + +\begin{enumerate}[leftmargin=42] +\def\labelenumi{(\arabic{enumi})} +\setcounter{enumi}{13} +\item + I am a philosophy student $\therefore$ \ puppies are cute. +\end{enumerate} + +\noindent This is clearly invalid. But if the conditional ``If I am a philosophy +student then puppies are cute'' is added as a suppressed premise to +(14), then we get the new valid argument: + +\begin{enumerate}[leftmargin=42] +\def\labelenumi{(\arabic{enumi})} +\setcounter{enumi}{14} +\item + If I am a philosophy student then puppies are cute, I am a +philosophy student $\therefore$ \ puppies are cute. +\end{enumerate} + +\noindent It appears there is no problem with adding this conditional if we take +the view that suppressed premises only need to be contingently true, and +not analytically true, because considered as a material conditional it +is true (the antecedent and consequent are true). This seems to be a +problem for the strategy, as it might allow for many arguments like +(14), that have true premises and true conclusions yet are not formally +or materially valid, to be valid by adding these conditionals as +suppressed premises. + +My response to this argument is to say that these conditionals are +indicative conditionals, not material conditionals, which means they +involve a different treatment. An indicative conditional is the +conditional of natural language, and the current discussion is about the +validity of natural-language arguments, so it makes sense to speak of +indicative conditionals rather than material conditionals. We may then +consider views of indicative conditionals which hold that their truth +values are different to those of material conditionals, and as such we +can formulate a view that holds that ``If I am a philosophy student then +puppies are cute'' is false. For instance, we might hold that an +indicative conditional is true iff it is assertable and is in turn +assertable iff it passes the Ramsey test. The Ramsey test is a test for +the assertability of a conditional, it holds that a conditional is +assertable if someone were to add the antecedent to her set of +suppositions, she would also have to add the consequent. ``If I am a +philosophy student then puppies are cute'' would clearly fail the Ramsey +test. Thus, we can still consider that the suppressed premise may be +true without the above presenting as a counterexample. + +I have only given a rough sketch of a possible response to the objection +suggested above, and while there are many problems with associating the +truth conditions of an indicative conditional with those of the material +conditional, there are still some who adopt this view. However, the +conditional suggested is one where the antecedent and the consequent are +both true and yet have nothing to do with each other. This sort of +conditional is itself a problem case for someone who holds this +truth-functional view of the indicative conditional, suggesting that +there is something wrong with equating the indicative conditional with +the material conditional. However, if the reader insists on the +indicative conditional and the material conditional having the same +truth value, even in cases where the antecedent and consequent have no +relation to each other, then this reader may simply choose to reject +this section on contingent truth and hold that the suppressed premise +must be an analytic truth. This does not detract from the fact that the +suppressed premise is not a logical truth. Of course, the reader may +still object to the idea of analytic truth. However, this paper defends +the interpretational account against the counterexample of material +valid arguments, which themselves rely heavily on the notion of +analyticity. So, if the reader places no importance on the analytic +connections between words, then there is no forceful objection to the +interpretational account and no need for SPS to begin with. + +A third objection connects to my answer to the second objection. I have +stated that the two arguments, the argument prior to the addition of the +suppressed premise and the argument after this addition, are two +different arguments. This may lead one to ask, ``what connects the two +arguments?'' The answer to this is simple: they both have the same aim. +The aim of an argument is an imprecise and informal notion; however, I +want to use it to capture an intuitive idea. The two arguments share the +same conclusion, and their aim is to use true (and very similar) +premises to arrive at this conclusion. Suppose that Jones is having a +discussion of Jill's profession; he would be just as happy receiving the +argument (11) as he would be receiving the argument (5), possibly even +happier receiving (11) if he is unaware that a paediatrician is a kind +of doctor (or if he is a logician who has a strong appreciation for +formal validity). However, Jones would be disappointed if instead of +receiving either of these arguments he received (3), for instance, which +clearly has nothing to do with Jill or her profession. The aim of the +arguments is informal, and the setting for which Jones might accept or +reject them, as described, is also informal. The arguments are connected +by this informality. The matter of validity in logic is strictly a +formal matter, and thus there is a distinct difference between (5) and +(11). + +\subsection*{The Problem of Modus Ponens} + +The final problem I shall explore in relation to SPS is the problem of +modus ponens. A modus ponens is a deductive argument of the following +form: + +\begin{enumerate}[leftmargin=42] +\def\labelenumi{(\arabic{enumi})} +\setcounter{enumi}{15} +\item + $A, \ A \rightarrow B \ \therefore \ B$ +\end{enumerate} + +\noindent Modus ponens is discussed by both Read and Timothy Smiley, in very +different ways.\footnote{Read, "Formal and Material Consequences," + 259-62; Timothy Smiley, "A Tale of Two Tortoises", \emph{Mind} 104, + no. 496, (1995): 727.} They both view modus ponens as having a similar +form to SPS but speak of different consequences related to this +similarity. Below, I address both in turn. + +The problem that Read notes with modus ponens is that the major premise +of this argument (16) is either false or redundant. While his discussion +of this problem is limited, he links it with SPS by arguing that in both +cases the additional premise ``adds psychological perspicuity +{[}\ldots{]} But at the same time, it is not essential''.\footnote{Read, + "Formal and Material Consequences," 262.} To some extent I disagree +with both points. Considering the second point, the suppressed premise +and the major premise in the modus ponens argument are vital in making +the argument valid, and thus are essential to the argument. On the first +point, there is some sense in which adding the suppressed premise and +the major modus ponens premise do add psychological perspicuity, but it +does not necessarily always do this or do this to the extent Read may be +suggesting. In cases where both parties implicitly know the suppressed +premise, its addition to the argument may not provide any psychological +clarity, only logical infallibility. This idea is strengthened when +considering that most of the arguments we make in everyday life have +suppressed premises and we do not seem to need to reveal these +suppressed premises for psychological reasons.\footnote{Smiley, "A Tale + of Two Tortoises," 727.} Rather we tend to reveal suppressed premises +for logical reasons. Given we are holding this discussion in the domain +of logic, we may accept the resemblance between SPS and modus ponens +while still rejecting Read's assertion of redundancy. + +Smiley's discussion of this matter refers to a paradox that seems to be +presented by modus ponens and the addition of the suppressed premises. +The paradox in question originated from Lewis Carroll, who wrote: + +\begin{quote} +If I grant (A) All men are mortal, and (B) Socrates is a man, but not +(C) The sequence "If all men are mortal, and if Socrates is a man, then +Socrates is mortal" is valid, then I do not grant (Z) Socrates is +mortal. Again, if I grant C, but not A and B, I still fail to grant Z. +Hence, before granting Z, I must grant A and B and C. {[}Now consider{]} +(D) If A and B and C be true, then Z is true.\footnote{Charles Lutwidge + Dodgson, \emph{Lewis Carroll's Symbolic Logic}, W. W. Bartley III, + ed., (Clarkson Potter, 1977), 472.} +\end{quote} + +\noindent This becomes paradoxical when we observe an infinite regress occurring +where we must grant (A), (B), (C), (D), and a further (E) If A and B and +C and D be true, then Z is true, yet we can think of an infinite number +of propositions that must be granted before it seems that Z is granted. +We can view (C), (D), etc, as suppressed premises of the argument that +Carroll reveals in his paradox. This leads Smiley to comment that +``Lewis Carroll was doomed to detect suppressed hypothetical premises +even in logically valid arguments, and incidentally to force them all +into the straitjacket of modus ponens''.\footnote{Smiley, "A Tale of Two + Tortoises," 727.} If these are considered to be suppressed premises +then there is a problem for SPS, for these can be added to any argument, +and make the argument paradoxical. In addition, this does not seem to be +what the strategy intends. To solve this, we can examine the +characteristics of the suppressed premise, which is that its addition +will make the argument formally valid. Yet the arguments that Lewis +Carroll imagines are already valid arguments, thus SPS should not be +employed in these cases. Smiley's examination of the problem also points +out that the specific wording of the paradox is crucial for its +paradoxical nature but is itself flawed. Lewis Carroll ``lacked any +distinct conception of a deduction as opposed to the assertion'', and it +is this confusion that leads to paradox. \footnote{Smiley, "A Tale of + Two Tortoises," 727.} By this Smiley means that (C) is not an +assertion but a deduction, and so it must be granted, but Carroll seems +to think that it is merely an assertion that can be accepted or denied. +Hence, this paradox does not show that even valid arguments might have +suppressed premises that lead to paradox, thus this objection presents +no issue to the use of SPS. + +The characterisation I have given of SPS prevents contradiction and I +have shown how it is able to overcome objections about the redundancy of +the suppressed premise, as well as more generally the nature of the +suppressed premise and the nature of the arguments to which it pertains. +Finally, I discussed the problem of Modus Ponens, showing two ways it +relates to SPS, and that this does not impact the use of the strategy. +Thus, SPS is a viable addition to the interpretational account and +explains the relation of material validity to validity, without a need +to adopt the representational account. Hence by defence of the +interpretational account succeeds and preferred to the representational +account. + +\newpage +\section*{Bibliography} + +\refsection + +\begin{hangparas}{\hangingindent}{1} +Adams, Robert Merrihew. ``Theories of Actuality.'' \emph{Noûs} 8, no. 3 +(1974), 211-231. + +Beall, Jc, Greg Restall, and Gil Sagi "Logical Consequence",~\emph{The +Stanford Encyclopedia of Philosophy~}(Summer 2024 Edition), Edward N. +Zalta \& Uri Nodelman~(eds.), +\newline +\url{https://plato.stanford.edu/archives/sum2024/entries/logical-consequence} + +Dodgson, Charles Lutwidge \emph{Lewis Carroll's Symbolic Logic}. W. W. +Bartley III, ed. Clarkson Potter, 1997. + +Griffiths, Owen, and Alexander Paseau. 2022. \emph{On True Logic: A +Monist Manifesto.} Oxford University Press, 2022. + +Lewis, David. \emph{On the Plurality of Worlds.} Basil Blackwell, 1986. + +Read, Stephen. ``Formal and Material Consequences.'' \emph{Journal of Philosophical Logic} 23, +no. 3 (1994): 247-265. + +Sainsbury, Mark. \emph{Logical Forms: An +Introduction to Philosophical Logic.} Blackwell, 2001. + +Smiley, Timothy. ``A Tale of Two Tortoises.'' \emph{Mind} 104, no. 416 (1995): +725-736. + +Stalnaker, Robert C. "Possible Worlds." \emph{Noûs} 10, no. 1, (1976): 65-75. + +Tarski, Alfred. ``What are Logical Notions?'' +\emph{History and Philosophy of Logic} 7, 1986): 143-154. +\end{hangparas}
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