The Purpose of Deductive Arguments

Graham Oppy's Suggestions for Better Rational Disagreement

In this post, I discuss the nature and purpose of argumentation and standards of success for philosophical arguments. These remarks are largely an exposition of the views of prominent Philosopher of Religion Graham Oppy who has extensively talked about¹ the use of deductive arguments in Philosophy of Religion and argued for better alternatives to the exchange of arguments.  

The man, the myth, the legend.

Argument

Argumentation is a general war of words between disputants who disagree about some claim. Here I am restricting myself to philosophical disagreement though some of these observations will extend to other kinds of disagreement. When disputants disagree about some claim arguments are provided as dialogical devices with the telos of persuading an interlocutor to adopt the argument’s conclusion (some position on the disputed claim.) 

In broad strokes, the structure of an argument consists of a set of claims called the premises which provide reasons to accept some central claim which is disputed amongst disagreeing parties. The conclusion of an argument is supposed to be supported by the premises given by a disputant.

In deductive arguments, the truth of the conclusion follows of (logical) necessity from the truth of the premises. That is, as a matter of logic, the premises cannot all be true and the conclusion false, that set is inconsistent, where the premises taken with the conclusion are consistent.

This piece is about what we can hope for when exchanging deductive arguments in disputes and highlights some pitfalls to look out for when we expect too much from them. It concludes by offering a better way forward for significant disputes.

Unsuccessful Deductive Argument  

Following Oppy², suppose we have two disputants; Pro and Con.

Pro believes: { p₁,p₂,…,p_n,CLAIM,q₁,q₂,…,q_n }
Con believes: {p₁,p₂,…,p_n,~CLAIM,~q₁,~q₂,…,~q_n}

Here ‘p_i’ {p1…pn} claims are claims agreed upon between the two parties. ‘CLAIM’ is disputed and shall be central to the disagreement, and ‘q_j’ {q1…qn} claims are disagreed upon between the two parties. 

Suppose that in the course of disagreement, the following interaction takes place. 

Pro: CLAIM 

Con: ~CLAIM. Therefore I win. 

Winning here is taken to be having provided reasons such that Pro is rationally obligated to abandon their position and adopt that being put forward by Con. It is obvious however that Con has done no such thing. Con has merely asserted the negation of a claim that Pro believes and Pro has no rational obligation to do anything in light of this. This is a case of an unsuccessful argument. 

Let p₁ be the claim that ‘~q₁⊃~CLAIM’, remembering that p₁ is from the set of claims to which Pro and Con agree. Consider the following interaction between Pro and Con. 

Pro: CLAIM 

Con:  p₁,~q₁. Therefore ~CLAIM. I win! 

Pro: No, that’s not right.
Con: This is a valid and sound deductive argument, so the conclusion must be true! I win!

These sorts of interactions in philosophical disputes can be incredibly frustrating. Having barely scratched the surface of the disagreement, but parroting some lines about logical validity and soundness, having provided a deductive argument, one party simply declares victory. The idea here is that offering a deductive argument is like some kind of gold standard of rationality and exerts knock down rational force on the opponent to commit to its conclusion.

However, stopping for just a second to think about this case shows how the declared victory is merely apparent, and shall provide us with some important reflections that will inform what we consider desiderata for good fruitful disputes. 

The argument provided by Con has the following form:
{ ‘~q₁⊃~CLAIM’, ‘~q₁’ } ⊨ { ‘~CLAIM’ }. That is to say that the set
T = { ‘~q₁⊃~CLAIM’, ‘~q₁’, ‘CLAIM’  } is inconsistent (using ‘T’ to denote the set as it stands for theory).

Anyone who believes all three statements is irrational as closure under entailment means that a person committed to T is committed to ‘CLAIM’ and ‘~CLAIM’ (a logical contradiction). To maintain rationality, someone committed to the inconsistent set T, must reject one of the claims that constitutes T.

Is this a problem for Pro? If we look at the dispute between Pro and Con it is true that Pro accepts ‘~q₁⊃~CLAIM’ and ‘CLAIM’. However, Pro doesn’t accept ‘~q₁’, the second premise of Cons argument. The upshot of this is that there isn’t any inconsistency in Pro’s commitments such that they ought to revise their beliefs. When it comes to rationality, Pro is unscathed by Con’s deductive arguments.

It is true that Con has an argument that they consider valid and sound, but that doesn’t do anything for Pro. Any insistence upon victory in light of having provided an argument without Pro’s agreeing to all the premises is rhetorical flourish; chest-thumping. Further, it can be noted that the provision of an argument in this interaction doesn’t do anything to improve the quality of this dispute over that of the first case where both parties simply asserted their disputed conclusions. We’re simply in the same spot where we have both parties asserting claims they disagree about.

When interlocutors offer contentious deductive arguments like this, though it may have the surface appearance of some deeper appeal to rationality and an attempt at reasonable persuasion, we shouldn’t consider it any different to simply asserting one’s contested conclusion and declaring victory. Not fruitful for dialectical progress.

Successful Deductive Argument 

Remember our toy model:

Pro believes: { p₁,p₂,…,p_n,CLAIM,q₁,q₂,…,q_n }
Con believes: {p₁,p₂,…,p_n,~CLAIM,~q₁,~q₂,…,~q_n}

Let p₂ be the claim that ‘q₂⊃~CLAIM’. Now we imagine the following interaction between Pro and Con. 

Pro: CLAIM
Con:  p₂,q₂. Therefore ~CLAIM
Pro: Hmmmm 

In this case, Con has provided Pro with the argument: { ‘q₂⊃~CLAIM’, ‘q₂’ } ⊨ { ‘~CLAIM’ }. This is a problem for Pro because they are now cognisant of the fact that they are committed to the inconsistent set
T* = { ‘q₂⊃~CLAIM’, ‘q₂’, ‘CLAIM’ }.
Under entailment { ‘CLAIM’, ‘~CLAIM’ } ⊂ T* which means that Pro is strictly irrational if they maintain their commitments to all of T*.  

In light of this Con has provided Pro with an argument that should compel Pro to belief revision. This constitutes a successful argument, in the sense that it successfully exerts rational pressure on Pro to revise their beliefs. What’s also interesting about this is that whilst it is a successful argument for Pro, this argument does nothing for Con, given their commitment to ‘~q2’. This type of argument is an internal critique of Pro’s position. It is useful because it highlights a tension within Pro’s beliefs that they should do something about.

Reductios

These observations show that successful arguments will be those which are comprised of premises that an interlocutor believes. In cases where interlocutors do not agree with the premises, arguments simply won’t go through. 

With respect to disputes about specific contested claims, successful arguments will show interlocutors that the negation of their reported commitment to a disputed claim can be derived from their other commitments. When successful, the result of these arguments will be the interlocutor’s awareness that they are committed to some contradictory sentences (i.e. CLAIM and ~CLAIM). Hence successful arguments will be reductios. 

Reasonable Belief Revision 

As successful arguments will be reductios of an interlocutor’s commitments the result of such arguments will be that interlocutors will be left in a state of awareness about their inconsistency which they must resolve.  

Is that it? Has Con finally won by providing a successful deductive argument, persuading Pro of ~CLAIM? No, not yet. The nature of deductive validity is such that all of T* is inconsistent, this doesn’t tell us anything about how Pro should revise their beliefs. People offer deductive arguments with the expectation that if their interlocutor dispute the conclusion they should die or something. However, Pro is within their rational rights to take their time, reflect on their broader commitments and decide to remain agnostic with respect to CLAIM, or reject any of the claims that make up T*, i.e. they could maintain their commitment to CLAIM and reject one of the premises.

One possible way for Pro to revise their beliefs is to reject p₂ or q₂. How Pro modifies their beliefs will be a product of how their holistic worldview hangs together. Belief adjustment will require Pro to assess how the claims in T* are related to the rest of their belief structure such that they can assess the relative overall costs of belief revision.

With this in mind, we can see how trading deductive arguments can be helpful. Where deductive argumentation highlights tensions in a view it can prompt reflection and belief revision. What we should also note is the fruitlessness of trading deductive arguments in a combative debate-like manner. Arguments will either be completely unsuccessful (given disagreement about the premises involved), or will prompt an interlocutor to belief revision which has no guaranteed outcome and logical consistency comes cheap!

Typically, when we are engaged in deep disagreements about contentious topics, we can assume that our disputants have achieved logical consistency in their worldview such that us telling them sets of claims we happen to believe that entail the falsehood of their view is dialectically flaccid and does little more than to reassure us that we are very logical and rational for believing what we believe (a form of intellectual pride which will trap us in false beliefs should we have them).

Further, we can note that when it comes to logic, there is nothing special about which claims appear in the premises or conclusion of an argument — there is no position of special logical privilege. Entailment runs both ways and if parties are sufficiently motivated to preserve commitment to their conclusions all that we achieve by trading deductive arguments is to turn philosophical enquiry into a sort of verbal performance where two people exchange Moorean shifts based on the contested point. Logic tells us nothing about whether the premises are held more firmly than the conclusion and, when it comes to the sorts of claims which people provide arguments for in disputes, they are often more committed to those claims than to the stories they tell about why they believe those claims. That is, frequently people do not believe the conclusion of an argument as a result of their commitment to the premises, but rather believe the premises of an argument as a result of their commitment to its conclusion.

Theory and Argument 

Oppy refers to the total set of beliefs a person has, closed under entailment, as a person’s theory. Following from the conclusions drawn in the previous section that conclusions often determine which premises we find plausible, Oppy makes the point that articulation of theory in as much detail as possible should come prior to argumentation. This is because our pre-existing commitments taken as a theory in this sense will wholly and properly determine which arguments we will view as sound (without the need for all the rhetorical garb of trading deductive arguments).

In light of this, Oppy³ suggests that in cases of disagreement, a more fruitful approach that avoids the pitfalls of trading deductive arguments highlighted above is for disputants to focus on holistic theory comparison. This is in contrast to getting bogged down in disagreement about which arguments are sound. 

Oppy offers a framework for disputants who wish to engage in deep philosophical disagreement more constructively with a three-step heuristic:

1. To fully spell out in as much detail as possible the best theories which embed the disputed claims.
   

2. To test the theories for internal consistency (especially under entailment).
   

3. To compare the surviving theories according to the theoretical virtues (broadly which theory maximises explanatory scope and minimises commitments).

Adopting Oppy’s framework for disputes can be a much more productive way of understanding and resolving differences in philosophy. Particularly when interlocutors have considerable and systematic disagreements — such as in metaethics, philosophy of religion, mind, language etc. Disputes in these arenas are rarely the result of parties disagreeing about some single claim which fits nicely in a modus ponens in favour of one of the contested views. These disputes are the result of significant and broad disagreements across multiple topics which need to be evaluated in the whole in order to assess their relative merits and figure out the best way forward with a clear mind.

Further references⁴⁵

1. Dr. Graham Oppy on the Nature of Arguments, Majesty of Reason ( 2020 )

2. Oppy, G., 2015. What derivations cannot do. Religious Studies, 51(3), pp.323-333.

3. Oppy, G., Pearce, K. and De Cruz, H., 2021. Is there a God?: A Debate. Routledge. pp.95.

4. Oppy, G., 2010. Arguing about gods. Cambridge: Cambridge University Press. pp.10 

5. McKay, T., 2000. Reasons, explanations, and decisions: guidelines for critical thinking. Australia: Wadsworth/Thomson Learning. pp.3