Why is argument the heart of logic

They are both valid and have true premises. A valid argument is an argument whose premises guarantee the truth of the conclusion. That is, if the premises are true, then it is impossible for the conclusion to be false. A valid deductive argument whose premises are all true is called a sound argument.

Hopefully, you can see that these arguments present a close connection between the premises and conclusion. It seems impossible to deny the conclusion while accepting that the premises are all true. This is what makes them valid deductive arguments. To show what happens when similar arguments employ false premises, consider the following examples:. You may recognize that these arguments have the same structure as the previous two arguments. That is, each expresses the same connection between the premises and conclusion, and they are all deductively valid.

However, these latter two arguments have at least one false premise and this false premise is the reason why these otherwise valid arguments reach a false conclusion. In the case of these arguments, the structure is good, but the evidence is bad. Deductive arguments are either valid or invalid because of the form or structure of the argument.

They are sound or unsound based on the form, plus the content. You might become familiar with some of the common forms of arguments many of them have names and once you do, you will be able to tell when a deductive argument is invalid.

These are arguments that have the wrong structure or form. Perhaps you have heard a playful argument like the following:. These arguments are examples of the fallacy of the undistributed middle term. The name is not important, but you may recognize what is going on here. The two types of objects in each conclusion are each a member of some third type, but they are not members of each other.

So, the premises are all true, but the conclusions are false. If you encounter an argument with this structure, you will know that it is invalid. But what do you do if you cannot immediately recognize when an argument is invalid? Philosophers look for counterexamples. A counterexample is a scenario in which the premises of the argument are true while the conclusion is clearly false.

So, a counterexample demonstrates that the argument is invalid. After all, validity requires that if the premises are all true, the conclusion cannot possibly be false. Consider the following argument, which is an example of a fallacy called affirming the consequent :. What if a water main broke and flooded the streets?

Then the streets would be wet, but it may not have rained. So, the scenario where a water main breaks demonstrates this argument is invalid. The counterexample method can also be applied to arguments where there is no clear scenario that makes the premises true and the conclusion false, but we will have to apply it a little differently. In these cases, we need to imagine another argument that has exactly the same structure as the argument in question but uses propositions that more easily produce a counterexample.

Suppose I made the following argument:. I can demonstrate this by imagining another argument with the same structure as this argument, but the premises of this argument are clearly true while its conclusion is false:. To review, deductive arguments purport to lead to a conclusion that must be true if all the premises are true. But there are many ways a deductive argument can go wrong. In order to evaluate a deductive argument, we must answer the following questions:.

Almost all of the formal logic taught to philosophy students is deductive. This is because we have a very well-established formal system, called first-order logic, that explains deductive validity. The problem is that the logic governing inductive and abductive inferences is significantly more complex and more difficult to formalize than deductive inferences.

The chief difference between deductive arguments and inductive or abductive arguments is that while the former arguments aim to guarantee the truth of the conclusion, the latter arguments only aim to ensure that the conclusion is more probable. Even the conclusions of the best inductive and abductive arguments may still turn out to be false.

Consequently, we do not refer to these arguments as valid or invalid. Instead, arguments with good inductive and abductive inferences are strong ; bad ones are weak. Similarly, strong inductive or abductive arguments with true premises are called cogent. Inductive inferences typically involve an appeal to past experience in order to infer some further claim directly related to that experience.

In its classic formulation, inductive inferences move from observed instances to unobserved instances, reasoning that what is not yet observed will resemble what has been observed before.

Generalizations, statistical inferences, and forecasts about the future are all examples of inductive inference. You might wonder why this conclusion is merely probable. Is there anything more certain than the fact that the Sun will rise tomorrow? Well, not much. But at some point in the future, the Sun, like all other stars, will die out and its light will become so faint that there will be no sunrise on the Earth. He hypothesizes 5 5 On the basis of grammatical-linguistic considerations Netz, , pp.

Again, this would suggest that the writing medium is not a necessary condition neither for regimentation nor for the use of schematic letters. In effect, the generality and arbitrariness brought in by the use of letters in diagrams "Let ABC be a triangle defined by the points A, B and C.

But notice that the concept of schematic letter placeholder must be sharply distinguished from the concept of a functional variable , which emerged much later within mathematics: a mathematical variable stands for an unknown but determinate value, whereas a schematic letter is a device of generality, indicating a range of possible instantiations.

Be that as it may, the use of schematic letters is arguably one of the main reasons why syllogistic is a logical system even by modern standards. It deals with schemata , so that the principles and rules stated are valid for any permissible instantiation of the schemata with specific terms. This feature also allows for meta-properties of the system to be investigated, as done in the first chapters of the Prior Analytics.

The use of schematic devices remained pervasive in the history of logic, in particular but not exclusively in connection with syllogistic.

In fact, a different ancient Greek tradition in logic just as remarkable, but having had significantly less historical influence also made extensive use of schematic devices: the Stoic tradition.

Modus ponens, for example, was thus formulated: "If the first, then the second; but the first; so the second" modus tollens: "If the first, then the second; but not the second; so not the first". However, the scope of application of the theory presented in the first chapters of the "Prior Analytics" would be extremely limited if it could indeed only be applied to arguments which already display the specific structure of syllogistic arguments.

Here are some relevant passages from chapter 32, where Aristotle makes general comments on the regimentation enterprise. Prior Analytics 47a It is evident from the things which have been said, then, what all demonstrations come from, and how, and what things one should look to in the case of each problem. But after these things, we must explain how we can lead deductions back into the figures stated previously, for this part of our inquiry still remains. For if we should study the origin of deductions, and also should have the power of finding them, and if, moreover, we could resolve those which have already been produced into the figures previously stated, then our initial project would have reached its goal.

First, then, one must try to pick out the two premises of the deduction [ For sometimes people who propose a universal premise do not take the premise included in it, either in writing or in speech. Or, they propose these premises but leave out what they are concluded through and instead ask for other useless things. One must therefore see whether something superfluous has been taken, and whether one of the necessary premises has been left out; and the one should be put in and the other taken away until the two premises are reached.

Instead, one must first get the two premises and next divide them in this way into terms, and that term which is stated in both the premises must be put as the middle for the middle must occur in both of them in all of the figures. Finding the middle term in particular is a very important step in the regimentation of an arbitrary argument into a recognized syllogistic structure.

The regimentation is nicely illustrated in a passage from the "Prior Analytics", in a reworked version as presented by Hodges , p. Journal of Philosophical Logic, 38, pp.

Aristotle's text needs careful dissection, and for a smooth exposition I've permuted some of his material. God does have right moments for action. Therefore some right moment for action is not a time that needs to be set aside for action. Aristotle would find the premises and the conclusion, and then write out the syllogistic terms together with letters to represent them:.

B: time needing to be set aside for action. C: right moment for action. Some right moment for action is a thing that God has. Therefore some right moment for action is not a time needing to be set aside for action. Why does Aristotle go through the trouble of reformulating an argument in this way?

Well, simply because the argument as originally formulated would not be amenable to analysis with the syllogistic machinery. But once it is re-written, the valid syllogistic mood 'No B is A. Some C is A. Therefore some C is not B. Thus, this style of regimentation marks the very birth of logic with Aristotle's syllogistic as a theory not only of the validity of argument patterns, but also of how to regiment arbitrary arguments into these patterns.

It is ironic that the very birth of syllogistic is in fact marked by a move away from ordinary language; it is widely acknowledged that the categorical sentences of syllogistic were rather contrived even from the point of view of the Greek spoken at the time. Hodges very appropriately asks: what is natural about so-called 'natural logic'? However, the obvious philosophical question which must be asked is: in what sense is the reworked, regimented version of the argument indeed the same argument as the original, or in any case a sufficiently close counterpart thereof?

One may be tempted to say that they have the same 'meaning', but this would presuppose that we have independent access to the meaning of the original argument, and are able to establish that the regimented version has the same meaning despite the linguistic dissimilarity what Stokhof STOKHOF, M. On formal and natural languages in semantics". Journal of Indian Philosophy, 35, pp. We may also say that the regimented version in fact depicts more conspicuously the actual, preexisting deep structure of the argument, which is there all along but in need to be discovered.

But how does the formalizer obtain access to the deep structure? Russell's king of France. None of these issues is explicitly discussed by Aristotle, but it is clear that they are all lurking in the background, as his general considerations in the passages quoted above suggest.

Besides syllogistic, another interesting example of theories of argument formalization are Latin medieval theories of supposition, which were intended to provide a systematic account of the semantic behavior of terms and sentences. Latin medieval logic in general is characterized by a high level of regimentation of the language used.

Latin medieval logicians adopted conventions related to word order so as to disambiguate some constructions: 'Every man loves a woman', for example, would be written 'A woman every man loves' if the intended reading was in anachronistic terms to assign wider scope to the existential quantifier. In fact, academic Latin was nobody's 'native language' at that point; it had become first and foremost a tool for intellectual inquiry, as well as for usage in other official contexts courts of law, the Church etc.

In: H. Lagerlund ed. Encyclopedia of Medieval Philosophy. Berlin: Springer, a. Berlin: Springer, Berlin: Springer, b. They were meant to codify the quantificational behavior of what we now refer to as quantifier expressions, which the medievals classified as syncategorematic terms.

Soundness A Valid patterns A Validity and relevance A Hidden Assumptions A Inductive Reasoning A Good Arguments A Argument mapping A Analogical Arguments A More valid patterns A Arguing with other people Quote of the page There are two ways to slide easily through life: to believe everything or to doubt everything. Popular pages What is critical thinking? What is logic? Hardest logic puzzle ever Free miniguide What is an argument?

Knights and knaves puzzles Logic puzzles What is a good argument? Improving critical thinking Analogical arguments. A crucial part of critical thinking is to identify, construct, and evaluate arguments. Before proceeding, read this page about statements.

Here is an example of an argument: If you want to find a good job, you should work hard. A few points to note: Dogmatic people tend to make assertions without giving reasons. When they are criticized they often fail to give arguments to defend their own opinions.

To improve our critical thinking skills, we should develop the habit of giving good arguments to support our opinions. This leads to some kinds of default or non-monotonic inference. The field of inductive consequence is difficult and important, but we shall leave that topic here and focus on deductive validity. See the entries on inductive logic and non-monotonic logic for more information on these topics.

The constraint of necessity is not sufficient to settle the notion of deductive validity, for the notion of necessity may also be fleshed out in a number of ways. To say that a conclusion necessarily follows from the premises is to say that the argument is somehow exceptionless , but there are many different ways to make that idea precise.

A first stab at the notion might use what we now call metaphysical necessity. Perhaps an argument is valid if it is metaphysically impossible for the premises to be true and the conclusion to be untrue, valid if—holding fixed the interpretations of premises and conclusion—in every possible world in which the premises hold, so does the conclusion.

If that claim is necessary, then the argument:. While there may be genuine discoveries of valid arguments that we had not previously recognised as such, it is another thing entirely to think that these discoveries require empirical investigation. An alternative line on the requisite sort of necessity turns to conceptual necessity. If metaphysical necessity is too coarse a notion to determine logical consequence since it may be taken to render too many arguments deductively valid , an appeal to conceptual or analytic necessity might seem to be a better route.

The trouble, as Quine argued, is that the distinction between analytic and synthetic and similarly, conceptual and non-conceptual truths is not as straightforward as we might have thought in the beginning of the 20th Century.

Furthermore many arguments seem to be truth-preserving on the basis of analysis alone:. Still, many have thought that 4 is not deductively valid, despite its credentials as truth-preserving on analytic or conceptual grounds.

It is not quite as general as it could be because it is not as formal as it could be. The argument succeeds only because of the particular details of family concepts involved. A further possibility for carving out the distinctive notion of necessity grounding logical consequence is the notion of apriority.

Deductively valid arguments, whatever they are, can be known to be so without recourse to experience, so they must be knowable a priori. A constraint of apriority certainly seems to rule argument 3 out as deductively valid, and rightly so.

However, it will not do to rule out argument 4. If we take arguments like 4 to turn not on matters of deductive validity but something else, such as an a priori knowable definition, then we must look elsewhere for a characterisation of logical consequence. The strongest and most widespread proposal for finding a narrower criterion for logical consequence is the appeal to formality.

What could the distinction between form and content mean? We mean to say that consequence is formal if it depends on the form and not the substance of the claims involved. But how is that to be understood? We will give at most a sketch, which, again, can be filled out in a number of ways.

The obvious first step is to notice that all presentations of the rules of logical consequence rely on schemes. Inference schemes, like the one above, display the structure of valid arguments. Perhaps to say that an argument is formally valid is to say that it falls under some general scheme of which every instance is valid, such as F e r io. That, too, is an incomplete specification of formality. The material argument 4 is an instance of:. We must say more to explain why some schemes count as properly formal and hence a sufficient ground for logical consequence and others do not.

A general answer will articulate the notion of logical form , which is an important issue in its own right involving the notion of logical constants , among other things.

Instead of exploring the details of different candidates for logical form, we will mention different proposals about the point of the exercise. What is the point in demanding that validity be underwritten by a notion of logical form? There are at least three distinct proposals for the required notion of formality, and each provides a different kind of answer to that question. We might take the formal rules of logic to be totally neutral with respect to particular features of objects.

Laws of logic, on this view, must abstract away from particular features of objects. Logic is formal in that it is totally general. One way to characterise what counts as a totally general notion is by way of permutations. Tarski proposed that an operation or predicate on a domain counted as general or logical if it was invariant under permutations of objects. A permutation of a collection of objects assigns for each object a unique object in that collection, such that no object is assigned more than once.

Defining this rigorously requires establishing how permutations operate on sentences, and this takes us beyond the scope of this article. A closely related analysis for formality is that formal rules are totally abstract. They abstract away from the semantic content of thoughts or claims, to leave only semantic structure. On this view, expressions such as propositional connectives and quantifiers do not add new semantic content to expressions, but instead add only ways to combine and structure semantic content.

Another way to draw the distinction or to perhaps to draw a different distinction is to take the formal rules of logic to be constitutitive norms for thought, regardless of its subject matter. It is plausible to hold that no matter what we think about, it makes sense to conjoin, disjoin and negate our thoughts to make new thoughts.

It might also make sense to quantify. The behaviour, then, of logical vocabulary may be used to structure and regulate any kind of theory, and the norms governing logical vocabulary apply totally universally.

The norms of valid argument, on this picture, are those norms that apply to thought irrespective of the particular content of that thought. Twentieth Century technical work on the notion of logical consequence has centered on two different mathematical tools, proof theory and model theory. Each of these can be seen as explicating different aspects of the concept of logical consequence, backed by different philosophical perspectives.

We have characterized logical consequence as necessary truth preservation in virtue of form. This idea can be explicated formally. One can use mathematical structures to account for the range of possibilities over which truth needs to be preserved. The formality of logical consequence can be explicated formally by giving a special role to the logical vocabulary, taken as constituting the forms of sentences.

Let us see how model theory attends to both these tasks. The model-centered approach to logical consequence takes the validity of an argument to be absence of counterexample. A counterexample to an argument is, in general, some way of manifesting the manner in which the premises of the argument fail to lead to a conclusion.

One way to do this is to provide an argument of the same form for which the premises are clearly true and the conclusion is clearly false. Another way to do this is to provide a circumstance in which the premises are true and the conclusion is false.

In the contemporary literature, the intuitive idea of a counterexample is developed into a theory of models. A model for an extensional first order language consists of a non-empty set which constitutes the domain , and an interpretation function , which assigns to each nonlogical term an extension over the domain—any extension agreeing with its semantic type individual constants are assigned elements of the domain, function symbols are assigned functions from the domain to itself, one-place first-order predicates are assigned subsets of the domain, etc.

The contemporary model-theoretic definition of logical consequence traces back to Tarski It builds on the definition of truth in a model given by Tarski in Tarski defines a true sentence in a model recursively, by giving truth or satisfaction conditions on the logical vocabulary. A conjunction, for example, is true in a model if and only if both conjuncts are true in that model.

Now we can define logical consequence as preservation of truth over models: an argument is valid if in any model in which the premises are true or in any interpretation of the premises according to which they are true , the conclusion is true too.

The model-theoretic definition is one of the most successful mathematical explications of a philosophical concept to date.