On the representational approach , each model is taken to represent a possible world. If an argument preserves truth over models, we are then guaranteed that it preserves truth over possible worlds, and if we accept the identification of necessity with truth in all possible worlds, we have the necessary truth preservation of logical consequence. The problem with this approach is that it identifies logical consequence with metaphysical consequence, and it gives no account of the formality of logical consequence.
On the representational approach, there is no basis for a distinction between the logical and the nonlogical vocabulary, and there is no explanation of why the interpretations of the nonlogical vocabulary are maximally varied.
The second perspective on models is afforded by the interpretational approach , by which each model assigns extensions to the nonlogical vocabulary from the actual world: what varies between models is not the world depicted but the meaning of the terms.
For instance, on the usual division of the vocabulary into logical and nonlogical, identity is considered a logical term, and can be used to form statements about the cardinality of the domain e. Each approach, as described here, is flawed with respect to our analysis of logical consequence as necessary and formal. The interpretational approach, by looking only at the actual world fails to account for necessity, and the representational approach fails to account for formality for details, see Etchemendy , Sher , and Shapiro , and for refinements see Etchemendy A possible response to Etchemendy would be to blend the representational and the interpretational perspectives, viewing each model as representing a possible world under a re-interpretation of the nonlogical vocabulary Shapiro , see also Sher and Hanson for alternative responses.
One of the main challenges set by the model-theoretic definition of logical consequence is to distinguish between the logical and the nonlogical vocabulary. The logical vocabulary is defined in all models by the recursive clauses such as those mentioned above for conjunction and the universal quantifier , and in that sense its meaning is fixed.
The choice of the logical vocabulary determines the class of models considered when evaluating validity, and thus it determines the class of the logically valid arguments.
Now, while each formal language is typically defined with a choice of a logical vocabulary, one can ask for a more principled characterization of logical vocabulary. Tarski left the question of a principled distinction open in his , and only gave the lines of a relativistic stance, by which different choices of the logical vocabulary may be admissible. Others have proposed criteria for logicality, demanding that logical constants be appropriately formal, general or topic neutral for references and details, see the entry on logical constants.
Note that a choice of the logical vocabulary is a special case of setting constraints on the class of models to be used. It has been suggested that the focus on criteria for the logical vocabulary misses this point, and that more generally the question is which semantic constraints should be adopted, limiting the admissible models for a language Sagi a, Zinke Another challenge faced by the model-theoretic account is due to the limitations of its set-theoretic basis.
Recall that models are sets. The worry is that truth-preservation over models might not guarantee necessary truth preservation—moreover, it might not even guarantee material truth preservation truth preservation in the actual world. One way of dealing with this worry is to employ external means, such as proof theory, in support of the model-theoretic definition. Another option is to use set-theoretic reflection principles. Generally speaking, reflection principles state that whatever is true of the universe of sets, is already true in an initial segment thereof which is always a set.
Explaining logical consequence in terms of truth in models is rather close to explaining logical consequence in terms of truth , and the analysis of truth-in-a-model is sometimes taken to be an explication of truth in terms of correspondence, a typically Realist notion. Some, however, view logical consequence as having an indispensable epistemic component, having to do with the way we establish the conclusion on the basis of the premises. On the proof-centered approach to logical consequence, the validity of an argument amounts to there being a proof of the conclusions from the premises.
Exactly what proofs are is a big issue but the idea is fairly plain at least if you have been exposed to some proof system or other.
Proofs are made up of small steps, the primitive inference principles of the proof system. The 20th Century has seen very many different kinds of proof systems, from so-called Hilbert proofs, with simple rules and complex axioms, to natural deduction systems, with few or even no axioms and very many rules.
The proof-centered approach highlights epistemic aspects of logical consequence. A proof does not merely attest to the validity of the argument: it provides the steps by which we can establish this validity. And so, if a reasoner has grounds for the premises of an argument, and they infer the conclusion via a series of applications of valid inference rules, they thereby obtain grounds for the conclusion see Prawitz One can go further and subscribe to inferentialism , the view by which the meaning of expressions is determined by their role in inference.
The idea is that our use of a linguistic expression is regulated by rules, and mastering the rules suffices for understanding the expression. This gives us a preliminary restriction on what semantic values of expressions can be: they cannot make any distinctions not accounted for by the rules. This view is favored by anti-realists about meaning, since meaning on this view is fully explained by what is knowable. The condition of necessity on logical consequence obtains a new interpretation in the proof-centered approach.
The condition can be reformulated thus: in a valid argument, the truth of the conclusion follows from the truth of the premises by necessity of thought Prawitz Let us parse this formulation. Truth is understood constructively : sentences are true in virtue of potential evidence for them, and the facts described by true sentences are thus conceived as constructed in terms of potential evidence.
Note that one can completely forgo reference to truth, and instead speak of assertibility or acceptance of sentences. Now, the necessity of thought by which an argument is valid is explained by the meaning of the terms involved, which compels us to accept the truth of the conclusion given the truth of the premises.
Meanings of expressions, in turn, are understood through the rules governing their use: the usual truth conditions give their way to proof conditions of formulas containing an expression.
One can thus provide a proof-theoretic semantics for a language Schroeder-Heister Under certain requirements, one can show that the elimination rule is validated by the introduction rule. One of the main challenges for the proof-centered approach is that of distinguishing between rules that are genuinely meaning-determining and those that are not. Some rules for connectives, if added to a system, would lead to triviality.
Some constraints have to posed on inference rules, and much of subsequent literature has been concerned with these constraints Belnap , Dummett , Prawitz To render the notions of proof and validity more systematized, Prawitz has introduced the notion of a canonical proof. A sentence might be proved in several different ways, but it is the direct, or canonical proof that is constitutive of its meaning. A canonical proof is a proof whose last step is an application of an introduction rule, and its immediate subproofs are canonical unless they have free variables or undischarged assumptions—for details see Prawitz A canonical proof is conceived as giving direct evidence for the sentence proved, as it establishes the truth of the sentence by the rule constitutive of the meaning of its connectives.
For more on canonical proofs and the ways other proofs can be reduced to them, see the entry on proof-theoretic semantics. We have indicated how the condition of necessity can be interpreted in the proof-centered approach.
The condition of formality can be accounted for as well. Note that on the present perspective as well, there is a division of the vocabulary into logical and nonlogical. This division can be used to define substitutions of an argument. A substitution of an argument is an argument obtained from the original one by replacing the nonlogical terms with terms of the same syntactic category in a uniform manner.
A definition of validity that respects the condition of formality will entail that an argument is valid if and only if all its substitutions are valid, and in the present context, this is a requirement that there is a proof of all its substitutions. This condition is satisfied in any proof system where rules are given only for the logical vocabulary.
Of course, in the proof-centered approach as well, there is a question of distinguishing the logical vocabulary see the entry on logical constants. Finally, it should be noted that a proof theoretic semantics can be given for classical logic as well as a variety of non-classical logics. However, due to the epistemic anti-realist attitude that lies at the basis of the proof-centered approach, its proponents have typically advocated intuitionistic logic see Dummett For more on the proof-centered perspective and on proof-theoretic semantics, see the entry on proof-theoretic semantics.
The proof-theoretic and model-theoretic perspectives have been considered as providing rival accounts of logical consequence. One can also note that the division between the model-theoretic and the proof-theoretic perspectives is a modern one, and it was only made possible when tools for metamathematical investigations were developed.
One can also ask what general features such a relation has independently of its analysis as proof-theoretic or model-theoretic. One way of answering this question goes back to Tarski , who introduced the notion of consequence operations.
For our purposes, we note only some features of such operations. See the entry on algebraic propositional logic for details. Among some of the minimal conditions one might impose on a consequence relation are the following two from Tarski :. Both of these conditions can be motivated from reflection on the model-theoretic and proof-theoretic approaches; and there are other such conditions too. For a general discussion, see the entry on algebraic propositional logic.
But as with many foundation issues e. For example, some might take condition 2 to be objectionable on the grounds that, for reasons of vagueness or more , important consequence relations over natural languages however formalized are not generally transitive in ways reflected in 2.
See Tennant , Cobreros et al , and Ripley , for philosophical motivations against transitive consequence. But we leave these issues for more advanced discussion. While the philosophical divide between Realists and Anti-realists remains vast, proof-centered and model-centered accounts of consequence have been united at least with respect to extension in many cases.
The great soundness and completeness theorems for different proof systems or, from the other angle, for different model-theoretic semantics show that, in an important sense, the two approaches often coincide, at least in extension. A proof system is sound with respect to a model-theoretic semantics if every argument that has a proof in the system is model-theoretically valid.
A proof system is complete with respect to a model-theoretic semantics if every model-theoretically valid argument has a proof in the system. While soundness is a principal condition on any proof system worth its name, completeness cannot always be expected. Leaving terminological issues aside, if a proof system is both sound and complete with respect to a model-theoretic semantics as, significantly, in the case of first order predicate logic , then the proof system and the model-theoretic semantics agree on which arguments are valid.
We have noted a weakness of the model-theoretic account: all models are sets, and so it might be that no model represents the actual world. Kreisel takes intuitive validity to be preservation of truth across all structures whether sets or not.
His analysis privileges the modal analysis of logical consequence—but note that the weakness we are addressing is that considering set-theoretic structures might not be enough. Another arena for the interaction between the proof-theoretic and the model-theoretic perspectives has to do with the definition of the logical vocabulary.
Carnap has famously shown that the classical inference rules allow non-standard interpretations of the logical expressions Carnap See also the entry on sentence connectives in formal logic. Finally, we should note that while model theory and proof theory are the most prominent contenders for the explication of logical consequence, there are alternative frameworks for formal semantics such as algebraic semantics , game-theoretic semantics and dynamic semantics see Wansig In fact, Aristotle focuses on arguments with exactly two premises the major premise and the minor premise , but nothing in his definition forbids arguments with three or more premises.
It is found by chaining together the two smaller arguments. If the two original arguments are formally valid, then so too is the longer argument from three premises. For such reasons, many have taken the relation of logical consequence to pair an arbitrary possibly infinite collection of premises with a single conclusion.
Using the indicator terms is particularly helpful because a conclusion may be stated first, last, or anywhere in between. People do all three when they write or talk in real life, so we cannot tell whether a statement is a conclusion simply by where it is positioned in the argument. First, the process helps you clearly see just what the other person is saying. It helps you identify the logical structure of the argument, which is necessary if you are to assess the strengths and weaknesses of the argument in order to know whether or not to accept it.
Second, you develop skills of analysis that you will need in order to organize and present arguments in support of a position that you may want to take on some question or issue.
Here are the basic moves that are required in order to create a clear diagram or outline of an argument. Identify all the claims made by the author. Since a sentence can contain multiple claims, rewrite statements so that you have one claim per sentence. Adopt some sort of numbering or labeling system for the claims—your instructor may have one that she wishes you to follow. Recognize that there may be sub-conclusions in addition to a final or main conclusion.
You may think of a sub-conclusion as the end point of a sub-argument nested inside the larger argument. Although the sub-conclusion is itself the conclusion of a nested argument, supported by premises, it also functions as a premise supporting the final or main conclusion.
Recognize that some premises are independent and others linked. If you were drawing or mapping the argument, you would be able to draw an arrow from an independent premise directly to the conclusion it supports.
Linked premises, however, are multiple statements that must be combined to provide support for a conclusion. If you were drawing or mapping the argument, you would have to find some way to show that the linked premises as a group support the conclusion. An author must organize her material to guide the audience through her argument.
One tool available to an author is the paragraph. The sentences clustered together in a paragraph should be tightly connected in terms of content. The paragraphs themselves should be placed in an order that reflects some overall plan so that the paragraphs reveal the steps or stages of the argument. The premises may be said to be key steps or stages in the argument. A well-constructed argument therefore may use each premise as a topic sentence for a paragraph.
Additionally, a premise may serve as the guiding idea for a group of paragraphs, each developing a subtopic. Look to see whether the author has used paragraphing-by-premise to organize her argument and outline its structure for the audience. You should also ask yourself whether any paragraphs are missing. That is, as you consider what premises serve as the foundations of the argument, be alert for the suppressed ones, the premises that the author assumes to be automatically true.
These unacknowledged premises may be ones that the author hopes the audience will not notice or question. In your analysis call her on it by determining where a paragraph on that premise should have appeared in the argument.
We would be bothered by reading an editorial in which someone stated a strong opinion on some public issue yet did nothing to justify that opinion. The argument now has a structure that can be outlined or diagrammed. The course provides enough information, together with examples and quizzes.
Highly recommend this to others. Category: FutureLearn Local , Learning. Category: Current Issues , General. Category: Digital Skills , Learning , What is. We offer a diverse selection of courses from leading universities and cultural institutions from around the world.
These are delivered one step at a time, and are accessible on mobile, tablet and desktop, so you can fit learning around your life. You can unlock new opportunities with unlimited access to hundreds of online short courses for a year by subscribing to our Unlimited package.
Build your knowledge with top universities and organisations. Learn more about how FutureLearn is transforming access to education. Learn more about this course. What is the standard form of an argument? View transcript. You should now know what statements and arguments are. Basically, statements are sentences that are either true or false. And arguments are ways of combining statements so as to make a point by providing premises, the reasons, intended to support a conclusion.
When you encounter arguments in the wild, it becomes difficult to isolate the premises from the conclusion and to isolate sentences that are actually part of the argument. We call this a standard form. Before you start evaluating arguments, your first task will be to put them in standard form.
The standard form of an argument is a way of presenting the argument which makes clear which statements are premises, how many premises there are, and which statements is the conclusion.
In standard form, the conclusion of the argument is listed last. A standard form looks like this— premise 1, premise 2, and so on for as many premises as there are— therefore, conclusion.
Premise 2— I only have bad days on Mondays. Therefore, conclusion— today is Monday. Animals from factory farming spend their entire lives in miserable conditions until the day they are slaughtered. Their suffering is unimaginable. Animals from factory farming are treated cruelly. Chickens get their beaks cut off with a burning hot blade and with no painkillers. And half of the chickens on farms, the cockerels, are slaughtered.
The fact is, eggs come from hens that are treated cruelly— all that so that you can enjoy bacon and eggs. But I only eat fish, some people say. The problem is that commercial fishing is destroying and emptying our oceans. When we put the argument in standard form, we have to isolate the statements that form the conclusion and premises, and we have to reorder them appropriately.
The conclusion here was explicitly stated— you should go vegan. And Justin clearly announced that he had three reasons in support of his conclusion. But he seems to have given us many more than three— or has he?
The first one is about factory farming and the maltreatment of animals.
0コメント