ValidityFrom Wikipedia, the free encyclopedia

Contents

1 Argument 11.1 Formal and informal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Standard types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Deductive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Inductive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Defeasible arguments and argumentation schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 By analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.7 Other kinds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.7.1 In informal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.7.2 World-disclosing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.8 Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.9 Fallacies and nonarguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.13 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 If and only if 102.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.3 Origin of i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Distinction from if and only if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 More general usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Interpretation (logic) 13

i

ii CONTENTS

3.1 Formal languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1.2 Logical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 General properties of truth-functional interpretations . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.1 Logical connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Interpretation of a theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Interpretations for propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5 First-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.5.1 Formal languages for rst-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5.2 Interpretations of a rst-order language . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.5.3 Example of a rst-order interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.5.4 Non-empty domain requirement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5.5 Interpreting equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5.6 Many-sorted rst-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.6 Higher-order predicate logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.7 Non-classical interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.8 Intended interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.8.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.9 Other concepts of interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Logic 214.1 The study of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.1.1 Logical form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.1.2 Deductive and inductive reasoning, and abductive inference . . . . . . . . . . . . . . . . . 224.1.3 Consistency, validity, soundness, and completeness . . . . . . . . . . . . . . . . . . . . . . 234.1.4 Rival conceptions of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Types of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3.1 Syllogistic logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3.2 Propositional logic (sentential logic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3.3 Predicate logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3.4 Modal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3.5 Informal reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3.6 Mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3.7 Philosophical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3.8 Computational logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3.9 Bivalence and the law of the excluded middle; non-classical logics . . . . . . . . . . . . . 284.3.10 Is logic empirical?" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3.11 Implication: strict or material? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

CONTENTS iii

4.3.12 Tolerating the impossible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3.13 Rejection of logical truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Validity 355.1 Validity of arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2 Valid formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.3 Validity of statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.4 Validity and soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.5 Satisability and validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.6 Preservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.7 n-Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.7.1 -Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.11 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.11.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.11.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.11.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Chapter 1

Argument

This article is about the subject as it is studied in logic and philosophy. For other uses, see Argument (disambiguation).

In logic and philosophy, an argument is a series of statements typically used to persuade someone of something orto present reasons for accepting a conclusion.[1][2] The general form of an argument in a natural language is that ofpremises (typically in the form of propositions, statements or sentences) in support of a claim: the conclusion.[3][4][5]The structure of some arguments can also be set out in a formal language, and formally dened arguments can bemade independently of natural language arguments, as in math, logic and computer science.In a typical deductive argument, the premises are meant to provide a guarantee of the truth of the conclusion, whilein an inductive argument, they are thought to provide reasons supporting the conclusions probable truth.[6] Thestandards for evaluating non-deductive arguments may rest on dierent or additional criteria than truth, for example,the persuasiveness of so-called indispensability claims in transcendental arguments,[7] the quality of hypotheses inretroduction, or even the disclosure of new possibilities for thinking and acting.[8]

The standards and criteria used in evaluating arguments and their forms of reasoning are studied in logic.[9] Waysof formulating arguments eectively are studied in rhetoric (see also: argumentation theory). An argument in aformal language shows the logical form of the symbolically represented or natural language arguments obtained byits interpretations.

1.1 Formal and informalFurther information: Informal logic and Formal logic

Informal arguments as studied in informal logic, are presented in ordinary language and are intended for every-day discourse. Conversely, formal arguments are studied in formal logic (historically called symbolic logic, morecommonly referred to as mathematical logic today) and are expressed in a formal language. Informal logic may besaid to emphasize the study of argumentation, whereas formal logic emphasizes implication and inference. Informalarguments are sometimes implicit. That is, the rational structure the relationship of claims, premises, warrants,relations of implication, and conclusion is not always spelled out and immediately visible and must sometimes bemade explicit by analysis.

1.2 Standard typesThere are several kinds of arguments in logic, the best-known of which are deductive and inductive. Deductivearguments are sometimes referred to as truth-preserving arguments, because the truth of the conclusion followsgiven that of the premises. A deductive argument asserts that the truth of the conclusion is a logical consequence ofthe premises. An inductive argument, on the other hand, asserts that the truth of the conclusion is otherwise supportedby the premises. Each premise and the conclusion are truth bearers or truth-candidates, capable of being either trueor false (and not both). While statements in an argument are referred to as being either true or false, arguments arereferred to as being valid or invalid (see logical truth). A deductive argument is valid if and only if the truth of the

1

2 CHAPTER 1. ARGUMENT

conclusion is entailed by (is a logical consequence of) the premises, and its corresponding conditional is therefore alogical truth. A sound argument is a valid argument with true premises; a valid argument may well have false premisesunder a given interpretation, however, the truth value of a conclusion cannot be determined by an unsound argument.

1.3 DeductiveMain article: Deductive argument

A deductive argument is one that, if valid, has a conclusion that is entailed by its premises. In other words, the truth ofthe conclusion is a logical consequence of the premisesif the premises are true, then the conclusion must be true.It would be self-contradictory to assert the premises and deny the conclusion, because the negation of the conclusionis contradictory to the truth of the premises.

1.3.1 ValidityMain article: Validity

Deductive arguments may be either valid or invalid. If an argument is valid, it is a valid deduction, and if its premisesare true, the conclusion must be true: a valid argument cannot have true premises and a false conclusion.An argument is formally valid if and only if the denial of the conclusion is incompatible with accepting all thepremises.The validity of an argument depends, however, not on the actual truth or falsity of its premises and conclusion, butsolely on whether or not the argument has a valid logical form. The validity of an argument is not a guarantee ofthe truth of its conclusion. Under a given interpretation, a valid argument may have false premises that render itinconclusive: the conclusion of a valid argument with one or more false premises may be either true or false.Logic seeks to discover the valid forms, the forms that make arguments valid. A form of argument is valid if andonly if the conclusion is true under all interpretations of that argument in which the premises are true. Since thevalidity of an argument depends solely on its form, an argument can be shown to be invalid by showing that its formis invalid. This can be done by giving a counter example of the same form of argument with premises that are trueunder a given interpretation, but a conclusion that is false under that interpretation. In informal logic this is called acounter argument.The form of argument can be shown by the use of symbols. For each argument form, there is a correspondingstatement form, called a corresponding conditional, and an argument form is valid if and only its correspondingconditional is a logical truth. A statement form which is logically true is also said to be a valid statement form. Astatement form is a logical truth if it is true under all interpretations. A statement form can be shown to be a logicaltruth by either (a) showing that it is a tautology or (b) by means of a proof procedure.The corresponding conditional of a valid argument is a necessary truth (true in all possible worlds) and so the con-clusion necessarily follows from the premises, or follows of logical necessity. The conclusion of a valid argument isnot necessarily true, it depends on whether the premises are true. If the conclusion, itself, just so happens to be anecessary truth, it is so without regard to the premises.Some examples:

Some Greeks are logicians; therefore, some logicians are Greeks. Valid argument; it would be self-contradictoryto admit that some Greeks are logicians but deny that some (any) logicians are Greeks.

All Greeks are human and all humans are mortal; therefore, all Greeks are mortal. : Valid argument; if thepremises are true the conclusion must be true.

Some Greeks are logicians and some logicians are tiresome; therefore, some Greeks are tiresome. Invalid argu-ment: the tiresome logicians might all be Romans (for example).

Either we are all doomed or we are all saved; we are not all saved; therefore, we are all doomed. Valid argument;the premises entail the conclusion. (Remember that this does not mean the conclusion has to be true; it is onlytrue if the premises are true, which they may not be!)

1.4. INDUCTIVE 3

Some men are hawkers. Some hawkers are rich. Therefore, some men are rich. Invalid argument. This can beeasier seen by giving a counter-example with the same argument form: Some people are herbivores. Some herbivores are zebras. Therefore, some people are zebras. Invalidargument, as it is possible that the premises be true and the conclusion false.

In the above second to last case (Some men are hawkers...), the counter-example follows the same logical form asthe previous argument, (Premise 1: Some X are Y. Premise 2: Some Y are Z. Conclusion: Some X are Z.) inorder to demonstrate that whatever hawkers may be, they may or may not be rich, in consideration of the premisesas such. (See also, existential import).The forms of argument that render deductions valid are well-established, however some invalid arguments can also bepersuasive depending on their construction (inductive arguments, for example). (See also, formal fallacy and informalfallacy).

1.3.2 SoundnessMain article: Soundness

A sound argument is a valid argument whose conclusion follows from its premise(s), and the premise(s) of whichis/are true.

1.4 InductiveMain article: Inductive argument

Non-deductive logic is reasoning using arguments in which the premises support the conclusion but do not entailit. Forms of non-deductive logic include the statistical syllogism, which argues from generalizations true for themost part, and induction, a form of reasoning that makes generalizations based on individual instances. An inductiveargument is said to be cogent if and only if the truth of the arguments premises would render the truth of theconclusion probable (i.e., the argument is strong), and the arguments premises are, in fact, true. Cogency can beconsidered inductive logic's analogue to deductive logic's "soundness. Despite its name, mathematical induction isnot a form of inductive reasoning. The lack of deductive validity is known as the problem of induction.

1.5 Defeasible arguments and argumentation schemesIn modern argumentation theories, arguments are regarded as defeasible passages from premises to a conclusion.Defeasibility means that when additional information (new evidence or contrary arguments) is provided, the premisesmay be no longer lead to the conclusion (non-monotonic reasoning). This type of reasoning is referred to as defeasiblereasoning. For instance we consider the famous Tweedy example:

Tweedy is a bird.Birds generally y.Therefore, Tweedy (probably) ies.

This argument is reasonable and the premises support the conclusion unless additional information indicating that thecase is an exception comes in. If Tweedy is a penguin, the inference is no longer justied by the premise. Defeasiblearguments are based on generalizations that hold only in the majority of cases, but are subject to exceptions anddefaults. In order to represent and assess defeasible reasoning, it is necessary to combine the logical rules (governingthe acceptance of a conclusion based on the acceptance of its premises) with rules of material inference, governinghow a premise can support a given conclusion (whether it is reasonable or not to draw a specic conclusion froma specic description of a state of aairs). Argumentation schemes have been developed to describe and assessthe acceptability or the fallaciousness of defeasible arguments. Argumentation schemes are stereotypical patternsof inference, combining semantic-ontological relations with types of reasoning and logical axioms and representing

4 CHAPTER 1. ARGUMENT

the abstract structure of the most common types of natural arguments.[10] The argumentation schemes provided in(Walton, Reed & Macagno, 2008) describe tentatively the patterns of the most typical arguments. However, the twolevels of abstraction are not distinguished. For this reason, under the label of argumentation schemes fall indistinctlypatterns of reasoning such as the abductive, analogical, or inductive ones, and types of argument such as the onesfrom classication or cause to eect. A typical example is the argument from expert opinion, which has two premisesand a conclusion.[11]

Each scheme is associated to a set of critical questions, namely criteria for assessing dialectically the reasonablenessand acceptability of an argument. The matching critical questions are the standard ways of casting the argument intodoubt.If an expert says that a proposition is true, this provides a reason for tentatively accepting it, in the absence of strongerreasons to doubt it. But suppose that evidence of nancial gain suggests that the expert is biased, for example byevidence showing that he will gain nancially from his claim.

1.6 By analogyArgument by analogy may be thought of as argument from the particular to particular. An argument by analogy mayuse a particular truth in a premise to argue towards a similar particular truth in the conclusion. For example, if A.Plato was mortal, and B. Socrates was like Plato in other respects, then asserting that C. Socrates was mortal is anexample of argument by analogy because the reasoning employed in it proceeds from a particular truth in a premise(Plato was mortal) to a similar particular truth in the conclusion, namely that Socrates was mortal.[12]

1.7 Other kindsOther kinds of arguments may have dierent or additional standards of validity or justication. For example, CharlesTaylor writes that so-called transcendental arguments are made up of a chain of indispensability claims that attemptto show why something is necessarily true based on its connection to our experience,[13] while Nikolas Kompridishas suggested that there are two types of fallible arguments: one based on truth claims, and the other based on thetime-responsive disclosure of possibility (see world disclosure).[14] The late French philosopher Michel Foucault issaid to have been a prominent advocate of this latter form of philosophical argument.[15]

1.7.1 In informal logicArgument is an informal calculus, relating an eort to be performed or sum to be spent, to possible future gain, eithereconomic or moral. In informal logic, an argument is a connexion between

1. an individual action2. through which a generally accepted good is obtained.

Ex :

1. (a) You should marry Jane (individual action, individual decision)(b) because she has the same temper as you. (generally accepted wisdom that marriage is good in itself, and

it is generally accepted that people with the same character get along well).2. (a) You should not smoke (individual action, individual decision)

(b) because smoking is harmful (generally accepted wisdom that health is good).

The argument is neither a) advice nor b) moral or economical judgement, but the connection between the two. Anargument always uses the connective because. An argument is not an explanation. It does not connect two events,cause and eect, which already took place, but a possible individual action and its benecial outcome. An argumentis not a proof. A proof is a logical and cognitive concept; an argument is a praxeologic concept. A proof changesour knowledge; an argument compels us to act.

1.8. EXPLANATIONS 5

Logical status

Argument does not belong to logic, because it is connected to a real person, a real event, and a real eort to be made.

1. If you, John, will buy this stock, it will become twice as valuable in a year.2. If you, Mary, study dance, you will become a famous ballet dancer.

The value of the argument is connected to the immediate circumstances of the person spoken to. If, in the rstcase,(1) John has no money, or will die the next year, he will not be interested in buying the stock. If, in the secondcase (2) she is too heavy, or too old, she will not be interested in studying and becoming a dancer. The argument isnot logical, but protable.

1.7.2 World-disclosingMain article: World disclosure

World-disclosing arguments are a group of philosophical arguments that are said to employ a disclosive approach,to reveal features of a wider ontological or cultural-linguistic understanding a world, in a specically ontologicalsense in order to clarify or transform the background ofmeaning and logical space onwhich an argument implicitlydepends.[16]

1.8 ExplanationsMain article: Explanation

While arguments attempt to show that something was, is, will be, or should be the case, explanations try to show whyor how something is or will be. If Fred and Joe address the issue of whether or not Freds cat has eas, Joe may state:Fred, your cat has eas. Observe, the cat is scratching right now. Joe has made an argument that the cat has eas.However, if Joe asks Fred, Why is your cat scratching itself?" the explanation, "...because it has eas. providesunderstanding.Both the above argument and explanation require knowing the generalities that a) eas often cause itching, and b) thatone often scratches to relieve itching. The dierence is in the intent: an argument attempts to settle whether or notsome claim is true, and an explanation attempts to provide understanding of the event. Note, that by subsuming thespecic event (of Freds cat scratching) as an instance of the general rule that animals scratch themselves when theyhave eas, Joe will no longer wonder why Freds cat is scratching itself. Arguments address problems of believe,explanations address problems of understanding. Also note that in the argument above, the statement, Freds cat haseas is up for debate (i.e. is a claim), but in the explanation, the statement, Freds cat has eas is assumed to betrue (unquestioned at this time) and just needs explaining.[17]

Arguments and explanations largely resemble each other in rhetorical use. This is the cause of much diculty inthinking critically about claims. There are several reasons for this diculty.

People often are not themselves clear on whether they are arguing for or explaining something. The same types of words and phrases are used in presenting explanations and arguments. The terms 'explain' or 'explanation,' et cetera are frequently used in arguments. Explanations are often used within arguments and presented so as to serve as arguments.[18]

Likewise, "...arguments are essential to the process of justifying the validity of any explanation as there areoften multiple explanations for any given phenomenon.[17]

Explanations and arguments are often studied in the eld of Information Systems to help explain user acceptance ofknowledge-based systems. Certain argument types may t better with personality traits to enhance acceptance byindividuals.[19]

6 CHAPTER 1. ARGUMENT

1.9 Fallacies and nonargumentsMain article: Formal fallacy

Fallacies are types of argument or expressions which are held to be of an invalid form or contain errors in reasoning.There is not as yet any general theory of fallacy or strong agreement among researchers of their denition or potentialfor application but the term is broadly applicable as a label to certain examples of error, and also variously applied toambiguous candidates.[20]

In Logic types of fallacy are rmly described thus: First the premises and the conclusion must be statements, capableof being true or false. Secondly it must be asserted that the conclusion follows from the premises. In English thewords therefore, so, because and hence typically separate the premises from the conclusion of an argument, but thisis not necessarily so. Thus: Socrates is a man, all men are mortal therefore Socrates is mortal is clearly an argument(a valid one at that), because it is clear it is asserted that Socrates is mortal follows from the preceding statements.However I was thirsty and therefore I drank is NOT an argument, despite its appearance. It is not being claimed thatI drank is logically entailed by I was thirsty. The therefore in this sentence indicates for that reason not it follows that.

Elliptical arguments

Often an argument is invalid because there is a missing premisethe supply of which would render it valid. Speakersand writers will often leave out a strictly necessary premise in their reasonings if it is widely accepted and the writerdoes not wish to state the blindingly obvious. Example: All metals expand when heated, therefore iron will expandwhen heated. (Missing premise: iron is a metal). On the other hand, a seemingly valid argument may be found tolack a premise a 'hidden assumption' which if highlighted can show a fault in reasoning. Example: A witnessreasoned: Nobody came out the front door except the milkman; therefore the murderer must have left by the back door.(Hidden assumptions- the milkman was not the murderer, and the murderer has left by the front or back door).

1.10 See also Abductive reasoning Argument map Argumentation theory Argumentative dialogue Belief bias Boolean logic Deductive reasoning Defeasible reasoning Evidence Evidence-based policy Fallacy Dialectic Formal fallacy Inductive reasoning Informal fallacy Inquiry Practical arguments

1.11. NOTES 7

Soundness theorem Soundness Truth Validity

1.11 Notes[1] Argument, Internet Encyclopedia of Philosophy. In everyday life, we often use the word argument to mean a verbal

dispute or disagreement. This is not the way this word is usually used in philosophy. However, the two uses are related.Normally, when two people verbally disagree with each other, each person attempts to convince the other that his/herviewpoint is the right one. Unless he or she merely results to name calling or threats, he or she typically presents anargument for his or her position, in the sense described above. In philosophy, arguments are those statements a personmakes in the attempt to convince someone of something, or present reasons for accepting a given conclusion.

[2] Ralph H. Johnson, Manifest Rationality: A pragmatic theory of argument (New Jersey: Laurence Erlbaum, 2000), 46-49.

[3] Ralph H. Johnson, Manifest Rationality: A pragmatic theory of argument (New Jersey: Laurence Erlbaum, 2000), 46.

[4] The Cambridge Dictionary of Philosophy, 2nd Ed. CUM, 1995 Argument: a sequence of statements such that some ofthem (the premises) purport to give reason to accept another of them, the conclusion

[5] Stanford Enc. Phil., Classical Logic

[6] Deductive and Inductive Arguments, Internet Encyclopedia of Philosophy.

[7] hCharles Taylor, The Validity of Transcendental Arguments, Philosophical Arguments (Harvard, 1995), 20-33. "[Tran-scendental] arguments consist of a string of what one could call indispensability claims. They move from their startingpoints to their conclusions by showing that the condition stated in the conclusion is indispensable to the feature identied atthe start Thus we could spell out Kants transcendental deduction in the rst edition in three stages: experience must havean object, that is, be of something; for this it must be coherent; and to be coherent it must be shaped by the understandingthrough the categories.

[8] Kompridis, Nikolas (2006). World Disclosing Arguments?". Critique and Disclosure. Cambridge: MIT Press. pp. 116124. ISBN 0262277425.

[9] Argument, Internet Encyclopedia of Philosophy.

[10] Macagno, Fabrizio; Walton, Douglas (2015). Classifying the patterns of natural arguments. Philosophy & Rhetoric . 48(1): 2653.

[11] Walton, Douglas; Reed, Chris; Macagno, Fabrizio (2008). Argumentation Schemes. New York: Cambridge UniversityPress. p. 310.

[12] Shaw 1922: p. 74.

[13] Charles Taylor, The Validity of Transcendental Arguments, Philosophical Arguments (Harvard, 1995), 20-33.

[14] Nikolas Kompridis, Two Kinds of Fallibilism, Critique and Disclosure (Cambridge: MIT Press, 2006), 180-183.

[15] In addition, Foucault said of his own approach that My role ... is to show people that they are much freer than they feel,that people accept as truth, as evidence, some themes which have been built up at a certain moment during history, andthat this so-called evidence can be criticized and destroyed. He also wrote that he was engaged in the process of puttinghistorico-critical reection to the test of concrete practices I continue to think that this task requires work on our limits,that is, a patient labor giving form to our impatience for liberty. (emphasis added) Hubert Dreyfus, "Being and Power:Heidegger and Foucault" and Michel Foucault, What is Enlightenment?"

[16] Nikolas Kompridis, World Disclosing Arguments?" in Critique and Disclosure, Cambridge:MIT Press (2006), 118-121.

[17] JONATHANF. OSBORNE, ALEXIS PATTERSONSchool of Education, Stanford University, Stanford, CA 94305, USAReceived 27 August 2010; revised 22 November 2010; accepted 29 November 2010 DOI 10.1002/sce.20438 Publishedonline 23 May 2011 in Wiley Online Library (wileyonlinelibrary.com)

[18] Critical Thinking, Parker and Moore

8 CHAPTER 1. ARGUMENT

[19] Justin Scott Giboney, Susan Brown, and Jay F. Nunamaker Jr. (2012). User Acceptance of Knowledge-Based SystemRecommendations: Explanations, Arguments, and Fit 45th Annual Hawaii International Conference on System Sciences,Hawaii, January 58.

[20]

1.12 References Shaw, Warren Choate (1922). The Art of Debate. Allyn and Bacon. p. 74. Robert Audi, Epistemology, Routledge, 1998. Particularly relevant is Chapter 6, which explores the relationshipbetween knowledge, inference and argument.

J. L. Austin How to Do Things With Words, Oxford University Press, 1976. H. P. Grice, Logic and Conversation in The Logic of Grammar, Dickenson, 1975. Vincent F. Hendricks, Thought 2 Talk: A Crash Course in Reection and Expression, New York: AutomaticPress / VIP, 2005, ISBN 87-991013-7-8

R. A. DeMillo, R. J. Lipton and A. J. Perlis, Social Processes and Proofs of Theorems and Programs, Commu-nications of the ACM, Vol. 22, No. 5, 1979. A classic article on the social process of acceptance of proofs inmathematics.

Yu. Manin, A Course in Mathematical Logic, Springer Verlag, 1977. A mathematical view of logic. This bookis dierent from most books on mathematical logic in that it emphasizes the mathematics of logic, as opposedto the formal structure of logic.

Ch. Perelman and L. Olbrechts-Tyteca, The New Rhetoric, Notre Dame, 1970. This classic was originallypublished in French in 1958.

Henri Poincar, Science and Hypothesis, Dover Publications, 1952 Frans van Eemeren and Rob Grootendorst, Speech Acts in Argumentative Discussions, Foris Publications, 1984. K. R. Popper Objective Knowledge; An Evolutionary Approach, Oxford: Clarendon Press, 1972. L. S. Stebbing, A Modern Introduction to Logic, Methuen and Co., 1948. An account of logic that covers theclassic topics of logic and argument while carefully considering modern developments in logic.

Douglas Walton, Informal Logic: A Handbook for Critical Argumentation, Cambridge, 1998. Walton, Douglas; Christopher Reed; Fabrizio Macagno, Argumentation Schemes, New York: Cambridge Uni-versity Press, 2008.

Carlos Chesevar, Ana Maguitman and Ronald Loui, Logical Models of Argument, ACM Computing Surveys,vol. 32, num. 4, pp. 337383, 2000.

T. Edward Damer. Attacking Faulty Reasoning, 5th Edition, Wadsworth, 2005. ISBN 0-534-60516-8 Charles Arthur Willard, A Theory of Argumentation. 1989. Charles Arthur Willard, Argumentation and the Social Grounds of Knowledge. 1982.

1.13 Further reading Salmon, Wesley C. Logic. New Jersey: Prentice-Hall (1963). Library of Congress Catalog Card no. 63-10528. Aristotle, Prior and Posterior Analytics. Ed. and trans. John Warrington. London: Dent (1964) Mates, Benson. Elementary Logic. NewYork: OUP (1972). Library of Congress Catalog Card no. 74-166004. Mendelson, Elliot. Introduction to Mathematical Logic. New York: Van Nostran Reinholds Company (1964). Frege, Gottlob. The Foundations of Arithmetic. Evanston, IL: Northwestern University Press (1980). Martin, Brian. The Controversy Manual (Sparsns, Sweden: Irene Publishing, 2014).

1.14. EXTERNAL LINKS 9

1.14 External links Argument at PhilPapers Argument at the Indiana Philosophy Ontology Project Argument entry in the Internet Encyclopedia of Philosophy

Chapter 2

If and only if

I redirects here. For other uses, see IFF (disambiguation)."" redirects here. It is not to be confused with Bidirectional trac.

Logical symbols representing iIn logic and related elds such as mathematics and philosophy, if and only if (shortened i) is a biconditional logicalconnective between statements.In that it is biconditional, the connective can be likened to the standard material conditional (only if, equal to if... then) combined with its reverse (if); hence the name. The result is that the truth of either one of the connectedstatements requires the truth of the other (i.e. either both statements are true, or both are false). It is controversialwhether the connective thus dened is properly rendered by the English if and only if, with its pre-existing meaning.There is nothing to stop one from stipulating that we may read this connective as only if and if, although this maylead to confusion.In writing, phrases commonly used, with debatable propriety, as alternatives to P if and only if Q include Q isnecessary and sucient for P, P is equivalent (or materially equivalent) to Q (comparematerial implication), P preciselyif Q, P precisely (or exactly) when Q, P exactly in case Q, and P just in case Q. Many authors regard i as unsuitablein formal writing;[1] others use it freely.[2]

In logic formulae, logical symbols are used instead of these phrases; see the discussion of notation.

2.1 Denition

The truth table of p q is as follows:[3]

Note that it is equivalent to that produced by the XNOR gate, and opposite to that produced by the XOR gate.

2.2 Usage

2.2.1 Notation

The corresponding logical symbols are "", "", and "", and sometimes i. These are usually treated as equiv-alent. However, some texts of mathematical logic (particularly those on rst-order logic, rather than propositionallogic) make a distinction between these, in which the rst, , is used as a symbol in logic formulas, while is usedin reasoning about those logic formulas (e.g., in metalogic). In ukasiewicz's notation, it is the prex symbol 'E'.Another term for this logical connective is exclusive nor.

10

2.3. DISTINCTION FROM IF AND ONLY IF 11

2.2.2 ProofsIn most logical systems, one proves a statement of the form P i Q by proving if P, then Q and if Q, then P.Proving this pair of statements sometimes leads to a more natural proof, since there are not obvious conditions inwhich one would infer a biconditional directly. An alternative is to prove the disjunction "(P and Q) or (not-P andnot-Q)", which itself can be inferred directly from either of its disjunctsthat is, because i is truth-functional, Pi Q follows if P and Q have both been shown true, or both false.

2.2.3 Origin of iUsage of the abbreviation i rst appeared in print in John L. Kelley's 1955 bookGeneral Topology.[4] Its inventionis often credited to Paul Halmos, who wrote I invented 'i,' for 'if and only if'but I could never believe I was reallyits rst inventor.[5]

2.3 Distinction from if and only if1. Madison will eat the fruit if it is an apple. (equivalent to Only if Madison will eat the fruit, is it an

apple;" or Madison will eat the fruit fruit is an apple)This states simply that Madison will eat fruits that are apples. It does not, however, exclude thepossibility that Madison might also eat bananas or other types of fruit. All that is known for certainis that she will eat any and all apples that she happens upon. That the fruit is an apple is a sucientcondition for Madison to eat the fruit.

2. Madison will eat the fruit only if it is an apple. (equivalent to If Madison will eat the fruit, then it isan apple or Madison will eat the fruit fruit is an apple)

This states that the only fruit Madison will eat is an apple. It does not, however, exclude the possi-bility that Madison will refuse an apple if it is made available, in contrast with (1), which requiresMadison to eat any available apple. In this case, that a given fruit is an apple is a necessary conditionfor Madison to be eating it. It is not a sucient condition since Madison might not eat all the applesshe is given.

3. Madison will eat the fruit if and only if it is an apple (equivalent to Madison will eat the fruit fruitis an apple)

This statement makes it clear that Madison will eat all and only those fruits that are apples. Shewill not leave any apple uneaten, and she will not eat any other type of fruit. That a given fruit is anapple is both a necessary and a sucient condition for Madison to eat the fruit.

Suciency is the inverse of necessity. That is to say, given PQ (i.e. if P then Q), P would be a sucient conditionfor Q, and Q would be a necessary condition for P. Also, given PQ, it is true that QP (where is the negationoperator, i.e. not). This means that the relationship between P and Q, established by PQ, can be expressed in thefollowing, all equivalent, ways:

P is sucient for QQ is necessary for PQ is sucient for PP is necessary for Q

As an example, take (1), above, which states PQ, where P is the fruit in question is an apple and Q is Madisonwill eat the fruit in question. The following are four equivalent ways of expressing this very relationship:

If the fruit in question is an apple, then Madison will eat it.Only if Madison will eat the fruit in question, is it an apple.If Madison will not eat the fruit in question, then it is not an apple.Only if the fruit in question is not an apple, will Madison not eat it.

12 CHAPTER 2. IF AND ONLY IF

So we see that (2), above, can be restated in the form of if...then as If Madison will eat the fruit in question, then itis an apple"; taking this in conjunction with (1), we nd that (3) can be stated as If the fruit in question is an apple,then Madison will eat it; AND if Madison will eat the fruit, then it is an apple.

2.4 More general usageI is used outside the eld of logic, wherever logic is applied, especially in mathematical discussions. It has the samemeaning as above: it is an abbreviation for if and only if, indicating that one statement is both necessary and sucientfor the other. This is an example of mathematical jargon. (However, as noted above, if, rather than i, is more oftenused in statements of denition.)The elements of X are all and only the elements of Y is used to mean: for any z in the domain of discourse, z is inX if and only if z is in Y.

2.5 See also Covariance Logical biconditional Logical equality Necessary and sucient condition Polysyllogism

2.6 Footnotes[1] E.g. Daepp, Ulrich; Gorkin, Pamela (2011), Reading, Writing, and Proving: A Closer Look at Mathematics, Undergraduate

Texts in Mathematics, Springer, p. 52, ISBN 9781441994790, While it can be a real time-saver, we don't recommend itin formal writing.

[2] Rothwell, Edward J.; Cloud, Michael J. (2014), Engineering Writing by Design: Creating Formal Documents of LastingValue, CRC Press, p. 98, ISBN 9781482234312, It is common in mathematical writing.

[3] p q. Wolfram|Alpha

[4] General Topology, reissue ISBN 978-0-387-90125-1

[5] Nicholas J. Higham (1998). Handbook of writing for the mathematical sciences (2nd ed.). SIAM. p. 24. ISBN 978-0-89871-420-3.

2.7 External links Language Log: Just in Case Southern California Philosophy for philosophy graduate students: Just in Case

Chapter 3

Interpretation (logic)

For other uses, see Interpretation (disambiguation).

An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages usedin mathematics, logic, and theoretical computer science are dened in solely syntactic terms, and as such do not haveany meaning until they are given some interpretation. The general study of interpretations of formal languages iscalled formal semantics.The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and forthese there are standard ways of presenting an interpretation. In these contexts an interpretation is a function thatprovides the extension of symbols and strings of symbols of an object language. For example, an interpretationfunction could take the predicate T (for tall) and assign it the extension {a} (for Abraham Lincoln). Note thatall our interpretation does is assign the extension {a} to the non-logical constant T, and does not make a claim aboutwhether T is to stand for tall and 'a' for Abraham Lincoln. Nor does logical interpretation have anything to say aboutlogical connectives like 'and', 'or' and 'not'. Though wemay take these symbols to stand for certain things or concepts,this is not determined by the interpretation function.An interpretation often (but not always) provides a way to determine the truth values of sentences in a language. If agiven interpretation assigns the value True to a sentence or theory, the interpretation is called a model of that sentenceor theory.

3.1 Formal languages

Main article: Formal language

A formal language consists of a xed collection of sentences (also called words or formulas, depending on the context)composed from a xed set of letters or symbols. The inventory fromwhich these letters are taken is called the alphabetover which the language is dened. The essential feature of a formal language is that its syntax can be dened withoutreference to interpretation. We can determine that (P or Q) is a well-formed formula even without knowing whetherit is true or false.To distinguish the strings of symbols that are in a formal language from arbitrary strings of symbols, the former aresometimes called well-formed formul (w).

3.1.1 Example

A formal languageW can be dened with the alphabet = {4 , }, and with a word being inW if it begins with4 and is composed solely of the symbols4 and .A possible interpretation ofW could assign the decimal digit '1' to4 and '0' to . Then44 would denote 101under this interpretation ofW .

13

14 CHAPTER 3. INTERPRETATION (LOGIC)

3.1.2 Logical constantsIn the specic cases of propositional logic and predicate logic, the formal languages considered have alphabets thatare divided into two sets: the logical symbols (logical constants) and the non-logical symbols. The idea behind thisterminology is that logical symbols have the same meaning regardless of the subject matter being studied, whilenon-logical symbols change in meaning depending on the area of investigation.Logical constants are always given the same meaning by every interpretation of the standard kind, so that only themeanings of the non-logical symbols are changed. Logical constants include quantier symbols (all) and (some), symbols for logical connectives (and), (or), (not), parentheses and other grouping symbols,and (in many treatments) the equality symbol =.

3.2 General properties of truth-functional interpretationsMany of the commonly studied interpretations associate each sentence in a formal language with a single truth value,either True or False. These interpretations are called truth functional; they include the usual interpretations of propo-sitional and rst-order logic. The sentences that are made true by a particular assignment are said to be satised bythat assignment.No sentence can be made both true and false by the same interpretation, but it is possible that the truth value ofthe same sentence can be dierent under dierent interpretations. A sentence is consistent if it is true under at leastone interpretation; otherwise it is inconsistent. A sentence is said to be logically valid if it is satised by everyinterpretation (if is satised by every interpretation that satises then is said to be a logical consequence of ).

3.2.1 Logical connectivesSome of the logical symbols of a language (other than quantiers) are truth-functional connectives that represent truthfunctions functions that take truth values as arguments and return truth values as outputs (in other words, theseare operations on truth values of sentences).The truth-functional connectives enable compound sentences to be built up from simpler sentences. In this way, thetruth value of the compound sentence is dened as a certain truth function of the truth values of the simpler sentences.The connectives are usually taken to be logical constants, meaning that the meaning of the connectives is always thesame, independent of what interpretations are given to the other symbols in a formula.This is how we dene logical connectives in propositional logic:

is True i is False. ( & ) is True i is True and is True. ( ) is True i ( & ) is True. ( ) is True i ( is True is True). ($ ) is True i ( ) is True and ( ) is True.

So under a given interpretation of all the sentence letters and (i.e., after assigning a truth-value to each sentenceletter), we can determine the truth-values of all formulas that have them as constituents, as a function of the logicalconnectives. The following table shows how this kind of thing looks. The rst two columns show the truth-values ofthe sentence letters as determined by the four possible interpretations. The other columns show the truth-values offormulas built from these sentence letters, with truth-values determined recursively.

Now its easier to see what makes a formula logically valid. Take the formula F: ( ~). If our interpretationfunction makes True, then ~ is made False by the negation connective. Since the disjunct of F is True underthat interpretation, F is True. Now the only other possible interpretation of makes it False, and if so, ~ is madeTrue by the negation function. That would make F True again, since one of Fs disjuncts, ~, would be true under thisinterpretation. Since these two interpretations for F are the only possible logical interpretations, and since F comesout True for both, we say that it is logically valid or tautologous.

3.3. INTERPRETATION OF A THEORY 15

3.3 Interpretation of a theory

Main article: Theory (mathematical logic)

An interpretation of a theory is the relationship between a theory and some subject matter when there is a many-to-onecorrespondence between certain elementary statements of the theory, and certain statements related to the subjectmatter. If every elementary statement in the theory has a correspondent it is called a full interpretation, otherwise itis called a partial interpretation.[1]

3.4 Interpretations for propositional logic

The formal language for propositional logic consists of formulas built up from propositional symbols (also calledsentential symbols, sentential variables, and propositional variables) and logical connectives. The only non-logicalsymbols in a formal language for propositional logic are the propositional symbols, which are often denoted by capitalletters. To make the formal language precise, a specic set of propositional symbols must be xed.The standard kind of interpretation in this setting is a function that maps each propositional symbol to one of the truthvalues true and false. This function is known as a truth assignment or valuation function. In many presentations, it isliterally a truth value that is assigned, but some presentations assign truthbearers instead.For a language with n distinct propositional variables there are 2n distinct possible interpretations. For any particularvariable a, for example, there are 21=2 possible interpretations: 1) a is assigned T, or 2) a is assigned F. For the paira, b there are 22=4 possible interpretations: 1) both are assigned T, 2) both are assigned F, 3) a is assigned T and bis assigned F, or 4) a is assigned F and b is assigned T.Given any truth assignment for a set of propositional symbols, there is a unique extension to an interpretation for allthe propositional formulas built up from those variables. This extended interpretation is dened inductively, usingthe truth-table denitions of the logical connectives discussed above.

3.5 First-order logic

Unlike propositional logic, where every language is the same apart from a choice of a dierent set of propositionalvariables, there are many dierent rst-order languages. Each rst-order language is dened by a signature. Thesignature consists of a set of non-logical symbols and an identication of each of these symbols as a constant symbol,a function symbol, or a predicate symbol. In the case of function and predicate symbols, a natural number arity isalso assigned. The alphabet for the formal language consists of logical constants, the equality relation symbol =, allthe symbols from the signature, and an additional innite set of symbols known as variables.For example, in the language of rings, there are constant symbols 0 and 1, two binary function symbols + and , andno binary relation symbols. (Here the equality relation is taken as a logical constant.)Again, we might dene a rst-order language L, as consisting of individual symbols a, b, and c; predicate symbolsF,G, H, I and J; variables x,y,z; no function letters; no sentential symbols.

3.5.1 Formal languages for rst-order logic

Given a signature , the corresponding formal language is known as the set of -formulas. Each -formula is built upout of atomic formulas bymeans of logical connectives; atomic formulas are built from terms using predicate symbols.The formal denition of the set of -formulas proceeds in the other direction: rst, terms are assembled from theconstant and function symbols together with the variables. Then, terms can be combined into an atomic formulausing a predicate symbol (relation symbol) from the signature or the special predicate symbol "=" for equality (seethe section "Interpreting equality below). Finally, the formulas of the language are assembled from atomic formulasusing the logical connectives and quantiers.

16 CHAPTER 3. INTERPRETATION (LOGIC)

3.5.2 Interpretations of a rst-order languageTo ascribe meaning to all sentences of a rst-order language, the following information is needed.

A domain of discourse[2] D, usually required to be non-empty (see below). For every constant symbol, an element of D as its interpretation. For every n-ary function symbol, an n-ary function from D to D as its interpretation (that is, a function Dn D).

For every n-ary predicate symbol, an n-ary relation on D as its interpretation (that is, a subset of Dn).

An object carrying this information is known as a structure (of signature , or -structure, or L-structure), or as amodel.The information specied in the interpretation provides enough information to give a truth value to any atomic for-mula, after each of its free variables, if any, has been replaced by an element of the domain. The truth value ofan arbitrary sentence is then dened inductively using the T-schema, which is a denition of rst-order semanticsdeveloped by Alfred Tarski. The T-schema interprets the logical connectives using truth tables, as discussed above.Thus, for example, & is satised if and only if both and are satised.This leaves the issue of how to interpret formulas of the form x (x) and x (x). The domain of discourse formsthe range for these quantiers. The idea is that the sentence x (x) is true under an interpretation exactly whenevery substitution instance of (x), where x is replaced by some element of the domain, is satised. The formula x(x) is satised if there is at least one element d of the domain such that (d) is satised.Strictly speaking, a substitution instance such as the formula (d) mentioned above is not a formula in the originalformal language of , because d is an element of the domain. There are two ways of handling this technical issue. Therst is to pass to a larger language in which each element of the domain is named by a constant symbol. The second isto add to the interpretation a function that assigns each variable to an element of the domain. Then the T-schema canquantify over variations of the original interpretation in which this variable assignment function is changed, insteadof quantifying over substitution instances.Some authors also admit propositional variables in rst-order logic, which must then also be interpreted. A proposi-tional variable can stand on its own as an atomic formula. The interpretation of a propositional variable is one of thetwo truth values true and false.[3]

Because the rst-order interpretations described here are dened in set theory, they do not associate each predicatesymbol with a property [4](or relation), but rather with the extension of that property (or relation). In other words,these rst-order interpretations are extensional [5] not intensional.

3.5.3 Example of a rst-order interpretationAn example of interpretation I of the language L described above is as follows.

Domain: A chess set Individual constants: a: The white King b: The black Queen c: The white Kings pawn F(x): x is a piece G(x): x is a pawn H(x): x is black I(x): x is white J(x, y): x can capture y

In the interpretation I of L:

the following are true sentences: F(a), G(c), H(b), I(a) J(b, c), the following are false sentences: J(a, c), G(a).

3.5. FIRST-ORDER LOGIC 17

3.5.4 Non-empty domain requirement

As stated above, a rst-order interpretation is usually required to specify a nonempty set as the domain of discourse.The reason for this requirement is to guarantee that equivalences such as

( _ 9x )$ 9x( _ )

where x is not a free variable of , are logically valid. This equivalence holds in every interpretation with a nonemptydomain, but does not always hold when empty domains are permitted. For example, the equivalence

[8y(y = y) _ 9x(x = x)] 9x[8y(y = y) _ x = x]

fails in any structure with an empty domain. Thus the proof theory of rst-order logic becomes more complicatedwhen empty structures are permitted. However, the gain in allowing them is negligible, as both the intended inter-pretations and the interesting interpretations of the theories people study have non-empty domains.[6][7]

Empty relations do not cause any problem for rst-order interpretations, because there is no similar notion of passinga relation symbol across a logical connective, enlarging its scope in the process. Thus it is acceptable for relationsymbols to be interpreted as being identically false. However, the interpretation of a function symbol must alwaysassign a well-dened and total function to the symbol.

3.5.5 Interpreting equality

The equality relation is often treated specially in rst order logic and other predicate logics. There are two generalapproaches.The rst approach is to treat equality as no dierent than any other binary relation. In this case, if an equality symbolis included in the signature, it is usually necessary to add various axioms about equality to axiom systems (for example,the substitution axiom saying that if a = b and R(a) holds then R(b) holds as well). This approach to equality is mostuseful when studying signatures that do not include the equality relation, such as the signature for set theory or thesignature for second-order arithmetic in which there is only an equality relation for numbers, but not an equalityrelation for set of numbers.The second approach is to treat the equality relation symbol as a logical constant that must be interpreted by the realequality relation in any interpretation. An interpretation that interprets equality this way is known as a normal model,so this second approach is the same as only studying interpretations that happen to be normal models. The advantageof this approach is that the axioms related to equality are automatically satised by every normal model, and so theydo not need to be explicitly included in rst-order theories when equality is treated this way. This second approachis sometimes called rst order logic with equality, but many authors adopt it for the general study of rst-order logicwithout comment.There are a few other reasons to restrict study of rst-order logic to normal models. First, it is known that any rst-order interpretation in which equality is interpreted by an equivalence relation and satises the substitution axioms forequality can be cut down to an elementarily equivalent interpretation on a subset of the original domain. Thus there islittle additional generality in studying non-normal models. Second, if non-normal models are considered, then everyconsistent theory has an innite model; this aects the statements of results such as the LwenheimSkolem theorem,which are usually stated under the assumption that only normal models are considered.

3.5.6 Many-sorted rst-order logic

A generalization of rst order logic considers languages with more than one sort of variables. The idea is dierentsorts of variables represent dierent types of objects. Every sort of variable can be quantied; thus an interpretationfor a many-sorted language has a separate domain for each of the sorts of variables to range over (there is an innitecollection of variables of each of the dierent sorts). Function and relation symbols, in addition to having arities, arespecied so that each of their arguments must come from a certain sort.

18 CHAPTER 3. INTERPRETATION (LOGIC)

One example of many-sorted logic is for planar Euclidean geometry. There are two sorts; points and lines. There isan equality relation symbol for points, an equality relation symbol for lines, and a binary incidence relation E whichtakes one point variable and one line variable. The intended interpretation of this language has the point variablesrange over all points on the Euclidean plane, the line variable range over all lines on the plane, and the incidencerelation E(p,l) holds if and only if point p is on line l.

3.6 Higher-order predicate logicsA formal language for higher-order predicate logic looks much the same as a formal language for rst-order logic.The dierence is that there are now many dierent types of variables. Some variables correspond to elements of thedomain, as in rst-order logic. Other variables correspond to objects of higher type: subsets of the domain, functionsfrom the domain, functions that take a subset of the domain and return a function from the domain to subsets of thedomain, etc. All of these types of variables can be quantied.There are two kinds of interpretations commonly employed for higher-order logic. Full semantics require that, oncethe domain of discourse is satised, the higher-order variables range over all possible elements of the correct type(all subsets of the domain, all functions from the domain to itself, etc.). Thus the specication of a full interpretationis the same as the specication of a rst-order interpretation. Henkin semantics, which are essentially multi-sortedrst-order semantics, require the interpretation to specify a separate domain for each type of higher-order variable torange over. Thus an interpretation in Henkin semantics includes a domainD, a collection of subsets ofD, a collectionof functions from D to D, etc. The relationship between these two semantics is an important topic in higher orderlogic.

3.7 Non-classical interpretationsThe interpretations of propositional logic and predicate logic described above are not the only possible interpreta-tions. In particular, there are other types of interpretations that are used in the study of non-classical logic (such asintuitionistic logic), and in the study of modal logic.Interpretations used to study non-classical logic include topological models, Boolean valued models, and Kripkemodels. Modal logic is also studied using Kripke models.

3.8 Intended interpretationsMany formal languages are associated with a particular interpretation that is used to motivate them. For example, therst-order signature for set theory includes only one binary relation, , which is intended to represent set membership,and the domain of discourse in a rst-order theory of the natural numbers is intended to be the set of natural numbers.The intended interpretation is called the standard model (a term introduced by Abraham Robinson in 1960).[8] In thecontext of Peano arithmetic, it consists of the natural numbers with their ordinary arithmetical operations. All modelsthat are isomorphic to the one just given are also called standard; these models all satisfy the Peano axioms. Thereare also non-standard models of the (rst-order version of the) Peano axioms, which contain elements not correlatedwith any natural number.While the intended interpretation can have no explicit indication in the syntactical rules since these rules mustbe strictly formal the authors intention respecting interpretation naturally aects her choice of the formation andtransformation rules of the syntactical system. For example, she chooses primitive signs in such a way that certainconcepts can be expressed; she chooses sentential formulas in such a way that their counterparts in the intended inter-pretation can appear as meaningful declarative sentences; her choice of primitive sentences must meet the requirementthat these primitive sentences come out as true sentences in the interpretation; her rules of inference must be suchthat, if by one of these rules the sentence Ij is directly derivable from a sentence Ii , then Ii ! Ij turns out to be atrue sentence (under the customary interpretation of as meaning implication). These requirements ensure that allprovable sentences also come out to be true.[9]

Most formal systems have many more models than they were intended to have (the existence of non-standard modelsis an example). When we speak about 'models in empirical sciences, we mean, if we want reality to be a model of ourscience, to speak about an intended model. A model in the empirical sciences is an intended factually-true descriptive

3.9. OTHER CONCEPTS OF INTERPRETATION 19

interpretation (or in other contexts: a non-intended arbitrary interpretation used to clarify such an intended factually-true descriptive interpretation.) All models are interpretations that have the same domain of discourse as the intendedone, but other assignments for non-logical constants.[10]

3.8.1 ExampleGiven a simple formal system (we shall call this one FS 0 ) whose alphabet consists only of three symbols { ,F, } and whose formation rule for formulas is:

'Any string of symbols of FS 0 which is at least 6 symbols long, and which is not innitely long, is aformula of FS 0 . Nothing else is a formula of FS 0 .'

The single axiom schema of FS 0 is:

" F * * " (where " * " is a metasyntactic variable standing for a nite string of " s )

A formal proof can be constructed as follows:

(1) F (2) F (3) F

In this example the theorem produced " F " can be interpreted as meaning One plus threeequals four. A dierent interpretation would be to read it backwards as Four minus three equals one.[11]

3.9 Other concepts of interpretationThere are other uses of the term interpretation that are commonly used, which do not refer to the assignment ofmeanings to formal languages.In model theory, a structure A is said to interpret a structure B if there is a denable subset D of A, and denablerelations and functions on D, such that B is isomorphic to the structure with domain D and these functions andrelations. In some settings, it is not the domain D that is used, but rather D modulo an equivalence relation denablein A. For additional information, see Interpretation (model theory).A theory T is said to interpret another theory S if there is a nite extension by denitions T of T such that S iscontained in T .

3.10 See also Free variables and Name binding Herbrand interpretation Interpretation (model theory) Logical system Lwenheim-Skolem theorem Modal logic Model (abstract) Model theory Satisable Truth

20 CHAPTER 3. INTERPRETATION (LOGIC)

3.11 References[1] Curry, Haskell, Foundations of Mathematical Logic p.48

[2] Sometimes called the universe of disourse

[3] Mates, Benson (1972), Elementary Logic, Second Edition, New York: Oxford University Press, p. 56, ISBN 0-19-501491-X

[4] The extension of a property (also called an attribute) is a set of individuals, so a property is a unary relation. E.g. Theproperties yellow and prime are unary relations.

[5] see also Extension (predicate logic)

[6] Hailperin, Theodore (1953), Quantication theory and empty individual-domains, The Journal of Symbolic Logic (As-sociation for Symbolic Logic) 18 (3): 197200, doi:10.2307/2267402, JSTOR 2267402, MR 0057820

[7] Quine, W. V. (1954), Quantication and the empty domain, The Journal of Symbolic Logic (Association for SymbolicLogic) 19 (3): 177179, doi:10.2307/2268615, JSTOR 2268615, MR 0064715

[8] Roland Mller (2009). The Notion of a Model. In Anthonie Meijers. Philosophy of technology and engineering sciences.Handbook of the Philosophy of Science 9. Elsevier. ISBN 978-0-444-51667-1.

[9] Rudolf Carnap, Introduction to Symbolic Logic and its Applications

[10] The Concept and the Role of the Model in Mathematics and Natural and Social Sciences

[11] Georey Hunter, Metalogic

3.12 External links Stanford Enc. Phil: Classical Logic, 4. Semantics mathworld.wolfram.com: FormalLanguage mathworld.wolfram.com: Connective mathworld.wolfram.com: Interpretation mathworld.wolfram.com: Propositional Calculus mathworld.wolfram.com: First Order Logic

Chapter 4

Logic

This article is about reasoning and its study. For other uses, see Logic (disambiguation).

Philosophical Logic (from the Ancient Greek: , logike)[1] is the use and study of valid reasoning.[2][3] Thestudy of logic features most prominently in the subjects of philosophy, mathematics, and computer science.Logic was studied in several ancient civilizations, including India,[4] China,[5] Persia and Greece. In the West, logicwas established as a formal discipline by Aristotle, who gave it a fundamental place in philosophy. The study oflogic was part of the classical trivium, which also included grammar and rhetoric. Logic was further extended byAl-Farabi who categorized it into two separate groups (idea and proof). Later, Avicenna revived the study of logicand developed relationship between temporalis and the implication. In the East, logic was developed by Hindus,Buddhists and Jains.Logic is often divided into three parts: inductive reasoning, abductive reasoning, and deductive reasoning.

4.1 The study of logicThe concept of logical form is central to logic, it being held that the validity of an argument is determined by itslogical form, not by its content. Traditional Aristotelian syllogistic logic and modern symbolic logic are examples offormal logics.

Informal logic is the study of natural language arguments. The study of fallacies is an especially importantbranch of informal logic. The dialogues of Plato[6] are good examples of informal logic.

Formal logic is the study of inference with purely formal content. An inference possesses a purely formalcontent if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is notabout any particular thing or property. The works of Aristotle contain the earliest known formal study of logic.Modern formal logic follows and expands on Aristotle.[7] In many denitions of logic, logical inference andinference with purely formal content are the same. This does not render the notion of informal logic vacuous,because no formal logic captures all of the nuances of natural language.

Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference.[8][9]Symbolic logic is often divided into two branches: propositional logic and predicate logic.

Mathematical logic is an extension of symbolic logic into other areas, in particular to the study of modeltheory, proof theory, set theory, and recursion theory.

4.1.1 Logical formMain article: Logical form

Logic is generally considered formal when it analyzes and represents the form of any valid argument type. The formof an argument is displayed by representing its sentences in the formal grammar and symbolism of a logical language

21

22 CHAPTER 4. LOGIC

to make its content usable in formal inference. If one considers the notion of form too philosophically loaded, onecould say that formalizing simply means translating English sentences into the language of logic.This is called showing the logical form of the argument. It is necessary because indicative sentences of ordinarylanguage show a considerable variety of form and complexity that makes their use in inference impractical. It requires,rst, ignoring those grammatical features irrelevant to logic (such as gender and declension, if the argument is inLatin), replacing conjunctions irrelevant to logic (such as but) with logical conjunctions like and and replacingambiguous, or alternative logical expressions (any, every, etc.) with expressions of a standard type (such as all,or the universal quantier ).Second, certain parts of the sentence must be replaced with schematic letters. Thus, for example, the expressionall As are Bs shows the logical form common to the sentences all men are mortals, all cats are carnivores, allGreeks are philosophers, and so on.That the concept of form is fundamental to logic was already recognized in ancient times. Aristotle uses variableletters to represent valid inferences in Prior Analytics, leading Jan ukasiewicz to say that the introduction of variableswas one of Aristotles greatest inventions.[10] According to the followers of Aristotle (such as Ammonius), only thelogical principles stated in schematic terms belong to logic, not those given in concrete terms. The concrete termsman, mortal, etc., are analogous to the substitution values of the schematic placeholders A, B, C, which werecalled the matter (Greek hyle) of the inference.The fundamental dierence between modern formal logic and traditional, or Aristotelian logic, lies in their dieringanalysis of the logical form of the sentences they treat.

In the traditional view, the form of the sentence consists of (1) a subject (e.g., man) plus a sign of quantity(all or some or no); (2) the copula, which is of the form is or is not"; (3) a predicate (e.g., mortal).Thus: all men are mortal. The logical constants such as all, no and so on, plus sentential connectives suchas and and or were called syncategorematic terms (from the Greek kategorei to predicate, and syn together with). This is a xed scheme, where each judgment has an identied quantity and copula, determiningthe logical form of the sentence.

According to the modern view, the fundamental form of a simple sentence is given by a recursive schema,involving logical connectives, such as a quantier with its bound variable, which are joined by juxtaposition toother sentences, which in turn may have logical structure.

The modern view is more complex, since a single judgement of Aristotles system involves two or more logicalconnectives. For example, the sentence All men are mortal involves, in term logic, two non-logical terms isa man (hereM) and is mortal (here D): the sentence is given by the judgement A(M,D). In predicate logic,the sentence involves the same two non-logical concepts, here analyzed as m(x) and d(x) , and the sentenceis given by 8x:(m(x)! d(x)) , involving the logical connectives for universal quantication and implication.

But equally, the modern view is more powerful. Medieval logicians recognized the problem of multiple gen-erality, where Aristotelian logic is unable to satisfactorily render such sentences as Some guys have all theluck, because both quantities all and some may be relevant in an inference, but the xed scheme thatAristotle used allows only one to govern the inference. Just as linguists recognize recursive structure in naturallanguages, it appears that logic needs recursive structure.

4.1.2 Deductive and inductive reasoning, and abductive inference

Deductive reasoning concerns what follows necessarily from given premises (if a, then b). However, inductive reason-ingthe process of deriving a reliable inference from observationsis often included in the study of logic. Similarly,it is important to distinguish deductive validity and inductive validity (called strength). An inference is deductivelyvalid if and only if there is no possible situation in which all the premises are true but the conclusion false. Aninference is inductively strong if and only if its premises give some degree of probability to its conclusion.The notion of deductive validity can be rigorously stated for systems of formal logic in terms of the well-understoodnotions of semantics. Inductive validity on the other hand requires us to dene a reliable generalization of some set ofobservations. The task of providing this denition may be approached in various ways, some less formal than others;some of these denitions may use mathematical models of probability. For the most part this discussion of logic dealsonly with deductive logic.

4.2. HISTORY 23

Abduction[11] is a form of logical inference that goes from observation to a hypothesis that accounts for the reliabledata (observation) and seeks to explain relevant evidence. The American philosopher Charles Sanders Peirce (18391914) rst introduced the term as guessing.[12] Peirce said that to abduce a hypothetical explanation a from anobserved surprising circumstance b is to surmise that a may be true because then b would be a matter of course.[13]Thus, to abduce a from b involves determining that a is sucient (or nearly sucient), but not necessary, for b .

4.1.3 Consistency, validity, soundness, and completenessAmong the important properties that logical systems can have:

Consistency, which means that no theorem of the system contradicts another.[14]

Validity, which means that the systems rules of proof never allow a false inference from true premises. Alogical system has the property of soundness when the logical system has the property of validity and uses onlypremises that prove true (or, in the case of axioms, are true by denition).[14]

Completeness, of a logical system, which means that if a formula is true, it can be proven (if it is true, it is atheorem of the system).

Soundness, the term soundness has multiple separate meanings, which creates a bit of confusion throughoutthe literature. Most commonly, soundness refers to logical systems, which means that if some formula can beproven in a system, then it is true in the relevant model/structure (if A is a theorem, it is true). This is theconverse of completeness. A distinct, peripheral use of soundness refers to arguments, which means that thepremises of a valid argument are true in the actual world.

Some logical systems do not have all four properties. As an example, Kurt Gdel's incompleteness theorems showthat suciently complex formal systems of arithmetic cannot be consistent and complete;[9] however, rst-orderpredicate logics not extended by specic axioms to be arithmetic formal systems with equality can be complete andconsistent.[15]

4.1.4 Rival conceptions of logicMain article: Denitions of logic

Logic arose (see below) from a concern with correctness of argumentation. Modern logicians usually wish to ensurethat logic studies just those arguments that arise from appropriately general forms of inference. For example, ThomasHofweber writes in the Stanford Encyclopedia of Philosophy that logic does not, however, cover good reasoning asa whole. That is the job of the theory of rationality. Rather it deals with inferences whose validity can be traced backto the formal features of the representations that are involved in that inference, be they linguistic, mental, or otherrepresentations.[16]

By contrast, Immanuel Kant argued that logic should be conceived as the science of judgement, an idea taken up inGottlob Frege's logical and philosophical work. But Freges work is ambiguous in the sense that it is both concernedwith the laws of thought as well as with the laws of truth, i.e. it both treats logic in the context of a theory of themind, and treats logic as the study of abstract formal structures.

4.2 HistoryMain article: History of logicIn Europe, logic was rst developed by Aristotle.[17] Aristotelian logic became widely accepted in science and mathe-matics and remained in wide use in theWest until the early 19th century.[18] Aristotles system of logic was responsiblefor the introduction of hypothetical syllogism,[19] temporal modal logic,[20][21] and inductive logic,[22] as well as in-uential terms such as terms, predicables, syllogisms and propositions. In Europe during the later medieval period,major eorts were made to show that Aristotles ideas were compatible with Christian faith. During the High MiddleAges, logic became a main focus of philosophers, who would engage in critical logical analyses of philosophicalarguments, often using variations of the methodology of scholasticism. In 1323, William of Ockham's inuential

24 CHAPTER 4. LOGIC

Aristotle, 384322 BCE.

Summa Logicae was released. By the 18th century, the structured approach to arguments had degenerated and fallenout of favour, as depicted in Holberg's satirical play Erasmus Montanus.The Chinese logical philosopher Gongsun Long (c. 325250 BCE) proposed the paradox One and one cannotbecome two, since neither becomes two.[23] In China, the tradition of scholarly investigation into logic, however,was repressed by the Qin dynasty following the legalist philosophy of Han Feizi.In India, innovations in the scholastic school, called Nyaya, continued from ancient times into the early 18th century

4.3. TYPES OF LOGIC 25

with the Navya-Nyaya school. By the 16th century, it developed theories resembling modern logic, such as GottlobFrege's distinction between sense and reference of proper names and his denition of number, as well as thetheory of restrictive conditions for universals anticipating some of the developments in modern set theory.[24] Since1824, Indian logic attracted the attention of many Western scholars, and has had an inuence on important 19th-century logicians such as Charles Babbage, Augustus De Morgan, and George Boole.[25] In the 20th century, Westernphilosophers like Stanislaw Schayer and Klaus Glasho have explored Indian logic more extensively.The syllogistic logic developed by Aristotle predominated in theWest until the mid-19th century, when interest in thefoundations of mathematics stimulated the development of symbolic logic (now called mathematical logic). In 1854,George Boole published An Investigation of the Laws of Thought on Which are Founded the Mathematical Theoriesof Logic and Probabilities, introducing symbolic logic and the principles of what is now known as Boolean logic.In 1879, Gottlob Frege published Begrisschrift, which inaugurated modern logic with the invention of quantiernotation. From 1910 to 1913, Alfred North Whitehead and Bertrand Russell published Principia Mathematica[8]on the foundations of mathematics, attempting to derive mathematical truths from axioms and inference rules insymbolic logic. In 1931, Gdel raised serious problems with the foundationalist program and logic ceased to focuson such issues.The development of logic since Frege, Russell, and Wittgenstein had a profound inuence on the practice of philos-ophy and the perceived nature of philosophical problems (see Analytic philosophy), and Philosophy of mathematics.Logic, especially sentential logic, is implemented in computer logic circuits and is fundamental to computer science.Logic is commonly taught by university philosophy departments, often as a compulsory discipline.

4.3 Types of logic

4.3.1 Syllogistic logic

Main article: Aristotelian logic

The Organon was Aristotle's body of work on logic, with the Prior Analytics constituting the rst explicit work informal logic, introducing the syllogistic.[26] The parts of syllogistic logic, also known by the name term logic, arethe analysis of the judgements into propositions consisting of two terms that are related by one of a xed number ofrelations, and the expression of inferences by means of syllogisms that consist of two propositions sharing a commonterm as premise, and a conclusion that is a proposition involving the two unrelated terms from the premises.Aristotles work was regarded in classical times and from medieval times in Europe and the Middle East as the verypicture of a fully worked out system. However, it was not alone: the Stoics proposed a system of propositional logicthat was studied by medieval logicians. Also, the problem of multiple generality was recognized in medieval times.Nonetheless, problems with syllogistic logic were not seen as being in need of revolutionary solutions.Today, some academics claim that Aristotles system is generally seen as having little more than historical value(though there is some current interest in extending term logics), regarded as made obsolete by the advent of propo-sitional logic and the predicate calculus. Others use Aristotle in argumentation theory to help develop and criticallyquestion argumentation schemes that are used in articial intelligence and legal arguments.

4.3.2 Propositional logic (sentential logic)

Main article: Propositional calculus

A propositional calculus or logic (also a sentential calculus) is a formal system in which formulae representing propo-sitions can be formed by combining atomic propositions using logical connectives, and in which a system of formalproof rules establishes certain formulae as theorems.

4.3.3 Predicate logic

Main article: Predicate logic

26 CHAPTER 4. LOGIC

Predicate logic is the generic term for symbolic formal systems such as rst-order logic, second-order logic, many-sorted logic, and innitary logic.Predicate logic provides an account of quantiers general enough to express a wide set of arguments occurring innatural language. Aristotelian syllogistic logic species a small number of forms that the relevant part of the involvedjudgements may take. Predicate logic allows sentences to be analysed into subject and argument in several additionalwaysallowing predicate logic to solve the problem of multiple generality that had perplexed medieval logicians.The development of predicate logic is usually attributed to Gottlob Frege, who is also credited as one of the foundersof analytical philosophy, but the formulation of predicate logic most often used today is the rst-order logic presentedin Principles of Mathematical Logic by David Hilbert and Wilhelm Ackermann in 1928. The analytical generalityof predicate logic allowed the formalization of mathematics, drove the investigation of set theory, and allowed thedevelopment of Alfred Tarski's approach to model theory. It provides the foundation of modern mathematical logic.Freges original system of predicate logic was second-order, rather than rst-order. Second-order logic is mostprominently defended (against the criticism of Willard Van Orman Quine and others) by George Boolos and StewartShapiro.

4.3.4 Modal logic

Main article: Modal logic

In languages, modality deals with the phenomenon that sub-parts of a sentence may have their semantics modiedby special verbs or modal particles. For example, "We go to the games" can be modied to give "We should go to thegames", and "We can go to the games" and perhaps "We will go to the games". More abstractly, we might say thatmodality aects the circumstances in which we take an assertion to be satis