Material conditionalFrom Wikipedia, the free encyclopedia

Contents

1 Entailment (pragmatics) 11.1 Types of entailment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Implication graph 22.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Implicational hierarchy 43.1 Phonology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4 Implicational propositional calculus 64.1 Virtual completeness as an operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.2 Axiom system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.3 Basic properties of derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.4 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.4.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.5 The BernaysTarski axiom system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.6 Testing whether a formula of the implicational propositional calculus is a tautology . . . . . . . . . 94.7 Adding an axiom schema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.8 An alternative axiomatization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5 Implicature 135.1 Types of implicature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.1.1 Conversational implicature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.1.2 Conventional implicature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.2 Implicature vs entailment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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5.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 Implicit 166.1 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.3 Other uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7 Linguistic universal 177.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.2 In semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

8 Material conditional 208.1 Denitions of the material conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

8.1.1 As a truth function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.1.2 As a formal connective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

8.2 Formal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.3 Philosophical problems with material conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8.4.1 Conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

9 Material implication 259.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

10 Modus ponens 2610.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.2 Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.3 Justication via truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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10.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Chapter 1

Entailment (pragmatics)

In pragmatics (linguistics), entailment is the relationship between two sentences where the truth of one (A) requiresthe truth of the other (B).For example, the sentence (A) The president was assassinated. entails (B) The president is dead. Notice also that if(B) is false, then (A) must necessarily be false. To show entailment, we must show that (A) being true forces (B) tobe true, or, equivalently, that (B) being false forces (A) to be false.Entailment diers from implicature (in their denitions for pragmatics), where the truth of one (A) suggests thetruth of the other (B), but does not require it. For example, the sentence (A) Mary had a baby and (B) got marriedimplicates that (A) she had a baby before (B) the wedding, but this is cancellable by adding not necessarily in thatorder. Entailments are not cancellable.Entailment also diers from presupposition in that in presupposition, the truth of what one is presupposing is takenfor granted. A simple test to dierentiate presupposition from entailment is negation. For example, both The king ofFrance is ill and The king of France is not ill presuppose that there is a king of France. However The president wasnot assassinated no longer entails The president is dead (nor its opposite, as the president could have died in anotherway). In this case, presupposition remains under negation, but entailment does not.

1.1 Types of entailmentThere are three types of entailment: formal or logical entailment, analytic entailment, synthetic entailment.

1.2 See also Compound question Downward entailing Loaded question

1.3 References

1.4 Further reading Entailment Regimes in SPARQL 1.1

1

Chapter 2

Implication graph

~x0

~x3

~x1~x5

x6

x5

~x6

~x4

~x2

x2

x4

x1

x3

x0

An implication graph representing the 2-satisability instance (x0_x2)^(x0_:x3)^(x1_:x3)^(x1_:x4)^(x2_:x4)^(x0_:x5)^(x1_:x5)^(x2_:x5)^(x3_x6)^(x4_x6)^(x5_x6):

In mathematical logic, an implication graph is a skew-symmetric directed graph G(V, E) composed of vertex setV and directed edge set E. Each vertex in V represents the truth status of a Boolean literal, and each directed edgefrom vertex u to vertex v represents the material implication If the literal u is true then the literal v is also true.Implication graphs were originally used for analyzing complex Boolean expressions.

2

2.1. APPLICATIONS 3

2.1 ApplicationsA 2-satisability instance in conjunctive normal form can be transformed into an implication graph by replacingeach of its disjunctions by a pair of implications. An instance is satisable if and only if no literal and its negationbelong to the same strongly connected component of its implication graph; this characterization can be used to solve2-satisability instances in linear time.[1]

2.2 References[1] Aspvall, Bengt; Plass, Michael F.; Tarjan, Robert E. (1979). A linear-time algorithm for testing the truth of certain

quantied boolean formulas. Information Processing Letters 8 (3): 121123. doi:10.1016/0020-0190(79)90002-4.

Chapter 3

Implicational hierarchy

Implicational hierarchy, in linguistics, is a chain of implicational universals. A set of chained universals is schemat-ically shown as in (1):(1) A < B < C < DIt can be reformulated in the following way: If a language has property D, then it also has properties A, B, and C;if a language has a property C, then it also has properties A and B, etc. In other words, the implicational hierarchydenes the possible combinations of properties A, B, C, and D as listed in matrix (2):Implicational hierarchies are a useful tool in capturing linguistic generalizations pertaining the dierent componentsof the language. They are found in all subelds of grammar.

3.1 Phonology(3) is an example of an implicational hierarchy concerning the distribution of nasal phonemes across languages, whichconcerns dental/alveolar, bilabial, and palatal voiced nasals, respectively:(3) /n/ < /m/ < //This hierarchy denes the following possible combinations of dental/alveolar, bilabial, and palatal voiced nasals inthe phoneme inventory of a language:(4)In other words, the hierarchy implies that there are no languages with // but without /m/ and /n/, or with // and /m/but without /n/.

3.2 MorphologyNumber marking provides an example of implicational hierarchies in morphology.(5) Number: singular < plural < dual < trial / paucalOn the one hand, the hierarchy implies that no language distinguishes a trial unless having a dual, and no languagehas dual without a plural. On the other hand, the hierarchy provides implications for the morphological marking: ifthe plural is coded with a certain number of morphemes, then the dual is coded with at least as many morphemes.

3.3 SyntaxImplicational hierarchies also play a role in syntactic phenomena. For instance, in some languages (e.g. Tangut) thetransitive verb agrees not with a subject, or the object, but with the syntactic argument which is higher on the personhierarchy.

4

3.4. BIBLIOGRAPHY 5

(5) Person: rst < second < third

See also: animacy.

3.4 Bibliography Comrie, B. (1989). Language universals and linguistic typology: Syntax and morphology. Oxford: Blackwell,2nd edn.

Croft, W. (1990). Typology and universals. Cambridge: Cambridge UP. Whaley, L.J. (1997). Introduction to typology: The unity and diversity of language. Newbury Park: Sage.

Chapter 4

Implicational propositional calculus

In mathematical logic, the implicational propositional calculus is a version of classical propositional calculus whichuses only one connective, called implication or conditional. In formulas, this binary operation is indicated by im-plies, if ..., then ..., "", "!", etc..

4.1 Virtual completeness as an operatorImplication alone is not functionally complete as a logical operator because one cannot form all other two-valuedtruth functions from it. However, if one has a propositional formula which is known to be false and uses that as if itwere a nullary connective for falsity, then one can dene all other truth functions. So implication is virtually completeas an operator. If P,Q, and F are propositions and F is known to be false, then:

P is equivalent to P F P Q is equivalent to (P (Q F)) F P Q is equivalent to (P Q) Q P Q is equivalent to ((P Q) ((Q P) F)) F

More generally, since the above operators are known to be functionally complete, it follows that any truth functioncan be expressed in terms of "" and "F", if we have a proposition F which is known to be false.It is worth noting that F is not denable from and arbitrary sentence variables: any formula constructed from and propositional variables must receive the value true when all of its variables are evaluated to true. It follows as acorollary that {} is not functionally complete. It cannot, for example, be used to dene the two-place truth functionthat always returns false.

4.2 Axiom system Axiom schema 1 is P (Q P). Axiom schema 2 is (P (Q R)) ((P Q) (P R)). Axiom schema 3 (Peirces law) is ((P Q) P) P. The one non-nullary rule of inference (modus ponens) is: from P and P Q infer Q.

Where in each case, P, Q, and R may be replaced by any formulas which contain only "" as a connective. If is aset of formulas and A a formula, then ` Ameans that A is derivable using the axioms and rules above and formulasfrom as additional hypotheses.ukasiewicz (1948) found an axiom system for the implicational calculus, which replaces the schemas 13 abovewith a single schema

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4.3. BASIC PROPERTIES OF DERIVATION 7

((P Q) R) ((R P) (S P)).

He also argued that there is no shorter axiom system.

4.3 Basic properties of derivationSince all axioms and rules of the calculus are schemata, derivation is closed under substitution:

If ` A; then () ` (A);

where is any substitution (of formulas using only implication).The implicational propositional calculus also satises the deduction theorem:

If ; A ` B , then ` A! B:

As explained in the deduction theorem article, this holds for any axiomatic extension of the system containing axiomschemas 1 and 2 above and modus ponens.

4.4 CompletenessThe implicational propositional calculus is semantically complete with respect to the usual two-valued semantics ofclassical propositional logic. That is, if is a set of implicational formulas, and A is an implicational formula entailedby , then ` A .

4.4.1 ProofA proof of the completeness theorem is outlined below. First, using the compactness theorem and the deductiontheorem, we may reduce the completeness theorem to its special case with empty , i.e., we only need to show thatevery tautology is derivable in the system.The proof is similar to completeness of full propositional logic, but it also uses the following idea to overcome thefunctional incompleteness of implication. If A and F are formulas, then A F is equivalent to (A*) F, whereA* is the result of replacing in A all, some, or none of the occurrences of F by falsity. Similarly, (A F) F isequivalent to A* F. So under some conditions, one can use them as substitutes for saying A* is false or A* is truerespectively.We rst observe some basic facts about derivability:

Indeed, we can derive A (B C) using Axiom 1, and then derive A C by modus ponens(twice) from Ax. 2.

This follows from (1) by the deduction theorem.

If we further assume C B, we can derive A B using (1), then we derive C by modusponens. This shows A ! C; (A ! B) ! C;C ! B ` C , and the deduction theoremgives A! C; (A! B)! C ` (C ! B)! C . We apply Ax. 3 to obtain (3).

8 CHAPTER 4. IMPLICATIONAL PROPOSITIONAL CALCULUS

Let F be an arbitrary xed formula. For any formula A, we dene A0 = (A F) and A1 = ((A F) F). Let usconsider only formulas in propositional variables p1, ..., pn. We claim that for every formula A in these variables andevery truth assignment e,

We prove (4) by induction on A. The base case A = pi is trivial. Let A = (B C). We distinguish three cases:

1. e(C) = 1. Then also e(A) = 1. We have

(C ! F )! F ` ((B ! C)! F )! F

by applying (2) twice to the axiom C (B C). Since we have derived (C F) F by theinduction hypothesis, we can infer ((B C) F) F.

2. e(B) = 0. Then again e(A) = 1. The deduction theorem applied to (3) gives

B ! F ` ((B ! C)! F )! F:

Since we have derived B F by the induction hypothesis, we can infer ((B C) F) F.

3. e(B) = 1 and e(C) = 0. Then e(A) = 0. We have

(B ! F )! F;C ! F;B ! C ` B ! F (1) by` F ponens, modus by

thus (B ! F )! F;C ! F ` (B ! C)! F by the deduction theorem. We have derived (BF) F and C F by the induction hypothesis, hence we can infer (B C) F. This completesthe proof of (4).

Now let A be a tautology in variables p1, ..., pn. We will prove by reverse induction on k = n,...,0 that for everyassignment e,

The base case k = n is a special case of (4). Assume that (5) holds for k + 1, we will show it for k. By applyingdeduction theorem to the induction hypothesis, we obtain

pe(p1)1 ; : : : ; p

e(pk)k ` (pk+1 ! F )! A1;

pe(p1)1 ; : : : ; p

e(pk)k ` ((pk+1 ! F )! F )! A1;

by rst setting e(pk) = 0 and second setting e(pk) = 1. From this we derive (5) using (3).For k = 0 we obtain that the formula A1, i.e., (A F) F, is provable without assumptions. Recall that F was anarbitrary formula, thus we can choose F = A, which gives us provability of the formula (A A) A. Since A Ais provable by the deduction theorem, we can infer A.This proof is constructive. That is, given a tautology, one could actually follow the instructions and create a proofof it from the axioms. However, the length of such a proof increases exponentially with the number of propositionalvariables in the tautology, hence it is not a practical method for any but the very shortest tautologies.

4.5. THE BERNAYSTARSKI AXIOM SYSTEM 9

4.5 The BernaysTarski axiom systemTheBernaysTarski axiom system is often used. In particular, ukasiewiczs paper derives the BernaysTarski axiomsfrom ukasiewiczs sole axiom as a means of showing its completeness.It diers from the axiom schemas above by replacing axiom schema 2, (P(QR))((PQ)(PR)), with

Axiom schema 2': (PQ)((QR)(PR))

which is called hypothetical syllogism. This makes derivation of the deduction meta-theorem a little more dicult,but it can still be done.We show that from P(QR) and PQ one can derive PR. This fact can be used in lieu of axiom schema 2 toget the meta-theorem.

1. P(QR) given2. PQ given3. (PQ)((QR))(PR)) ax 2'4. (QR)(PR) mp 2,35. (P(QR))(((QR)(PR))(P(PR))) ax 2'6. ((QR)(PR))(P(PR)) mp 1,57. P(PR) mp 4,68. (P(PR))(((PR)R)(PR)) ax 2'9. ((PR)R)(PR) mp 7,810. (((PR)R)(PR))(PR) ax 311. PR mp 9,10 qed

4.6 Testing whether a formula of the implicational propositional calculusis a tautology

Main articles: Tautology (logic) Ecient verication and the Boolean satisability problem and Boolean satisa-bility problem Algorithms for solving SAT

In this case, a useful technique is to presume that the formula is not a tautology and attempt to nd a valuation whichmakes it false. If one succeeds, then it is indeed not a tautology. If one fails, then it is a tautology.Example of a non-tautology:Suppose [(AB)((CA)E)]([F((CD)E)][(AF)(DE)]) is false.Then (AB)((CA)E) is true; F((CD)E) is true; AF is true; D is true; and E is false.Since D is true, CD is true. So the truth of F((CD)E) is equivalent to the truth of FE.Then since E is false and FE is true, we get that F is false.Since AF is true, A is false. Thus AB is true and (CA)E is true.CA is false, so C is true.The value of B does not matter, so we can arbitrarily choose it to be true.Summing up, the valuationwhich setsB,C andD to be true andA,E andF to be false will make [(AB)((CA)E)]([F((CD)E)][(AF)(DE)])false. So it is not a tautology.Example of a tautology:

10 CHAPTER 4. IMPLICATIONAL PROPOSITIONAL CALCULUS

Suppose ((AB)C)((CA)(DA)) is false.Then (AB)C is true; CA is true; D is true; and A is false.Since A is false, AB is true. So C is true. Thus A must be true, contradicting the fact that it is false.Thus there is no valuation which makes ((AB)C)((CA)(DA)) false. Consequently, it is a tautology.

4.7 Adding an axiom schemaWhat would happen if another axiom schema were added to those listed above? There are two cases: (1) it is atautology; or (2) it is not a tautology.If it is a tautology, then the set of theorems remains the set of tautologies as before. However, in some cases it maybe possible to nd signicantly shorter proofs for theorems. Nevertheless, the minimum length of proofs of theoremswill remain unbounded, that is, for any natural number n there will still be theorems which cannot be proved in n orfewer steps.If the new axiom schema is not a tautology, then every formula becomes a theorem (which makes the concept ofa theorem useless in this case). What is more, there is then an upper bound on the minimum length of a proof ofevery formula, because there is a common method for proving every formula. For example, suppose the new axiomschema were ((BC)C)B. Then ((A(AA))(AA))A is an instance (one of the new axioms) and also nota tautology. But [((A(AA))(AA))A]A is a tautology and thus a theorem due to the old axioms (using thecompleteness result above). Applying modus ponens, we get that A is a theorem of the extended system. Then allone has to do to prove any formula is to replace A by the desired formula throughout the proof of A. This proof willhave the same number of steps as the proof of A.

4.8 An alternative axiomatizationThe axioms listed above primarily work through the deductionmetatheorem to arrive at completeness. Here is anotheraxiom system which aims directly at completeness without going through the deduction metatheorem.First we have axiom schemas which are designed to eciently prove the subset of tautologies which contain only onepropositional variable.

aa 1: AA aa 2: (AB)(A(CB)) aa 3: A((BC)((AB)C)) aa 4: A(BA)

The proof of each such tautology would begin with two parts (hypothesis and conclusion) which are the same. Theninsert additional hypotheses between them. Then insert additional tautological hypotheses (which are true even whenthe sole variable is false) into the original hypothesis. Then add more hypotheses outside (on the left). This procedurewill quickly give every tautology containing only one variable. (The symbol "" in each axiom schema indicates wherethe conclusion used in the completeness proof begins. It is merely a comment, not a part of the formula.)Consider any formula which may contain A, B, C1, ..., Cn and ends with A as its nal conclusion. Then we take

aa 5: ()

as an axiom schema where is the result of replacing B by A throughout and is the result of replacing B by(AA) throughout . This is a schema for axiom schemas since there are two level of substitution: in the rst issubstituted (with variations); in the second, any of the variables (including both A and B) may be replaced by arbitraryformulas of the implicational propositional calculus. This schema allows one to prove tautologies with more than onevariable by considering the case when B is false and the case when B is true .If the variable which is the nal conclusion of a formula takes the value true, then the whole formula takes the valuetrue regardless of the values of the other variables. Consequently if A is true, then , , and () are

4.8. AN ALTERNATIVE AXIOMATIZATION 11

all true. So without loss of generality, we may assume that A is false. Notice that is a tautology if and only if both and are tautologies. But while has n+2 distinct variables, and both have n+1. So the question ofwhether a formula is a tautology has been reduced to the question of whether certain formulas with one variable eachare all tautologies. Also notice that () is a tautology regardless of whether is, because if is false theneither or will be false depending on whether B is false or true.Examples:Deriving Peirces law

1. [((PP)P)P]([((P(PP))P)P][((PQ)P)P]) aa 5

2. PP aa 1

3. (PP)((PP)(((PP)P)P)) aa 3

4. (PP)(((PP)P)P) mp 2,3

5. ((PP)P)P mp 2,4

6. [((P(PP))P)P][((PQ)P)P] mp 5,1

7. P(PP) aa 4

8. (P(PP))((PP)(((P(PP))P)P)) aa 3

9. (PP)(((P(PP))P)P) mp 7,8

10. ((P(PP))P)P mp 2,9

11. ((PQ)P)P mp 10,6 qed

Deriving ukasiewicz' sole axiom

1. [((PQ)P)((PP)(SP))]([((PQ)(PP))(((PP)P)(SP))][((PQ)R)((RP)(SP))])aa 5

2. [((PP)P)((PP)(SP))]([((P(PP))P)((PP)(SP))][((PQ)P)((PP)(SP))])aa 5

3. P(SP) aa 4

4. (P(SP))(P((PP)(SP))) aa 2

5. P((PP)(SP)) mp 3,4

6. PP aa 1

7. (PP)((P((PP)(SP)))[((PP)P)((PP)(SP))]) aa 3

8. (P((PP)(SP)))[((PP)P)((PP)(SP))] mp 6,7

9. ((PP)P)((PP)(SP)) mp 5,8

10. [((P(PP))P)((PP)(SP))][((PQ)P)((PP)(SP))] mp 9,2

11. P(PP) aa 4

12. (P(PP))((P((PP)(SP)))[((P(PP))P)((PP)(SP))]) aa 3

13. (P((PP)(SP)))[((P(PP))P)((PP)(SP))] mp 11,12

14. ((P(PP))P)((PP)(SP)) mp 5,13

15. ((PQ)P)((PP)(SP)) mp 14,10

16. [((PQ)(PP))(((PP)P)(SP))][((PQ)R)((RP)(SP))] mp 15,1

12 CHAPTER 4. IMPLICATIONAL PROPOSITIONAL CALCULUS

17. (PP)((P(SP))[((PP)P)(SP)]) aa 3

18. (P(SP))[((PP)P)(SP)] mp 6,1719. ((PP)P)(SP) mp 3,18

20. (((PP)P)(SP))[((PQ)(PP))(((PP)P)(SP))] aa 421. ((PQ)(PP))(((PP)P)(SP)) mp 19,20

22. ((PQ)R)((RP)(SP)) mp 21,16 qed

Using a truth table to verify ukasiewicz' sole axiom would require consideration of 16=24 cases since it contains 4distinct variables. In this derivation, we were able to restrict consideration to merely 3 cases: R is false and Q is false,R is false and Q is true, and R is true. However because we are working within the formal system of logic (instead ofoutside it, informally), each case required much more eort.

4.9 See also Deduction theorem List of logic systems#Implicational propositional calculus Peirces law Propositional calculus Tautology (logic) Truth table Valuation (logic)

4.10 References Mendelson, Elliot (1997) Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall. ukasiewicz, Jan (1948) The shortest axiom of the implicational calculus of propositions, Proc. Royal IrishAcademy, vol. 52, sec. A, no. 3, pp. 2533.

Chapter 5

Implicature

Implicature is a technical term in the pragmatics subeld of linguistics, coined by H. P. Grice, which refers to whatis suggested in an utterance, even though neither expressed nor strictly implied (that is, entailed) by the utterance.[1]For example, the sentence "Mary had a baby and got married" strongly suggests that Mary had the baby before thewedding, but the sentence would still be strictly true if Mary had her baby after she got married. Further, if we add thequalication " not necessarily in that order" to the original sentence, then the implicature is cancelled even thoughthe meaning of the original sentence is not altered.Implicature is an alternative to "implication, which has additional meanings in logic and informal language.

5.1 Types of implicature

5.1.1 Conversational implicaturePaul Grice identied three types of general conversational implicatures:1. The speaker deliberately outs a conversational maxim to convey an additional meaning not expressed literally.For instance, a speaker responds to the question How did you like the guest lecturer?" with the following utterance:

Well, Im sure he was speaking English.

If the speaker is assumed to be following the cooperative principle,[2] in spite of outing the Maxim of Quantity,then the utterance must have an additional nonliteral meaning, such as: The content of the lecturers speech wasconfusing.2. The speakers desire to fulll two conicting maxims results in his or her outing one maxim to invoke the other.For instance, a speaker responds to the question Where is John?" with the following utterance:

Hes either in the cafeteria or in his oce.

In this case, the Maxim of Quantity and the Maxim of Quality are in conict. A cooperative speaker does not want tobe ambiguous but also does not want to give false information by giving a specic answer in spite of his uncertainty.By outing the Maxim of Quantity, the speaker invokes the Maxim of Quality, leading to the implicature that thespeaker does not have the evidence to give a specic location where he believes John is.3. The speaker invokes a maxim as a basis for interpreting the utterance. In the following exchange:

Do you know where I can get some gas?Theres a gas station around the corner.

The second speaker invokes the Maxim of Relevance, resulting in the implicature that the gas station is open andone can probably get gas there.

13

14 CHAPTER 5. IMPLICATURE

Scalar implicature

According to Grice (1975), another form of conversational implicature is also known as a scalar implicature. Thisconcerns the conventional uses of words like all or some in conversation.

I ate some of the pie.

This sentence implies I did not eat all of the pie. While the statement I ate some pie is still true if the entire piewas eaten, the conventional meaning of the word some and the implicature generated by the statement is not all.

5.1.2 Conventional implicatureConventional implicature is independent of the cooperative principle and its four maxims. A statement always carriesits conventional implicature.

Donovan is poor but happy.

This sentence implies poverty and happiness are not compatible but in spite of this Donovan is still happy. Theconventional interpretation of the word but will always create the implicature of a sense of contrast. So Donovanis poor but happy will always necessarily imply Surprisingly Donovan is happy in spite of being poor.

5.2 Implicature vs entailmentThis can be contrasted with cases of entailment. For example, the statement The president was assassinated notonly suggests that The president is dead is true, but requires that it be true. The rst sentence could not be true ifthe second were not true; if the president were not dead, then whatever it is that happened to him would not havecounted as a (successful) assassination. Similarly, unlike implicatures, entailments cannot be cancelled; there is noqualication that one could add to The president was assassinated which would cause it to cease entailing Thepresident is dead while also preserving the meaning of the rst sentence.

5.3 See also Allofunctional implicature Cooperative principle Gricean maxims Entailment, or implication, in logic Entailment (pragmatics) Explicature Indirect speech act Intrinsic and extrinsic properties Presupposition

5.4 References[1] Blackburn 1996, p. 189.

[2] Kordi 1991, pp. 8992.

5.5. BIBLIOGRAPHY 15

5.5 Bibliography Blackburn, Simon (1996). implicature, The Oxford Dictionary of Philosophy, Oxford, pp. 188-89. P. Cole (1975) The synchronic and diachronic status of conversational implicature. In Syntax and Semantics,3: Speech Acts (New York: Academic Press) ed. P. Cole & J. L. Morgan, pp. 257288. ISBN 0-12-785424-X.

A. Davison (1975) Indirect speech acts and what to do with them. ibid, pp. 143184. G. M. Green (1975) How to get people to do things with words. ibid, pp. 107141. New York: AcademicPress

H. P. Grice (1975) Logic and conversation. ibid. Reprinted in Studies in the Way of Words, ed. H. P. Grice,pp. 2240. Cambridge, MA: Harvard University Press (1989) ISBN 0-674-85270-2.

Michael Hancher (1978) Grices Implicature and Literary Interpretation: Background and Preface Twen-tieth Annual Meeting Midwest Modern Language Association

Kordi, Snjeana (1991). Konverzacijske implikature [Conversational implicatures]. Suvremena lingvistika(in Serbo-Croatian) 17 (31-32): 8796. ISSN 0586-0296. Archived from the original (PDF) on 2 September2012. Retrieved 6 September 2012.

John Searle (1975) Indirect speech acts. ibid. Reprinted in Pragmatics: A Reader, ed. S. Davis, pp. 265277.Oxford: Oxford University Press. (1991) ISBN 0-19-505898-4.

5.6 Further reading Kent, Bach (2006). The Top 10 Misconceptions about Implicature (PDF). in: Birner, B.; Ward, G. AFestschrift for Larry Horn. Amsterdam: John Benjamins.

5.7 External links Implicature in the Stanford Encyclopedia of Philosophy The Top 10 Misconceptions about Implicature by Kent Bach (2005)

Chapter 6

Implicit

Implicit may refer to:

6.1 MathematicsA function dened by an equation in several variables or the equation dening this function, as in

Implicit function Implicit function theorem Implicit curve Implicit surface Implicit dierential equation

6.2 Computer science Implicit type conversion

6.3 Other uses Implicit solvation Implicit Association Test Implicit learning Implicit memory Implicit and explicit atheism

6.4 See also Implication (disambiguation)

16

Chapter 7

Linguistic universal

A linguistic universal is a pattern that occurs systematically across natural languages, potentially true for all of them.For example, All languages have nouns and verbs, or If a language is spoken, it has consonants and vowels. Researchin this area of linguistics is closely tied to the study of linguistic typology, and intends to reveal generalizations acrosslanguages, likely tied to cognition, perception, or other abilities of the mind. The eld was largely pioneered by thelinguist Joseph Greenberg, who derived a set of forty-ve basic universals, mostly dealing with syntax, from a studyof some thirty languages.

7.1 TerminologyLinguists distinguish between two kinds of universals: absolute (opposite: statistical, often called tendencies) andimplicational (opposite non-implicational). Absolute universals apply to every known language and are quite few innumber; an example is All languages have pronouns. An implicational universal applies to languages with a particularfeature that is always accompanied by another feature, such as If a language has trial grammatical number, it alsohas dual grammatical number, while non-implicational universals just state the existence (or non-existence) of oneparticular feature.Also in contrast to absolute universals are tendencies, statements that may not be true for all languages, but never-theless are far too common to be the result of chance.[1] They also have implicational and non-implicational forms.An example of the latter would be The vast majority of languages have nasal consonants.[2] However, most ten-dencies, like their universal counterparts, are implicational. For example, With overwhelmingly greater-than-chancefrequency, languages with normal SOV order are postpositional. Strictly speaking, a tendency is not a kind of univer-sal, but exceptions to most statements called universals can be found. For example, Latin is an SOV language withprepositions. Often it turns out that these exceptional languages are undergoing a shift from one type of language toanother. In the case of Latin, its descendant Romance languages switched to SVO, which is a much more commonorder among prepositional languages.Universals may also be bidirectional or unidirectional. In a bidirectional universal two features each imply theexistence of each other. For example, languages with postpositions usually have SOV order, and likewise SOVlanguages usually have postpositions. The implication works both ways, and thus the universal is bidirectional. Bycontrast, in a unidirectional universal the implication works only one way. Languages that place relative clauses beforethe noun they modify again usually have SOV order, so pre-nominal relative clauses imply SOV. On the other hand,SOV languages worldwide show little preference for pre-nominal relative clauses, and thus SOV implies little aboutthe order of relative clauses. As the implication works only one way, the proposed universal is a unidirectional one.Linguistic universals in syntax are sometimes held up as evidence for universal grammar (although epistemologicalarguments are more common). Other explanations for linguistic universals have been proposed, for example, thatlinguistic universals tend to be properties of language that aid communication. If a language were to lack one of theseproperties, it has been argued, it would probably soon evolve into a language having that property.[3]

Michael Halliday has argued for a distinction between descriptive and theoretical categories in resolving the matterof the existence of linguistic universals, a distinction he takes from J.R. Firth and Louis Hjelmslev. He argues thattheoretical categories, and their inter-relations construe an abstract model of language...; they are interlocking andmutually dening. Descriptive categories, by contrast, are those set up to describe particular languages. He argues

17

18 CHAPTER 7. LINGUISTIC UNIVERSAL

that When people ask about 'universals, they usually mean descriptive categories that are assumed to be found in alllanguages. The problem is there is no mechanism for deciding how much alike descriptive categories from dierentlanguages have to be before they are said to be 'the same thing'" [4]

7.2 In semanticsIn the domain of semantics, research into linguistic universals has taken place in a number of ways. Some linguists,starting with Leibniz, have pursued the search for a hypothetic irreducible semantic core of all languages. A modernvariant of this approach can be found in the Natural Semantic Metalanguage of Wierzbicka and associates.[5] Otherlines of research suggest cross-linguistic tendencies to use body part terms metaphorically as adpositions,[6] or tenden-cies to have morphologically simple words for cognitively salient concepts.[7] The human body, being a physiologicaluniversal, provides an ideal domain for research into semantic and lexical universals. In a seminal study, Cecil H.Brown (1976) proposed a number of universals in the semantics of body part terminology, including the following:in any language, there will be distinct terms for BODY, HEAD, ARM, EYES, NOSE, andMOUTH; if there is a distinctterm for FOOT, there will be a distinct term for HAND; similarly, if there are terms for INDIVIDUAL TOES, thenthere are terms for INDIVIDUAL FINGERS. Subsequent research has shown that most of these features have to beconsidered cross-linguistic tendencies rather than true universals. Several languages, for example Tidore and KuukThaayorre, lack a general term meaning 'body'. On the basis of such data it has been argued that the highest level inthe partonomy of body part terms would be the word for 'person'.[8]

7.3 See also Greenbergs linguistic universals Cultural universal

7.4 Notes[1] Dryer (1998)

[2] Lushootseed and Rotokas are examples of the rare languages which truly lack nasal consonants as normal speech sounds.

[3] Daniel everett: Language the cultural tool

[4] Halliday, M.A.K. 2002. A personal perspective. In On Grammar, Volume 1 in the Collected Works of M.A.K. Halliday.London and New York: Continuumm p12.

[5] see for example Goddard & Wierzbicka (1994) and Goddard (2002).

[6] Heine (1997)

[7] Rosch et al. (1976)

[8] Wilkins (1993), Eneld et al. 2006:17.

7.5 References Brown, Cecil H. (1976) General principles of human anatomical partonomy and speculations on the growthof partonomic nomenclature. American Ethnologist 3, no. 3, Folk Biology, pp. 400424

Comrie, Bernard (1981) Language Universals and Linguistic Typology. Chicago: University of Chicago Press. Croft, W. (2002). Typology and Universals. Cambridge: Cambridge UP. 2nd ed. ISBN 0-521-00499-3 Dryer, Matthew S. (1998) Why Statistical Universals are Better Than Absolute Universals Chicago LinguisticSociety 33: The Panels, pp. 123145.

7.6. EXTERNAL LINKS 19

Eneld, Nick J. & Asifa Majid & Miriam van Staden (2006) 'Cross-linguistic categorisation of the body:Introduction' (special issue of Language Sciences).

Ferguson, Charles A. (1968) 'Historical background of universals research'. In: Greenberg, Ferguson, &Moravcsik, Universals of human languages, pp. 731.

Goddard, Cli and Wierzbicka, Anna (eds.). 1994. Semantic and Lexical Universals - Theory and EmpiricalFindings. Amsterdam/Philadelphia: John Benjamins.

Goddard, Cli (2002) The search for the shared semantic core of all languages. In Goddard & Wierzbicka(eds.) Meaning and Universal Grammar - Theory and Empirical Findings volume 1, pp. 540, Amster-dam/Philadelphia: John Benjamins.

Greenberg, Joseph H. (ed.) (1963) Universals of Language. Cambridge, Mass.: MIT Press. Greenberg, Joseph H. (ed.) (1978a) Universals of Human Language Vol. 4: Syntax. Stanford, California:Stanford University Press.

Greenberg, Joseph H. (ed.) (1978b) Universals of Human Language Vol. 3: Word Structure. Stanford, Cali-fornia: Stanford University Press.

Heine, Bernd (1997) Cognitive Foundations of Grammar. New York/Oxford: Oxford University Press. Song, Jae Jung (2001) Linguistic Typology: Morphology and Syntax. Harlow, UK: Pearson Education (Long-man).

Song, Jae Jung (ed.) (2011) Oxford Handbook of Linguistic Typology. Oxford: Oxford University Press. Rosch, E. &Mervis, C.B. &Gray,W.D.& Johnson, D.M.&Boyes-Braem, P. (1976) 'Basic Objects In NaturalCategories, Cognitive Psychology 8-3, 382-439.

Wilkins, David P. (1993) From part to person: natural tendencies of semantic change and the search forcognates, Working paper No. 23, Cognitive Anthropology Research Group at the Max Planck Institute forPsycholinguistics.

7.6 External links Some Universals of Grammar with Particular Reference to the Order of Meaningful Elements by Joseph H.Greenberg

The Universals Archive by the University of Konstanz

Chapter 8

Material conditional

Logical conditional redirects here. For other related meanings, see Conditional statement.Not to be confused with material inference.Thematerial conditional (also known as "material implication", "material consequence", or simply "implication",

Venn diagram of A! B .If a member of the set described by this diagram (the red areas) is a member of A , it is in the intersection of A and B , and ittherefore is also in B .

"implies" or "conditional") is a logical connective (or a binary operator) that is often symbolized by a forward ar-row "". The material conditional is used to form statements of the form "pq" (termed a conditional statement)which is read as if p then q or p only if q and conventionally compared to the English construction If...then....But unlike the English construction, the material conditional statement "pq" does not specify a causal relationshipbetween p and q and is to be understood to mean if p is true, then q is also true such that the statement "pq"is false only when p is true and q is false.[1] Intuitively, consider that a given p being true and q being false wouldprove an if p is true, q is always also true statement false, even when the if p then q does not represent a causal

20

8.1. DEFINITIONS OF THE MATERIAL CONDITIONAL 21

relationship between p and q. Instead, the statement describes p and q as each only being true when the other istrue, and makes no claims that p causes q. However, note that such a general and informal way of thinking aboutthe material conditional is not always acceptable, as will be discussed. As such, the material conditional is also to bedistinguished from logical consequence.The material conditional is also symbolized using:

1. p q (Although this symbol may be used for the superset symbol in set theory.);2. p ) q (Although this symbol is often used for logical consequence (i.e. logical implication) rather than for

material conditional.)

With respect to the material conditionals above, p is termed the antecedent, and q the consequent of the conditional.Conditional statements may be nested such that either or both of the antecedent or the consequent may themselvesbe conditional statements. In the example "(pq) (rs)" both the antecedent and the consequent are conditionalstatements.In classical logic p! q is logically equivalent to :(p ^ :q) and by De Morgans Law logically equivalent to :p _ q.[2] Whereas, in minimal logic (and therefore also intuitionistic logic) p! q only logically entails :(p^:q) ; and inintuitionistic logic (but not minimal logic) :p _ q entails p! q .

8.1 Denitions of the material conditionalLogicians have many dierent views on the nature of material implication and approaches to explain its sense.[3]

8.1.1 As a truth functionIn classical logic, the compound pq is logically equivalent to the negative compound: not both p and not q. Thusthe compound pq is false if and only if both p is true and q is false. By the same stroke, pq is true if and only ifeither p is false or q is true (or both). Thus is a function from pairs of truth values of the components p, q to truthvalues of the compound pq, whose truth value is entirely a function of the truth values of the components. Hence,this interpretation is called truth-functional. The compound pq is logically equivalent also to pq (either not p, orq (or both)), and to qp (if not q then not p). But it is not equivalent to pq, which is equivalent to qp.

Truth table

The truth table associated with the material conditional pq is identical to that of pq and is also denoted by Cpq.It is as follows:It may also be useful to note that in Boolean algebra, true and false can be denoted as 1 and 0 respectively with anequivalent table.

8.1.2 As a formal connectiveThe material conditional can be considered as a symbol of a formal theory, taken as a set of sentences, satisfying allthe classical inferences involving , in particular the following characteristic rules:

1. Modus ponens;2. Conditional proof;3. Classical contraposition;4. Classical reductio ad absurdum.

Unlike the truth-functional one, this approach to logical connectives permits the examination of structurally identi-cal propositional forms in various logical systems, where somewhat dierent properties may be demonstrated. Forexample, in intuitionistic logic which rejects proofs by contraposition as valid rules of inference, (p q) p qis not a propositional theorem, but the material conditional is used to dene negation.

22 CHAPTER 8. MATERIAL CONDITIONAL

8.2 Formal propertiesWhen studying logic formally, the material conditional is distinguished from the semantic consequence relation j= .We say A j= B if every interpretation that makes A true also makes B true. However, there is a close relationshipbetween the two in most logics, including classical logic. For example, the following principles hold:

If j= then ? j= ('1 ^ ^ 'n ! ) for some '1; : : : ; 'n 2 . (This is a particular form of thededuction theorem. In words, it says that if models this means that can be deduced just from somesubset of the theorems in .)

The converse of the above

Both! and j= are monotonic; i.e., if j= then [ j= , and if '! then (' ^ )! for any ,. (In terms of structural rules, this is often referred to as weakening or thinning.)

These principles do not hold in all logics, however. Obviously they do not hold in non-monotonic logics, nor do theyhold in relevance logics.Other properties of implication (the following expressions are always true, for any logical values of variables):

distributivity: (s! (p! q))! ((s! p)! (s! q))

transitivity: (a! b)! ((b! c)! (a! c))

reexivity: a! a

totality: (a! b) _ (b! a)

truth preserving: The interpretation under which all variables are assigned a truth value of 'true' produces atruth value of 'true' as a result of material implication.

commutativity of antecedents: (a! (b! c)) (b! (a! c))

Note that a ! (b ! c) is logically equivalent to (a ^ b) ! c ; this property is sometimes called un/currying.Because of these properties, it is convenient to adopt a right-associative notation for where a ! b ! c denotesa! (b! c) .Comparison of Boolean truth tables shows that a! b is equivalent to :a _ b , and one is an equivalent replacementfor the other in classical logic. See material implication (rule of inference).

8.3 Philosophical problems with material conditionalOutside of mathematics, it is a matter of some controversy as to whether the truth function for material implica-tion provides an adequate treatment of conditional statements in English (a sentence in the indicative mood with aconditional clause attached, i.e., an indicative conditional, or false-to-fact sentences in the subjunctive mood, i.e., acounterfactual conditional).[4] That is to say, critics argue that in some non-mathematical cases, the truth value ofa compound statement, if p then q", is not adequately determined by the truth values of p and q.[4] Examples ofnon-truth-functional statements include: "q because p", "p before q" and it is possible that p".[4] [Of] the sixteenpossible truth-functions of A and B, material implication is the only serious candidate. First, it is uncontroversial thatwhen A is true and B is false, If A, B" is false. A basic rule of inference is modus ponens: from If A, B" and A, wecan infer B. If it were possible to have A true, B false and If A, B" true, this inference would be invalid. Second, it isuncontroversial that If A, B" is sometimes true when A and B are respectively (true, true), or (false, true), or (false,false) Non-truth-functional accounts agree that If A, B" is false when A is true and B is false; and they agree thatthe conditional is sometimes true for the other three combinations of truth-values for the components; but they denythat the conditional is always true in each of these three cases. Some agree with the truth-functionalist that when Aand B are both true, If A, B" must be true. Some do not, demanding a further relation between the facts that A andthat B.[4]

8.4. SEE ALSO 23

The truth-functional theory of the conditional was integral to Frege's new logic (1879). It was takenup enthusiastically by Russell (who called it material implication), Wittgenstein in the Tractatus, andthe logical positivists, and it is now found in every logic text. It is the rst theory of conditionals whichstudents encounter. Typically, it does not strike students as obviously correct. It is logics rst surprise.Yet, as the textbooks testify, it does a creditable job in many circumstances. And it has many defenders.It is a strikingly simple theory: If A, B" is false when A is true and B is false. In all other cases, If A,B" is true. It is thus equivalent to "~(A&~B)" and to "~A or B". "A B" has, by stipulation, these truthconditions.

Dorothy Edgington, The Stanford Encyclopedia of Philosophy, Conditionals[4]

The meaning of the material conditional can sometimes be used in the natural language English if condition thenconsequence" construction (a kind of conditional sentence), where condition and consequence are to be lled withEnglish sentences. However, this construction also implies a reasonable connection between the condition (protasis)and consequence (apodosis) (see Connexive logic).The material conditional can yield some unexpected truths when expressed in natural language. For example, anymaterial conditional statement with a false antecedent is true (see vacuous truth). So the statement if 2 is odd then 2is even is true. Similarly, any material conditional with a true consequent is true. So the statement if I have a pennyin my pocket then Paris is in France is always true, regardless of whether or not there is a penny in my pocket. Theseproblems are known as the paradoxes of material implication, though they are not really paradoxes in the strict sense;that is, they do not elicit logical contradictions. These unexpected truths arise because speakers of English (and othernatural languages) are tempted to equivocate between the material conditional and the indicative conditional, or otherconditional statements, like the counterfactual conditional and the material biconditional. It is not surprising that arigorously dened truth-functional operator does not correspond exactly to all notions of implication or otherwiseexpressed by 'if...then...' sentences in English (or their equivalents in other natural languages). For an overview ofsome the various analyses, formal and informal, of conditionals, see the References section below.

8.4 See also

8.4.1 Conditionals Counterfactual conditional Indicative conditional Corresponding conditional Strict conditional

8.5 References[1] Magnus, P.D (January 6, 2012). forallx: An Introduction to Formal Logic (PDF). Creative Commons. p. 25. Retrieved

28 May 2013.

[2] Teller, Paul (January 10, 1989). A Modern Formal Logic Primer: Sentence Logic Volume 1 (PDF). Prentice Hall. p.54. Retrieved 28 May 2013.

[3] Clarke, Matthew C. (March 1996). A Comparison of Techniques for Introducing Material Implication. Cornell Univer-sity. Retrieved March 4, 2012.

[4] Edgington, Dorothy (2008). Edward N. Zalta, ed. Conditionals. The Stanford Encyclopedia of Philosophy (Winter 2008ed.).

8.6 Further reading Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations, 1st edition, KluwerAcademic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.

24 CHAPTER 8. MATERIAL CONDITIONAL

Edgington, Dorothy (2001), Conditionals, in Lou Goble (ed.), The Blackwell Guide to Philosophical Logic,Blackwell.

Quine, W.V. (1982), Methods of Logic, (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, HarvardUniversity Press, Cambridge, MA.

Stalnaker, Robert, Indicative Conditionals, Philosophia, 5 (1975): 269286.

8.7 External links Conditionals entry by Edgington, Dorothy in the Stanford Encyclopedia of Philosophy

Chapter 9

Material implication

Material implication may refer to:

Material conditional, a logical connective Material implication (rule of inference), a rule of replacement for some propositional logic

9.1 See also Implication (disambiguation) Conditional statement (disambiguation)

25

Chapter 10

Modus ponens

In propositional logic, modus ponendo ponens (Latin for the way that arms by arming"; often abbreviated toMP ormodus ponens[1][2][3][4]) or implication elimination is a valid, simple argument form and rule of inference.[5]It can be summarized as "P implies Q; P is asserted to be true, so therefore Q must be true. The history of modusponens goes back to antiquity.[6]

While modus ponens is one of the most commonly used concepts in logic it must not be mistaken for a logicallaw; rather, it is one of the accepted mechanisms for the construction of deductive proofs that includes the rule ofdenition and the rule of substitution.[7] Modus ponens allows one to eliminate a conditional statement from alogical proof or argument (the antecedents) and thereby not carry these antecedents forward in an ever-lengtheningstring of symbols; for this reasonmodus ponens is sometimes called the rule of detachment.[8] Enderton, for example,observes that modus ponens can produce shorter formulas from longer ones,[9] andRussell observes that the processof the inference cannot be reduced to symbols. Its sole record is the occurrence of q [the consequent] . . . aninference is the dropping of a true premise; it is the dissolution of an implication.[10]

A justication for the trust in inference is the belief that if the two former assertions [the antecedents] are not inerror, the nal assertion [the consequent] is not in error.[11] In other words: if one statement or proposition impliesa second one, and the rst statement or proposition is true, then the second one is also true. If P implies Q and P istrue, then Q is true.[12] An example is:

If it is raining, I will meet you at the theater.It is raining.Therefore, I will meet you at the theater.

Modus ponens can be stated formally as:

P ! Q; P) Q

where the rule is that whenever an instance of "P Q" and "P" appear by themselves on lines of a logical proof,Q can validly be placed on a subsequent line; furthermore, the premise P and the implication dissolves, their onlytrace being the symbol Q that is retained for use later e.g. in a more complex deduction.It is closely related to another valid form of argument, modus tollens. Both have apparently similar but invalid formssuch as arming the consequent, denying the antecedent, and evidence of absence. Constructive dilemma is thedisjunctive version of modus ponens. Hypothetical syllogism is closely related to modus ponens and sometimesthought of as double modus ponens.

10.1 Formal notationThe modus ponens rule may be written in sequent notation:

26

10.2. EXPLANATION 27

P ! Q; P ` Q

where is a metalogical symbol meaning that Q is a syntactic consequence of P Q and P in some logical system;or as the statement of a truth-functional tautology or theorem of propositional logic:

((P ! Q) ^ P )! Q

where P, and Q are propositions expressed in some formal system.

10.2 ExplanationThe argument form has two premises (hypothesis). The rst premise is the ifthen or conditional claim, namelythat P implies Q. The second premise is that P, the antecedent of the conditional claim, is true. From these twopremises it can be logically concluded that Q, the consequent of the conditional claim, must be true as well. Inarticial intelligence, modus ponens is often called forward chaining.An example of an argument that ts the form modus ponens:

If today is Tuesday, then John will go to work.Today is Tuesday.Therefore, John will go to work.

This argument is valid, but this has no bearing on whether any of the statements in the argument are true; for modusponens to be a sound argument, the premises must be true for any true instances of the conclusion. An argumentcan be valid but nonetheless unsound if one or more premises are false; if an argument is valid and all the premisesare true, then the argument is sound. For example, John might be going to work on Wednesday. In this case, thereasoning for Johns going to work (because it isWednesday) is unsound. The argument is not only sound on Tuesdays(when John goes to work), but valid on every day of the week. A propositional argument using modus ponens is saidto be deductive.In single-conclusion sequent calculi, modus ponens is the Cut rule. The cut-elimination theorem for a calculus saysthat every proof involving Cut can be transformed (generally, by a constructive method) into a proof without Cut,and hence that Cut is admissible.The CurryHoward correspondence between proofs and programs relates modus ponens to function application: if fis a function of type P Q and x is of type P, then f x is of type Q.

10.3 Justication via truth tableThe validity of modus ponens in classical two-valued logic can be clearly demonstrated by use of a truth table.In instances of modus ponens we assume as premises that p q is true and p is true. Only one line of the truthtablethe rstsatises these two conditions (p and p q). On this line, q is also true. Therefore, whenever p q is true and p is true, q must also be true.

10.4 See also Condensed detachment

What the Tortoise Said to Achilles

28 CHAPTER 10. MODUS PONENS

10.5 References[1] Stone, Jon R. (1996). Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language. London, UK: Routledge: 60.

[2] Copi and Cohen

[3] Hurley

[4] Moore and Parker

[5] Enderton 2001:110

[6] Susanne Bobzien (2002). The Development of Modus Ponens in Antiquity, Phronesis 47.

[7] Alfred Tarski 1946:47. Also Enderton 2001:110.

[8] Tarski 1946:47

[9] Enderton 2001:111

[10] Whitehead and Russell 1927:9

[11] Whitehead and Russell 1927:9

[12] Jago, Mark (2007). Formal Logic. Humanities-Ebooks LLP. ISBN 978-1-84760-041-7.

10.6 Sources Alfred Tarski 1946 Introduction to Logic and to theMethodology of theDeductive Sciences 2ndEdition, reprintedby Dover Publications, Mineola NY. ISBN 0-486-28462-X (pbk).

Alfred North Whitehead and Bertrand Russell 1927 Principia Mathematica to *56 (Second Edition) paperbackedition 1962, Cambridge at the University Press, London UK. No ISBN, no LCCCN.

Herbert B. Enderton, 2001, A Mathematical Introduction to Logic Second Edition, Harcourt Academic Press,Burlington MA, ISBN 978-0-12-238452-3.

10.7 External links Hazewinkel, Michiel, ed. (2001), Modus ponens, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Modus ponens at PhilPapers Modus ponens at Wolfram MathWorld

10.8. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 29

10.8 Text and image sources, contributors, and licenses10.8.1 Text

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30 CHAPTER 10. MODUS PONENS

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Entailment (pragmatics)Types of entailmentSee alsoReferencesFurther reading

Implication graphApplicationsReferences

Implicational hierarchyPhonologyMorphologySyntaxBibliography

Implicational propositional calculusVirtual completeness as an operatorAxiom systemBasic properties of derivationCompletenessProof

The BernaysTarski axiom system Testing whether a formula of the implicational propositional calculus is a tautology Adding an axiom schema An alternative axiomatization See alsoReferences

ImplicatureTypes of implicatureConversational implicatureConventional implicature

Implicature vs entailmentSee alsoReferencesBibliographyFurther readingExternal links

ImplicitMathematics Computer science Other uses See also

Linguistic universalTerminologyIn semanticsSee alsoNotesReferencesExternal links

Material conditionalDefinitions of the material conditionalAs a truth functionAs a formal connective

Formal propertiesPhilosophical problems with material conditional See alsoConditionals

ReferencesFurther reading External links

Material implicationSee also

Modus ponensFormal notation Explanation Justification via truth tableSee alsoReferences Sources External links Text and image sources, contributors, and licensesTextImagesContent license