Logical truthWikipedia

Contents

1 Logical truth 11.1 Logical truths and analytic truths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Truth values and tautologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Logical truth and logical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Logical truth and rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Non-classical logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Direct proof 42.1 History and etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 The sum of two even integers equals an even integer . . . . . . . . . . . . . . . . . . . . . 52.2.2 Pythagoras Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.3 If n is an odd integer, n2 is also an odd integer. . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Formal proof 93.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 Formal language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.2 Formal grammar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.3 Formal systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.4 Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Formation rule 114.1 Formal language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Formal systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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4.3 Propositional and predicate logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5 Logical constant 135.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Logical form 146.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.2 Example of argument form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.3 Importance of argument form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7 Proof (truth) 177.1 On proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

8 Rule of inference 198.1 The standard form of rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.2 Axiom schemas and axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.3 Example: Hilbert systems for two propositional logics . . . . . . . . . . . . . . . . . . . . . . . . 208.4 Admissibility and derivability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

9 Satisability 239.1 Reduction of validity to satisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.2 Propositional satisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.3 Satisability in rst-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.4 Satisability in model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.5 Finite satisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.6 Numerical constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

10 Substitution (logic) 2610.1 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

CONTENTS iii

10.1.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.1.2 Tautologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

10.2 First-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

11 Tautology (logic) 2911.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3011.3 Denition and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3011.4 Verifying tautologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111.5 Tautological implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111.6 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111.7 Ecient verication and the Boolean satisability problem . . . . . . . . . . . . . . . . . . . . . . 3211.8 Tautologies versus validities in rst-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3211.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

11.9.1 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3311.9.2 Related logical topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

11.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3311.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

12 Tautology (rhetoric) 3412.1 Rhetorical tautology vs. circular reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

12.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3412.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3412.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

13 Two Dogmas of Empiricism 3613.1 Analyticity and circularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3613.2 Reductionism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3713.3 Quines holism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.4 Critique and inuence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

14 Validity 4014.1 Validity of arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.2 Valid formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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14.3 Validity of statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114.4 Validity and soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114.5 Satisability and validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.6 Preservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.7 n-Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

14.7.1 -Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4314.11Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 44

14.11.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4414.11.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4514.11.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Chapter 1

Logical truth

Logical truth is one of the most fundamental concepts in logic, and there are dierent theories on its nature. Alogical truth is a statement which is true, and remains true under all reinterpretations of its components other than itslogical constants. It is a type of analytic statement. All of philosophical logic can be thought of as providing accountsof the nature of logical truth, as well as logical consequence.[1]

Logical truths (including tautologies) are truths which are considered to be necessarily true. This is to say that theyare considered to be such that they could not be untrue and no situation could arise which would cause us to reject alogical truth. However, it is not universally agreed that there are any statements which are necessarily true.A logical truth is considered by some philosophers to be a statement which is true in all possible worlds. This iscontrasted with facts (which may also be referred to as contingent claims or synthetic claims) which are true in thisworld, as it has historically unfolded, but which is not true in at least one possible world, as it might have unfolded.The proposition If p and q, then p and the proposition All married people are married are logical truths becausethey are true due to their inherent structure and not because of any facts of the world. Later, with the rise of formallogic a logical truth was considered to be a statement which is true under all possible interpretations.The existence of logical truths has been put forward by rationalist philosophers as an objection to empiricism becausethey hold that it is impossible to account for our knowledge of logical truths on empiricist grounds. Empiricistscommonly respond to this objection by arguing that logical truths (which they usually deem to be mere tautologies),are analytic and thus do not purport to describe the world.

1.1 Logical truths and analytic truths

Main article: Analyticsynthetic distinction

Logical truths, being analytic statements, do not contain any information about any matters of fact. Other than logicaltruths, there is also a second class of analytic statements, typied by No bachelor is married. The characteristicof such a statement is that it can be turned into a logical truth by substituting synonyms for synonyms salva veritate.No bachelor is married. can be turned into No unmarried man is married. by substituting 'unmarried man' for itssynonym 'bachelor.'In his essay, Two Dogmas of Empiricism, the philosopher W.V.O. Quine called into question the distinction betweenanalytic and synthetic statements. It was this second class of analytic statements that caused him to note that theconcept of analyticity itself stands in need of clarication, because it seems to depend on the concept of synonymy,which stands in need of clarication. In his conclusion, Quine rejects that logical truths are necessary truths. Insteadhe posits that the truth-value of any statement can be changed, including logical truths, given a re-evaluation of thetruth-values of every other statement in ones complete theory.

1

2 CHAPTER 1. LOGICAL TRUTH

1.2 Truth values and tautologiesMain article: Tautology (logic)

Considering dierent interpretations of the same statement leads to the notion of truth value. Simplest approach totruth values means that the statement may be true in one case, but false in another. In one sense of the termtautology, it is any type of formula or proposition which turns out to be true under any possible interpretation ofits terms (may also be called a valuation or assignment depending upon the context). This is synonymous to logicaltruth.However, the term tautology is also commonly used to refer to what could more specically be called truth-functional tautologies. Whereas a tautology or logical truth is true solely because of the logical terms it containsin general (e.g. "every", "some", and is), a truth-functional tautology is true because of the logical terms it containswhich are logical connectives (e.g. "or", "and", and "nor"). Not all logical truths are tautologies of such kind.

1.3 Logical truth and logical constantsMain article: Logical constant

Logical constants, including logical connectives and quantiers, can all be reduced conceptually to logical truth. Forinstance, two statements or more are logically incompatible if, and only if their conjunction is logically false. Onestatement logically implies another when it is logically incompatible with the negation of the other. A statement islogically false if, and only if its negation is logically true, etc. In this way all logical connectives can be expressed interms of preserving logical truth.

1.4 Logical truth and rules of inferenceThe concept of logical truth is closely connected to the concept of a rule of inference.[2]

1.5 Non-classical logicsMain article: Non-classical logic

Non-classical logic is the name given to formal systems which dier in a signicant way from standard logical systemssuch as propositional and predicate logic. There are several ways in which this is done, including by way of extensions,deviations, and variations. The aim of these departures is to make it possible to construct dierent models of logicalconsequence and logical truth.[3]

1.6 See also Contradiction

False (logic)

Satisability

Tautology (logic) (for symbolism of logical truth)

Theorem

Validity

1.7. REFERENCES 3

1.7 References[1] Quine, Willard Van Orman, Philosophy of logic

[2] Alfred Ayer, Language, Truth, and Logic

[3] Theodore Sider, Logic for philosophy

1.8 External links Logical truth entry in the Stanford Encyclopedia of Philosophy Logical truth at the Indiana Philosophy Ontology Project Logical truth at PhilPapers

Chapter 2

Direct proof

In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by astraightforward combination of established facts, usually axioms, existing lemmas and theorems, without making anyfurther assumptions.[1] In order to directly prove a conditional statement of the form If p, then q", it suces toconsider the situations in which the statement p is true. Logical deduction is employed to reason from assumptionsto conclusion. The type of logic employed is almost invariably rst-order logic, employing the quantiers for all andthere exists. Common proof rules used are modus ponens and universal instantiation.[2]

In contrast, an indirect proof may begin with certain hypothetical scenarios and then proceed to eliminate the uncer-tainties in each of these scenarios until an inescapable conclusion is forced. For example instead of showing directlyp q, one proves its contrapositive ~q ~p (one assumes ~q and shows that it leads to ~p). Since p q and ~q~p are equivalent by the principle of transposition (see law of excluded middle), p q is indirectly proved. Proofmethods that are not direct include proof by contradiction, including proof by innite descent. Direct proof methodsinclude proof by exhaustion and proof by induction.

2.1 History and etymologyA direct proof is the simplest form of proof there is. The word proof comes from the Latin word probare,[3] whichmeans to test. The earliest use of proofs was prominent in legal proceedings. A person with authority, such as anobleman, was said to have probity, which means that the evidence was by his relative authority, which outweighedempirical testimony. In days gone by, mathematics and proof was often intertwined with practical questions withpopulations like the Egyptians and the Greeks showing an interest in surveying land.[4] This lead to a natural curiositywith regards to geometry and trigonometry particularly triangles and rectangles. These were the shapes whichprovided the most questions in terms of practical things, so early geometrical concepts were focused on these shapes,for example, the likes of buildings and pyramids used these shapes in abundance. Another shape which is crucial inthe history of direct proof is the circle, which was crucial for the design of arenas and water tanks. This meant thatancient geometry (and Euclidean Geometry) discussed circles.The earliest form of mathematics was phenomenological. For example, if someone could draw a reasonable picture,or give a convincing description, then that met all the criteria for something to be described as a mathematical fact.On occasion, analogical arguments took place, or even by invoking the gods. The idea that mathematical statementscould be proven had not been developed yet, so these were the earliest forms of the concept of proof, despite notbeing actual proof at all.Proof as we know it came about with one specic question: what is a proof? Traditionally, a proof is a platformwhich convinces someone beyond reasonable doubt that a statement is mathematically true. Naturally, one wouldassume that the best way to prove the truth of something like this (B) would be to draw up a comparison withsomething old (A) that has already been proven as true. Thus was created the concept of deriving a new result froman old result.

2.2 Examples

4

2.2. EXAMPLES 5

Geometric Constructions

2.2.1 The sum of two even integers equals an even integer

Consider two even integers x and y. Since they are even, they can be written as

x = 2a

y = 2b

respectively for integers a and b. Then the sum can be written as

x+ y = 2a+ 2b = 2(a+ b)

From this it is clear x + y has 2 as a factor and therefore is even, so the sum of any two even integers is even.

2.2.2 Pythagoras Theorem

Observe that we have four right-angled triangles and a square packed into a large square. Each of the triangles hassides a and b and hypotenuse c. The area of a square is dened as the square of the length of its sides - in this case,(a + b)2. However, the area of the large square can also be expressed as the sum of the areas of its components. Inthis case, that would be the sum of the areas of the four triangles and the small square in the middle.[5]

We know that the area of the large square is equal to (a + b)2

The area of a triangle is equal to 12ab

6 CHAPTER 2. DIRECT PROOF

New result from an old result

We know that the area of the large square is also equal to the sum of the areas of the triangles, plus the area of thesmall square, and thus the area of the large square equals 4( 12ab) + c2

These are equal, and so:

(a+ b)2 = 4(1/2ab) + c2

After some simplifying:

a2 + 2ab+ b2 = 2ab+ c2

Removing the ab that appears on both sides gives

a2 + b2 = c2

Which proves Pythagoras theorem.

2.2.3 If n is an odd integer, n2 is also an odd integer.By denition, if n is an odd integer, it can be expressed as:

2.2. EXAMPLES 7

Diagram of Pythagoras Theorem

n = 2k + 1

for some integer k. Thus:

n2 = (2k + 1)2

= (2k + 1)(2k + 1)

= 4k2 + 2k + 2k + 1

= 4k2 + 4k + 1

= 2(2k2 + 2k) + 1

As (2k2 + 2k) is an integer, our answer can be expressed as:

2k + 1

And hence we have shown that n2 is odd.

8 CHAPTER 2. DIRECT PROOF

2.3 References[1] Cupillari, Antonella. The Nuts and Bolts of Proofs. Academic Press, 2001. Page 3.

[2] C. Gupta, S. Singh, S. Kumar Advanced Discrete Structure. I.K. International Publishing House Pvt. Ltd., 2010. Page 127.

[3] New Shorter Oxford English Dictionary

[4] Krantz, Steven G. The History and Concept of Mathematical Proof. February 5, 2007.

[5] Krantz, Steven G. The Proof is the Pudding. Springer, 2010. Page 43.

2.4 Sources Franklin, J.; A. Daoud (2011). Proof in Mathematics: An Introduction. Sydney: Kew Books. ISBN 0-646-54509-4. (Ch. 1.)

2.5 External links Direct Proof from Larry W. Cusicks How To Write Proofs. Direct Proofs from Patrick Keef and David Guichards Introduction to Higher Mathematics. Direct Proof section of Richard Hammacks Book of Proof.

Chapter 3

Formal proof

A formal proof or derivation is a nite sequence of sentences (called well-formed formulas in the case of a formallanguage) each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by arule of inference. The last sentence in the sequence is a theorem of a formal system. The notion of theorem is notin general eective, therefore there may be no method by which we can always nd a proof of a given sentence ordetermine that none exists. The concept of natural deduction is a generalization of the concept of proof.[1]

The theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof. For a well-formedformula to qualify as part of a proof, it must be the result of applying a rule of the deductive apparatus of some formalsystem to the previous well-formed formulae in the proof sequence.Formal proofs often are constructed with the help of computers in interactive theorem proving. Signicantly, theseproofs can be checked automatically, also by computer. Checking formal proofs is usually simple, while the problemof nding proofs (automated theorem proving) is usually computationally intractable and/or only semi-decidable,depending upon the formal system in use.

3.1 Background

3.1.1 Formal languageMain article: Formal language

A formal language is a set of nite sequences of symbols. Such a language can be dened without reference to anymeanings of any of its expressions; it can exist before any interpretation is assigned to it that is, before it has anymeaning. Formal proofs are expressed in some formal language.

3.1.2 Formal grammarMain articles: Formal grammar and Formation rule

A formal grammar (also called formation rules) is a precise description of the well-formed formulas of a formallanguage. It is synonymous with the set of strings over the alphabet of the formal language which constitute wellformed formulas. However, it does not describe their semantics (i.e. what they mean).

3.1.3 Formal systemsMain article: Formal system

A formal system (also called a logical calculus, or a logical system) consists of a formal language together with adeductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformation

9

10 CHAPTER 3. FORMAL PROOF

rules (also called inference rules) or a set of axioms, or have both. A formal system is used to derive one expressionfrom one or more other expressions.

3.1.4 InterpretationsMain articles: Formal semantics (logic) and Interpretation (logic)

An interpretation of a formal system is the assignment of meanings to the symbols, and truth-values to the sentencesof a formal system. The study of interpretations is called formal semantics. Giving an interpretation is synonymouswith constructing a model.

3.2 See also Proof (truth) Mathematical proof Proof theory Axiomatic system

3.3 References[1] The Cambridge Dictionary of Philosophy, deduction

3.4 External links A Special Issue on Formal Proof. Notices of the American Mathematical Society. December 2008. 2ix.com: Logic Part of a series of articles covering mathematics and logic.

Chapter 4

Formation rule

In mathematical logic, formation rules are rules for describing which strings of symbols formed from the alphabet ofa formal language are syntactically valid within the language. These rules only address the location and manipulationof the strings of the language. It does not describe anything else about a language, such as its semantics (i.e. whatthe strings mean). (See also formal grammar).

4.1 Formal languageMain article: Formal language

A formal language is an organized set of symbols the essential feature being that it can be precisely dened in termsof just the shapes and locations of those symbols. Such a language can be dened, then, without any reference to anymeanings of any of its expressions; it can exist before any interpretation is assigned to itthat is, before it has anymeaning. A formal grammar determines which symbols and sets of symbols are formulas in a formal language.

4.2 Formal systemsMain article: Formal system

A formal system (also called a logical calculus, or a logical system) consists of a formal language together with adeductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformationrules (also called inference rules) or a set of axioms, or have both. A formal system is used to derive one expressionfrom one or more other expressions. Propositional and predicate calculi are examples of formal systems.

4.3 Propositional and predicate logicThe formation rules of a propositional calculus may, for instance, take a form such that;

if we take to be a propositional formula we can also take to be a formula;

if we take and to be a propositional formulas we can also take ( & ), ( ), ( ) and ($ )to also be formulas.

A predicate calculus will usually include all the same rules as a propositional calculus, with the addition of quantierssuch that if we take to be a formula of propositional logic and as a variable then we can take () and () eachto be formulas of our predicate calculus.

11

12 CHAPTER 4. FORMATION RULE

4.4 See also Finite state automaton

Chapter 5

Logical constant

In logic, a logical constant of a languageL is a symbol that has the same semantic value under every interpretation ofL . Two important types of logical constants are logical connectives and quantiers. The equality predicate (usuallywritten '=') is also treated as a logical constant in many systems of logic. One of the fundamental questions in thephilosophy of logic is What is a logical constant?"; that is, what special feature of certain constants makes themlogical in nature?[1]

Some symbols that are commonly treated as logical constants are:Many of these logical constants are sometimes denoted by alternate symbols (e.g., the use of the symbol "&" ratherthan "" to denote the logical and).

5.1 See also Non-logical symbol Logical value Logical connective

5.2 References[1] Carnap

5.3 External links Stanford Encyclopedia of Philosophy entry on logical constants

13

Chapter 6

Logical form

This article is about the term as used in logic. For the linguistics term, see Logical Form

The logical form of a sentence (or proposition or statement or truthbearer) or set of sentences is the form obtainedby abstracting from the subject matter of its content terms or by regarding the content terms as mere placeholdersor blanks on a form. In an ideal logical language, the logical form can be determined from syntax alone; formallanguages used in formal sciences are examples of such languages. Logical form however should not be confusedwith the mere syntax used to represent it; there may be more than one string that represents the same logical form ina given language.[1]

The logical form of an argument is called the argument form or test form of the argument.

6.1 HistoryThe importance of the concept of form to logic was already recognized in ancient times. Aristotle, in the PriorAnalytics, was probably the rst to employ variable letters to represent valid inferences. Therefore, ukasiewiczclaims that the introduction of variables was 'one of Aristotles greatest inventions.According to the followers of Aristotle like Ammonius, only the logical principles stated in schematic terms belong tologic, and not those given in concrete terms. The concrete terms man, mortal, etc., are analogous to the substitutionvalues of the schematic placeholders 'A', 'B', 'C', which were called the 'matter' (Greek hyle, Latin materia) of theargument.The term logical form itself was introduced by Bertrand Russell in 1914, in the context of his program to formalizenatural language and reasoning, which he called philosophical logic. Russell wrote: Some kind of knowledge oflogical forms, though with most people it is not explicit, is involved in all understanding of discourse. It is thebusiness of philosophical logic to extract this knowledge from its concrete integuments, and to render it explicit andpure. [2][3]

6.2 Example of argument formTo demonstrate the important notion of the form of an argument, substitute letters for similar items throughout thesentences in the original argument.

Original argument All humans are mortal.Socrates is human.Therefore, Socrates is mortal.Argument Form All H are M.S is H.

14

6.3. IMPORTANCE OF ARGUMENT FORM 15

Therefore, S is M.

All we have done in the Argument form is to put 'H' for 'human' and 'humans, 'M' for 'mortal', and 'S' for 'Socrates;what results is the form of the original argument. Moreover, each individual sentence of the Argument form is thesentence form of its respective sentence in the original argument.[4]

6.3 Importance of argument formAttention is given to argument and sentence form, because form is what makes an argument valid or cogent. Someexamples of valid argument forms are modus ponens, modus tollens, disjunctive syllogism, hypothetical syllogismand dilemma. Two invalid argument forms are arming the consequent and denying the antecedent.A logical argument, seen as an ordered set of sentences, has a logical form that derives from the form of its constituentsentences; the logical form of an argument is sometimes called argument form.[5] Some authors only dene logicalform with respect to whole arguments, as the schemata or inferential structure of the argument.[6] In argumentationtheory or informal logic, an argument form is sometimes seen as a broader notion than the logical form.[7]

It consists of stripping out all spurious grammatical features from the sentence (such as gender, and passive forms), andreplacing all the expressions specic to the subject matter of the argument by schematic variables. Thus, for example,the expression 'all As are Bs' shows the logical form which is common to the sentences 'all men are mortals, 'all catsare carnivores, 'all Greeks are philosophers and so on.

6.4 See also Argument map Fallacy

Logical fallacy Informal fallacy

Categorial grammar Sense and reference Analytic proposition List of invalid argument forms List of valid argument forms Synthetic proposition

6.5 References[1] The Cambridge Dictionary of Philosophy, CUP 1999, pp. 511-512

[2] Russell, Bertrand. 1914(1993). Our Knowledge of the External World: as a eld for scientic method in philosophy. NewYork: Routledge. p. 53

[3] Ernie Lepore, Kirk Ludwig (2002). What is logical form?". In Gerhard Preyer, Georg Peter. Logical form and language.Clarendon Press. p. 54. ISBN 978-0-19-924555-0. preprint

[4] Hurley, Patrick J. (1988). A concise introduction to logic. Belmont, Calif.: Wadsworth Pub. Co. ISBN 0-534-08928-3.[5] J. C. Beall (2009). Logic: the Basics. Taylor & Francis. p. 18. ISBN 978-0-415-77498-7.

[6] Paul Tomassi (1999). Logic. Routledge. p. 386. ISBN 978-0-415-16696-6.[7] Robert C. Pinto (2001). Argument, inference and dialectic: collected papers on informal logic. Springer. p. 84. ISBN

978-0-7923-7005-5.

16 CHAPTER 6. LOGICAL FORM

6.6 Further reading RichardMark Sainsbury (2001). Logical forms: an introduction to philosophical logic. Wiley-Blackwell. ISBN978-0-631-21679-7.

Gerhard Preyer, Georg Peter, ed. (2002). Logical form and language. Clarendon Press. ISBN 978-0-19-924555-0.

Gila Sher (1991). The bounds of logic: a generalized viewpoint. MIT Press. ISBN 978-0-262-19311-5.

6.7 External links Logical form at PhilPapers Logical Form entry by Paul Pietroski in the Stanford Encyclopedia of Philosophy Logical form at the Indiana Philosophy Ontology Project Beaney, Michael, Analysis, The Stanford Encyclopedia of Philosophy (Summer 2009 Edition), Edward N.Zalta (ed.)

IEP, Validity and Soundness

Chapter 7

Proof (truth)

For other uses, see Proof.

A proof is sucient evidence or an argument for the truth of a proposition.[1][2][3][4]

The concept is applied in a variety of disciplines, with both the nature of the evidence or justication and the criteriafor suciency being area-dependent. In the area of oral and written communication such as conversation, dialog,rhetoric, etc., a proof is a persuasive perlocutionary speech act, which demonstrates the truth of a proposition.[5]In any area of mathematics dened by its assumptions or axioms, a proof is an argument establishing a theorem ofthat area via accepted rules of inference starting from those axioms and other previously established theorems.[6]The subject of logic, in particular proof theory, formalizes and studies the notion of formal proof.[7] In the areas ofepistemology and theology, the notion of justication plays approximately the role of proof,[8] while in jurisprudencethe corresponding term is evidence,[9] with burden of proof as a concept common to both philosophy and law.

7.1 On proof

In most disciplines, evidence is required to prove something. Evidence is drawn from experience of the world aroundus, with science obtaining its evidence from nature,[10] law obtaining its evidence from witnesses and forensic investi-gation,[11] and so on. A notable exception is mathematics, whose proofs are drawn from a mathematical world begunwith axioms and further developed and enriched by theorems proved earlier.Exactly what evidence is sucient to prove something is also strongly area-dependent, usually with no absolute thresh-old of suciency at which evidence becomes proof.[12][13][14] In law, the same evidence that may convince one jurymay not persuade another. Formal proof provides the main exception, where the criteria for proofhood are ironcladand it is impermissible to defend any step in the reasoning as obvious";[15] for a well-formed formula to qualify aspart of a formal proof, it must be the result of applying a rule of the deductive apparatus of some formal system tothe previous well-formed formulae in the proof sequence.[16]

Proofs have been presented since antiquity. Aristotle used the observation that patterns of nature never display themachine-like uniformity of determinism as proof that chance is an inherent part of nature.[17] On the other hand,Thomas Aquinas used the observation of the existence of rich patterns in nature as proof that nature is not ruled bychance.[18]

Proofs need not be verbal. Before Galileo, people took the apparent motion of the Sun across the sky as proof thatthe Sun went round the Earth.[19] Suitably incriminating evidence left at the scene of a crime may serve as proof ofthe identity of the perpetrator. Conversely, a verbal entity need not assert a proposition to constitute a proof of thatproposition. For example, a signature constitutes direct proof of authorship; less directly, handwriting analysis maybe submitted as proof of authorship of a document.[20] Privileged information in a document can serve as proof thatthe documents author had access to that information; such access might in turn establish the location of the authorat certain time, which might then provide the author with an alibi.

17

18 CHAPTER 7. PROOF (TRUTH)

7.2 See also Mathematical proof Proof theory Proof of concept Provability logic Evidence, information which tends to determine or demonstrate the truth of a proposition Proof procedure Proof complexity Standard of proof

7.3 References[1] Proof and other dilemmas: mathematics and philosophy by Bonnie Gold, Roger A. Simons 2008 ISBN 0883855674 pages

1220[2] Philosophical Papers, Volume 2 by Imre Lakatos, John Worrall, Gregory Currie, ISBN Philosophical Papers, Volume 2 by

Imre Lakatos, John Worrall, Gregory Currie 1980 ISBN 0521280303 pages 6063[3] Evidence, proof, and facts: a book of sources by Peter Murphy 2003 ISBN 0199261954 pages 12[4] Logic in Theology And Other Essays by Isaac Taylor 2010 ISBN 1445530139 pages 515[5] John Langshaw Austin: How to Do Things With Words. Cambridge (Mass.) 1962 Paperback: Harvard University Press,

2nd edition, 2005, ISBN 0-674-41152-8.[6] Cupillari, Antonella. The Nuts and Bolts of Proofs. Academic Press, 2001. Page 3.[7] Alfred Tarski, Introduction to Logic and to the Methodology of the Deductive Sciences (ed. Jan Tarski). 4th Edition.

Oxford Logic Guides, No. 24. New York and Oxford: Oxford University Press, 1994, xxiv + 229 pp. ISBN 0-19-504472-X

[8] http://plato.stanford.edu/entries/justep-foundational/[9] http://dictionary.reference.com/browse/proof[10] Reference Manual on Scientic Evidence, 2nd Ed. (2000), p. 71. Accessed May 13, 2007.[11] John Henry Wigmore, A Treatise on the System of Evidence in Trials at Common Law, 2nd ed., Little, Brown, and Co.,

Boston, 1915[12] Simon, Rita James, and Mahan, Linda. (1971). Quantifying Burdens of ProofA View from the Bench, the Jury, and

the Classroom. Law and Society Review 5 (3): 319330. doi:10.2307/3052837. JSTOR 3052837.[13] Katie Evans, David Osthus, Ryan G. Spurrier. Distributions of Interest for Quantifying Reasonable Doubt and Their

Applications (PDF). Retrieved 2007-01-14.[14] The Principle of Sucient Reason: A Reassessment by Alexander R. Pruss[15] A. S. Troelstra, H. Schwichtenberg (1996). Basic Proof Theory. In series Cambridge Tracts in Theoretical Computer

Science, Cambridge University Press, ISBN 0-521-77911-1.[16] Hunter, Georey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California

Pres, 1971[17] Aristotles Physics: a Guided Study, Joe Sachs, 1995 ISBN 0813521920 p. 70[18] The treatise on the divine nature: Summa theologiae I, 113, by Saint Thomas Aquinas, Brian J. Shanley, 2006 ISBN

0872208052 p. 198[19] Thomas S. Kuhn, The Copernican Revolution, pp. 520[20] Trial tactics by Stephen A. Saltzburg, 2007 ISBN 159031767X page 47

Chapter 8

Rule of inference

In logic, a rule of inference, inference rule, or transformation rule is a logical form consisting of a function whichtakes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of inferencecalled modus ponens takes two premises, one in the form If p then q and another in the form p, and returns theconclusion q. The rule is valid with respect to the semantics of classical logic (as well as the semantics of many othernon-classical logics), in the sense that if the premises are true (under an interpretation), then so is the conclusion.Typically, a rule of inference preserves truth, a semantic property. In many-valued logic, it preserves a generaldesignation. But a rule of inferences action is purely syntactic, and does not need to preserve any semantic property:any function from sets of formulae to formulae counts as a rule of inference. Usually only rules that are recursiveare important; i.e. rules such that there is an eective procedure for determining whether any given formula is theconclusion of a given set of formulae according to the rule. An example of a rule that is not eective in this sense isthe innitary -rule.[1]

Popular rules of inference in propositional logic includemodus ponens, modus tollens, and contraposition. First-orderpredicate logic uses rules of inference to deal with logical quantiers.

8.1 The standard form of rules of inference

In formal logic (and many related areas), rules of inference are usually given in the following standard form:Premise#1Premise#2...Premise#nConclusionThis expression states that whenever in the course of some logical derivation the given premises have been obtained,the specied conclusion can be taken for granted as well. The exact formal language that is used to describe bothpremises and conclusions depends on the actual context of the derivations. In a simple case, one may use logicalformulae, such as in:ABABThis is the modus ponens rule of propositional logic. Rules of inference are often formulated as schemata employingmetavariables.[2] In the rule (schema) above, the metavariables A and B can be instantiated to any element of theuniverse (or sometimes, by convention, a restricted subset such as propositions) to form an innite set of inferencerules.A proof system is formed from a set of rules chained together to form proofs, also called derivations. Any derivationhas only one nal conclusion, which is the statement proved or derived. If premises are left unsatised in the derivation,then the derivation is a proof of a hypothetical statement: "if the premises hold, then the conclusion holds.

19

20 CHAPTER 8. RULE OF INFERENCE

8.2 Axiom schemas and axioms

Inference rules may also be stated in this form: (1) zero or more premises, (2) a turnstile symbol ` , which meansinfers, proves, or concludes, and (3) a conclusion. This form usually embodies the relational (as opposed tofunctional) view of a rule of inference, where the turnstile stands for a deducibility relation holding between premisesand conclusion.An inference rule containing no premises is called an axiom schema or, if it contains no metavariables, simply anaxiom.[2]

Rules of inference must be distinguished from axioms of a theory. In terms of semantics, axioms are valid assertions.Axioms are usually regarded as starting points for applying rules of inference and generating a set of conclusions. Or,in less technical terms:Rules are statements about the system, axioms are statements in the system. For example:

The rule that from ` p you can infer ` Provable(p) is a statement that says if you've proven p , then it isprovable that p is provable. This rule holds in Peano arithmetic, for example.

The axiom p ! Provable(p) would mean that every true statement is provable. This axiom does not hold inPeano arithmetic.

Rules of inference play a vital role in the specication of logical calculi as they are considered in proof theory, suchas the sequent calculus and natural deduction.

8.3 Example: Hilbert systems for two propositional logics

In a Hilbert system, the premises and conclusion of the inference rules are simply formulae of some language, usuallyemploying metavariables. For graphical compactness of the presentation and to emphasize the distinction betweenaxioms and rules of inference, this section uses the sequent notation () instead of a vertical presentation of rules.The formal language for classical propositional logic can be expressed using just negation (), implication () andpropositional symbols. A well-known axiomatization, comprising three axiom schema and one inference rule (modusponens), is:(CA1) A (B A)(CA2) (A (B C)) ((A B) (A C))(CA3) (A B) (B A)(MP) A, A B BIt may seem redundant to have two notions of inference in this case, and . In classical propositional logic, theyindeed coincide; the deduction theorem states that A B if and only if A B. There is however a distinction worthemphasizing even in this case: the rst notation describes a deduction, that is an activity of passing from sentences tosentences, whereas A B is simply a formula made with a logical connective, implication in this case. Without aninference rule (like modus ponens in this case), there is no deduction or inference. This point is illustrated in LewisCarroll's dialogue called "What the Tortoise Said to Achilles".[3]

For some non-classical logics, the deduction theorem does not hold. For example, the three-valued logic 3 ofukasiewicz can be axiomatized as:[4]

(CA1) A (B A)(LA2) (A B) ((B C) (A C))(CA3) (A B) (B A)(LA4) ((A A) A) A(MP) A, A B BThis sequence diers from classical logic by the change in axiom 2 and the addition of axiom 4. The classicaldeduction theorem does not hold for this logic, however a modied form does hold, namely A B if and only if A (A B).[5]

8.4. ADMISSIBILITY AND DERIVABILITY 21

8.4 Admissibility and derivabilityMain article: Admissible rule

In a set of rules, an inference rule could be redundant in the sense that it is admissible or derivable. A derivablerule is one whose conclusion can be derived from its premises using the other rules. An admissible rule is onewhose conclusion holds whenever the premises hold. All derivable rules are admissible. To appreciate the dierence,consider the following set of rules for dening the natural numbers (the judgment n nat asserts the fact that n is anatural number):

0 natn nat

s(n) nat

The rst rule states that 0 is a natural number, and the second states that s(n) is a natural number if n is. In this proofsystem, the following rule, demonstrating that the second successor of a natural number is also a natural number, isderivable:

n nats(s(n)) nat

Its derivation is the composition of two uses of the successor rule above. The following rule for asserting the existenceof a predecessor for any nonzero number is merely admissible:

s(n) natn nat

This is a true fact of natural numbers, as can be proven by induction. (To prove that this rule is admissible, assume aderivation of the premise and induct on it to produce a derivation of n nat .) However, it is not derivable, because itdepends on the structure of the derivation of the premise. Because of this, derivability is stable under additions to theproof system, whereas admissibility is not. To see the dierence, suppose the following nonsense rule were added tothe proof system:

s(3) natIn this new system, the double-successor rule is still derivable. However, the rule for nding the predecessor is nolonger admissible, because there is no way to derive 3 nat . The brittleness of admissibility comes from the way itis proved: since the proof can induct on the structure of the derivations of the premises, extensions to the system addnew cases to this proof, which may no longer hold.Admissible rules can be thought of as theorems of a proof system. For instance, in a sequent calculus where cutelimination holds, the cut rule is admissible.

8.5 See also Inference objection Immediate inference Law of thought List of rules of inference Logical truth structural rule

22 CHAPTER 8. RULE OF INFERENCE

8.6 References[1] Boolos, George; Burgess, John; Jerey, Richard C. (2007). Computability and logic. Cambridge: Cambridge University

Press. p. 364. ISBN 0-521-87752-0.

[2] John C. Reynolds (2009) [1998]. Theories of Programming Languages. Cambridge University Press. p. 12. ISBN 978-0-521-10697-9.

[3] Kosta Dosen (1996). Logical consequence: a turn in style. In Maria Luisa Dalla Chiara, Kees Doets, Daniele Mundici,Johan van Benthem. Logic and Scientic Methods: Volume One of the Tenth International Congress of Logic, Methodologyand Philosophy of Science, Florence, August 1995. Springer. p. 290. ISBN 978-0-7923-4383-7. preprint (with dierentpagination)

[4] Bergmann, Merrie (2008). An introduction to many-valued and fuzzy logic: semantics, algebras, and derivation systems.Cambridge University Press. p. 100. ISBN 978-0-521-88128-9.

[5] Bergmann, Merrie (2008). An introduction to many-valued and fuzzy logic: semantics, algebras, and derivation systems.Cambridge University Press. p. 114. ISBN 978-0-521-88128-9.

Chapter 9

Satisability

In mathematical logic, satisability and validity are elementary concepts of semantics. A formula is satisable ifit is possible to nd an interpretation (model) that makes the formula true.[1] A formula is valid if all interpretationsmake the formula true. The opposites of these concepts are unsatisability and invalidity, that is, a formula isunsatisable if none of the interpretations make the formula true, and invalid if some such interpretation makes theformula false. These four concepts are related to each other in a manner exactly analogous to Aristotle's square ofopposition.The four concepts can be raised to apply to whole theories: a theory is satisable (valid) if one (all) of the inter-pretations make(s) each of the axioms of the theory true, and a theory is unsatisable (invalid) if all (one) of theinterpretations make(s) each of the axioms of the theory false.It is also possible to consider only interpretations that make all of the axioms of a second theory true. This general-ization is commonly called satisability modulo theories.The question whether a sentence in propositional logic is satisable is a decidable problem. In general, the questionwhether sentences in rst-order logic are satisable is not decidable. In universal algebra and equational theory, themethods of term rewriting, congruence closure and unication are used to attempt to decide satisability. Whether aparticular theory is decidable or not depends whether the theory is variable-free or on other conditions.[2]

9.1 Reduction of validity to satisability

For classical logics, it is generally possible to reexpress the question of the validity of a formula to one involvingsatisability, because of the relationships between the concepts expressed in the above square of opposition. Inparticular is valid if and only if is unsatisable, which is to say it is not true that is satisable. Put anotherway, is satisable if and only if is invalid.For logics without negation, such as the positive propositional calculus, the questions of validity and satisability maybe unrelated. In the case of the positive propositional calculus, the satisability problem is trivial, as every formula issatisable, while the validity problem is co-NP complete.

9.2 Propositional satisability

Main article: Propositional satisability

In the case of classical propositional logic, satisability is decidable for propositional formulae. In particular, satis-ability is an NP-complete problem, and is one of the most intensively studied problems in computational complexitytheory.

23

24 CHAPTER 9. SATISFIABILITY

9.3 Satisability in rst-order logicSatisability is undecidable and indeed it isn't even a semidecidable property of formulae in rst-order logic (FOL).[3]This fact has to do with the undecidability of the validity problem for FOL. The question of the status of the validityproblem was posed rstly by David Hilbert, as the so-called Entscheidungsproblem. The universal validity of aformula is a semi-decidable problem. If satisability were also a semi-decidable problem, then the problem of theexistence of counter-models would be too (a formula has counter-models i its negation is satisable). So the problemof logical validity would be decidable, which contradicts the Church-Turing theorem, a result stating the negativeanswer for the Entscheidungsproblem.

9.4 Satisability in model theoryIn model theory, an atomic formula is satisable if there is a collection of elements of a structure that render theformula true.[4] If A is a structure, is a formula, and a is a collection of elements, taken from the structure, thatsatisfy , then it is commonly written that

A [a]

If has no variables, that is, if is an atomic sentence, and it is satised by A, then one writes

A

In this case, one may also say that A is a model for , or that is true in A. If T is a collection of atomic sentences(a theory) satised by A, one writes

A T

9.5 Finite satisabilityA problem related to satisability is that of nite satisability, which is the question of determining whether aformula admits a nite model that makes it true. For a logic that has the nite model property, the problems ofsatisability and nite satisability coincide, as a formula of that logic has a model if and only if it has a nite model.This question is important in the mathematical eld of nite model theory.Nevertheless, nite satisability and satisability need not coincide in general. For instance, consider the rst-orderlogic formula obtained as the conjunction of the following sentences, where a and b are constants:

R(a0; a0)

R(a0; a1)

8xy(R(x; y)! 9zR(y; z))

8xyz(R(y; x) ^R(z; x)! x = z))

The resulting formula has the innite model R(a0; a0); R(a0; a1); R(a1; a2); : : : , but it can be shown that it has nonite model (starting at the fact R(a; b) and following the chain of R atoms that must exist by the third axiom, theniteness of a model would require the existence of a loop, which would violate the fourth axiom, whether it loopsback on a0 or on a dierent element).The computational complexity of deciding satisability for an input formula in a given logic may dier from that ofdeciding nite satisability; in fact, for some logics, only one of them is decidable.

9.6. NUMERICAL CONSTRAINTS 25

9.6 Numerical constraintsFurther information: Satisability modulo theories and Constraint satisfaction problem

Numerical constraints often appear in the eld ofmathematical optimization, where one usually wants tomaximize (orminimize) an objective function subject to some constraints. However, leaving aside the objective function, the basicissue of simply deciding whether the constraints are satisable can be challenging or undecidable in some settings.The following table summarizes the main cases.Table source: Bockmayr and Weispfenning.[5]:754

For linear constraints, a fuller picture is provided by the following table.Table source: Bockmayr and Weispfenning.[5]:755

9.7 See also 2-satisability Boolean satisability problem Circuit satisability Karps 21 NP-complete problems Validity Constraint satisfaction Satiscing

9.8 Notes[1] See, for example, Boolos and Jerey, 1974, chapter 11.

[2] Franz Baader; Tobias Nipkow (1998). Term Rewriting and All That. Cambridge University Press. pp. 5892. ISBN0-521-77920-0.

[3] Baier, Christel (2012). Chapter 1.3 Undecidability of FOL (PDF). Lecture Notes Advanced Logics. TechnischeUniversitt Dresden Institute for Technical Computer Science. pp. 2832. Retrieved 21 July 2012.

[4] Wilifrid Hodges (1997). A Shorter Model Theory. Cambridge University Press. p. 12. ISBN 0-521-58713-1.

[5] Alexander Bockmayr, Volker Weispfenning (2001). Solving Numerical Constraints. In John Alan Robinson and AndreiVoronkov. Handbook of Automated Reasoning Volume I. Elsevier and MIT Press. ISBN 0-444-82949-0 (Elsevier) ISBN0-262-18221-1 (MIT Press).

9.9 References Boolos and Jerey, 1974. Computability and Logic. Cambridge University Press.

9.10 Further reading Daniel Kroening; Ofer Strichman (2008). Decision Procedures: An Algorithmic Point of View. Springer Science& Business Media. ISBN 978-3-540-74104-6.

A. Biere, M. Heule, H. van Maaren, T. Walsh, ed. (2009). Handbook of Satisability. IOS Press. ISBN978-1-60750-376-7.

Chapter 10

Substitution (logic)

Not to be confused with Substitution (algebra).

Substitution is a fundamental concept in logic. A substitution is a syntactic transformation on formal expressions.To apply a substitution to an expression means to consistently replace its variable, or placeholder, symbols by otherexpressions. The resulting expression is called a substitution instance of the original expression.

10.1 Propositional logic

10.1.1 Denition

Where and represent formulas of propositional logic, is a substitution instance of if and only if maybe obtained from by substituting formulas for symbols in , always replacing an occurrence of the same symbolby an occurrence of the same formula. For example:

(R S) & (T S)

is a substitution instance of:

P & Q

and

(A A) (A A)

is a substitution instance of:

(A A)

In some deduction systems for propositional logic, a new expression (a proposition) may be entered on a line of aderivation if it is a substitution instance of a previous line of the derivation (Hunter 1971, p. 118). This is how newlines are introduced in some axiomatic systems. In systems that use rules of transformation, a rule may include theuse of a substitution instance for the purpose of introducing certain variables into a derivation.In rst-order logic, every closed propositional formula that can be derived from an open propositional formula a bysubstitution is said to be a substitution instance of a. If a is a closed propositional formula we count a itself as its onlysubstitution instance.

26

10.2. FIRST-ORDER LOGIC 27

10.1.2 Tautologies

A propositional formula is a tautology if it is true under every valuation (or interpretation) of its predicate symbols.If is a tautology, and is a substitution instance of , then is again a tautology. This fact implies the soundnessof the deduction rule described in the previous section.

10.2 First-order logicIn rst-order logic, a substitution is a total mapping : V T from variables to terms; the notation { x1 t1, ...,xk tk } [note 1] refers to a substitution mapping each variable xi to the corresponding term ti, for i=1,...,k, and everyother variable to itself; the xi must be pairwise distinct. Applying that substitution to a term t is written in postxnotation as t { x1 t1, ..., xk tk }; it means to (simultaneously) replace every occurrence of each xi in t by ti.[note 2] The result t of applying a substitution to a term t is called an instance of that term t. For example, applyingthe substitution { x z, z h(a,y) } to the termThe domain dom() of a substitution is commonly dened as the set of variables actually replaced, i.e. dom() ={ x V | x x }. A substitution is called a ground substitution if it maps all variables of its domain to ground, i.e.variable-free, terms. The substitution instance t of a ground substitution is a ground term if all of t's variables arein 's domain, i.e. if vars(t) dom(). A substitution is called a linear substitution if t is a linear term for some(and hence every) term t containing just the variables of 's domain, i.e. with vars(t) = dom(). A substitution iscalled a at substitution if x is a variable for every variable x. A substitution is called a renaming substitution, ifit is a permutation on the set of all variables. Like every permutation, a renaming substitution always has an inversesubstitution 1, such that t1 = t = t1 for every term t. However, it is not possible to dene an inverse for anarbitrary substitution.For example, { x 2, y 3+4 } is a ground substitution, { x x1, y y2+4 } is non-ground and non-at, butlinear, { x y2, y y2+4 } is non-linear and non-at, { x y2, y y2 } is at, but non-linear, { x x1, y y2 } is both linear and at, but not a renaming, since is maps both y and y2 to y2; each of these substitutions has theset {x,y} as its domain. An example for a renaming substitution is { x x1, x1 y, y y2, y2 x }, it has theinverse { x y2, y2 y, y x1, x1 x }. The at substitution { x z, y z } cannot have an inverse, since e.g.(x+y) { x z, y z } = z+z, and the latter term cannot be transformed back to x+y, as the information about theorigin a z stems from is lost. The ground substitution { x 2 } cannot have an inverse due to a similar loss of origininformation e.g. in (x+2) { x 2 } = 2+2, even if replacing constants by variables was allowed by some ctitiouskind of generalized substitutions.Two substitutions are considered equal if they map each variable to structurally equal result terms, formally: = ifx = x for each variable x V. The composition of two substitutions = { x1 t1, ..., xk tk } and = { y1 u1, ..., ylul } is obtained by removing from the substitution { x1 t1, ..., xk tk, y1 u1, ..., yl ul } thosepairs yi ui for which yi { x1, ..., xk }. The composition of and is denoted by . Composition is an associativeoperation, and is compatible with substitution application, i.e. () = (), and (t) = t(), respectively, for everysubstitutions , , , and every term t. The identity substitution, which maps every variable to itself, is the neutralelement of substitution composition. A substitution is called idempotent if = , and hence t = t for everyterm t. The substitution { x1 t1, ..., xk tk } is idempotent if and only if none of the variables xi occurs in any ti.Substitution composition is not commutative, that is, may be dierent from , even if and are idempotent.[1][2]

For example, { x 2, y 3+4 } is equal to { y 3+4, x 2 }, but dierent from { x 2, y 7 }. Thesubstitution { x y+y } is idempotent, e.g. ((x+y) {xy+y}) {xy+y} = ((y+y)+y) {xy+y} = (y+y)+y, while thesubstitution { x x+y } is non-idempotent, e.g. ((x+y) {xx+y}) {xx+y} = ((x+y)+y) {xx+y} = ((x+y)+y)+y.An example for non-commuting substitutions is { x y } { y z } = { x z, y z }, but { y z} { x y} = {x y, y z }.

10.3 See also Substitution property in Equality (mathematics)#Some basic logical properties of equality First-order logic#Rules of inference Rule of universal substitution

28 CHAPTER 10. SUBSTITUTION (LOGIC)

Lambda calculus#Substitution Truth-value semantics Unication (computer science) Metavariable Mutatis mutandis Rule of replacement

10.4 Notes[1] some authors use [ t1/x1, ..., tk/xk ] to denote that substitution, e.g. M. Wirsing (1990). Jan van Leeuwen, ed. Algebraic

Specication. Handbook of Theoretical Computer Science B. Elsevier. pp. 675788., here: p.682;

[2] From a term algebra point of view, the set T of terms is the free term algebra over the set V of variables, hence for eachsubstitution mapping : V T there is a unique homomorphism : T T that agrees with on V T ; the above-denedapplication of to a term t is then viewed as applying the function to the argument t.

10.5 References Hunter, G. (1971). Metalogic: An Introduction to the Metatheory of Standard First Order Logic. University ofCalifornia Press. ISBN 0-520-01822-2

Kleene, S. C. (1967). Mathematical Logic. Reprinted 2002, Dover. ISBN 0-486-42533-9

[1] David A. Duy (1991). Principles of Automated Theorem Proving. Wiley.; here: p.73-74

[2] Franz Baader, Wayne Snyder (2001). Alan Robinson and Andrei Voronkov, ed. Unication Theory (PDF). Elsevier. pp.439526.; here: p.445-446

10.6 External links Substitution in nLab

Chapter 11

Tautology (logic)

In logic, a tautology (from the Greek word ) is a formula that is true in every possible interpretation.Philosopher Ludwig Wittgenstein rst applied the term to redundancies of propositional logic in 1921. (It had beenused earlier to refer to rhetorical tautologies, and continues to be used in that alternate sense.) A formula is satisableif it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisable. Unsat-isable statements, both through negation and armation, are known formally as contradictions. A formula that isneither a tautology nor a contradiction is said to be logically contingent. Such a formula can be made either true orfalse based on the values assigned to its propositional variables. The double turnstile notation S is used to indicatethat S is a tautology. Tautology is sometimes symbolized by Vpq", and contradiction by Opq". The tee symbol >is sometimes used to denote an arbitrary tautology, with the dual symbol ? (falsum) representing an arbitrary con-tradiction; in any symbolism, a tautology may be substituted for the truth value "true, as symbolized, for instance,by 1.Tautologies are a key concept in propositional logic, where a tautology is dened as a propositional formula that istrue under any possible Boolean valuation of its propositional variables. A key property of tautologies in propositionallogic is that an eective method exists for testing whether a given formula is always satised (or, equivalently, whetherits negation is unsatisable).The denition of tautology can be extended to sentences in predicate logic, which may contain quantiers, unlikesentences of propositional logic. In propositional logic, there is no distinction between a tautology and a logically validformula. In the context of predicate logic, many authors dene a tautology to be a sentence that can be obtained bytaking a tautology of propositional logic and uniformly replacing each propositional variable by a rst-order formula(one formula per propositional variable). The set of such formulas is a proper subset of the set of logically validsentences of predicate logic (which are the sentences that are true in every model).

11.1 HistoryThe word tautologywas used by the ancient Greeks to describe a statement that was true merely by virtue of saying thesame thing twice, a pejorative meaning that is still used for rhetorical tautologies. Between 1800 and 1940, the wordgained new meaning in logic, and is currently used in mathematical logic to denote a certain type of propositionalformula, without the pejorative connotations it originally possessed.In 1800, Immanuel Kant wrote in his book Logic:

The identity of concepts in analytical judgments can be either explicit (explicita) or non-explicit (im-plicita). In the former case analytic propositions are tautological."

Here analytic proposition refers to an analytic truth, a statement in natural language that is true solely because of theterms involved.In 1884, Gottlob Frege proposed in his Grundlagen that a truth is analytic exactly if it can be derived using logic.But he maintained a distinction between analytic truths (those true based only on the meanings of their terms) andtautologies (statements devoid of content).

29

30 CHAPTER 11. TAUTOLOGY (LOGIC)

In 1921, in his Tractatus Logico-Philosophicus, Ludwig Wittgenstein proposed that statements that can be deducedby logical deduction are tautological (empty of meaning) as well as being analytic truths. Henri Poincar had madesimilar remarks in Science and Hypothesis in 1905. Although Bertrand Russell at rst argued against these remarksby Wittgenstein and Poincar, claiming that mathematical truths were not only non-tautologous but were synthetic,he later spoke in favor of them in 1918:

Everything that is a proposition of logic has got to be in some sense or the other like a tautology. It hasgot to be something that has some peculiar quality, which I do not know how to dene, that belongs tological propositions but not to others.

Here logical proposition refers to a proposition that is provable using the laws of logic.During the 1930s, the formalization of the semantics of propositional logic in terms of truth assignments was devel-oped. The term tautology began to be applied to those propositional formulas that are true regardless of the truthor falsity of their propositional variables. Some early books on logic (such as Symbolic Logic by C. I. Lewis andLangford, 1932) used the term for any proposition (in any formal logic) that is universally valid. It is common inpresentations after this (such as Stephen Kleene 1967 and Herbert Enderton 2002) to use tautology to refer to a logi-cally valid propositional formula, but to maintain a distinction between tautology and logically valid in the context ofrst-order logic (see below).

11.2 BackgroundMain article: propositional logic

Propositional logic begins with propositional variables, atomic units that represent concrete propositions. A for-mula consists of propositional variables connected by logical connectives in a meaningful way, so that the truth ofthe overall formula can be uniquely deduced from the truth or falsity of each variable. A valuation is a functionthat assigns each propositional variable either T (for truth) or F (for falsity). So, for example, using the propositionalvariables A and B, the binary connectives _ and ^ representing disjunction and conjunction respectively, and theunary connective : representing negation, the following formula can be obtained:: (A ^ B) _ (:A) _ (:B) . Avaluation here must assign to each of A and B either T or F. But no matter how this assignment is made, the overallformula will come out true. For if the rst conjunction (A ^B) is not satised by a particular valuation, then one ofA and B is assigned F, which will cause the corresponding later disjunct to be T.

11.3 Denition and examplesA formula of propositional logic is a tautology if the formula itself is always true regardless of which valuation isused for the propositional variables.There are innitely many tautologies. Examples include:

(A _ :A) ("A or not A"), the law of the excluded middle. This formula has only one propositional variable,A. Any valuation for this formula must, by denition, assign A one of the truth values true or false, and assign: A the other truth value.

(A ! B) , (:B ! :A) (if A implies B then not-B implies not-A", and vice versa), which expresses thelaw of contraposition.

((:A! B)^ (:A! :B))! A (if not-A implies both B and its negation not-B, then not-Amust be false,then A must be true), which is the principle known as reductio ad absurdum.

:(A ^B), (:A _ :B) (if not both A and B, then not-A or not-B", and vice versa), which is known as deMorgans law.

((A ! B) ^ (B ! C)) ! (A ! C) (if A implies B and B implies C, then A implies C"), which is theprinciple known as syllogism.

11.4. VERIFYING TAUTOLOGIES 31

((A _ B) ^ (A ! C) ^ (B ! C)) ! C (if at least one of A or B is true, and each implies C, then C mustbe true as well), which is the principle known as proof by cases.

A minimal tautology is a tautology that is not the instance of a shorter tautology.

(A _B)! (A _B) is a tautology, but not a minimal one, because it is an instantiation of C ! C .

11.4 Verifying tautologiesThe problem of determining whether a formula is a tautology is fundamental in propositional logic. If there are nvariables occurring in a formula then there are 2n distinct valuations for the formula. Therefore the task of determiningwhether or not the formula is a tautology is a nite, mechanical one: one need only evaluate the truth value of theformula under each of its possible valuations. One algorithmic method for verifying that every valuation causes thissentence to be true is to make a truth table that includes every possible valuation.For example, consider the formula

((A ^B)! C), (A! (B ! C)):

There are 8 possible valuations for the propositional variables A, B, C, represented by the rst three columns of thefollowing table. The remaining columns show the truth of subformulas of the formula above, culminating in a columnshowing the truth value of the original formula under each valuation.Because each row of the nal column shows T, the sentence in question is veried to be a tautology.It is also possible to dene a deductive system (proof system) for propositional logic, as a simpler variant of thedeductive systems employed for rst-order logic (see Kleene 1967, Sec 1.9 for one such system). A proof of atautology in an appropriate deduction system may be much shorter than a complete truth table (a formula with npropositional variables requires a truth table with 2n lines, which quickly becomes infeasible as n increases). Proofsystems are also required for the study of intuitionistic propositional logic, in which the method of truth tables cannotbe employed because the law of the excluded middle is not assumed.

11.5 Tautological implicationMain article: Tautological consequence

A formula R is said to tautologically imply a formula S if every valuation that causes R to be true also causes S tobe true. This situation is denoted R j= S . It is equivalent to the formula R! S being a tautology (Kleene 1967 p.27).For example, let S be A^ (B _:B) . Then S is not a tautology, because any valuation that makes A false will makeS false. But any valuation that makes A true will make S true, because B _ :B is a tautology. Let R be the formulaA ^ C . Then R j= S , because any valuation satisfying R makes A true and thus makes S true.It follows from the denition that if a formula R is a contradiction then R tautologically implies every formula, becausethere is no truth valuation that causes R to be true and so the denition of tautological implication is trivially satised.Similarly, if S is a tautology then S is tautologically implied by every formula.

11.6 SubstitutionMain article: Substitution instance

There is a general procedure, the substitution rule, that allows additional tautologies to be constructed from a giventautology (Kleene 1967 sec. 3). Suppose that S is a tautology and for each propositional variable A in S a xed

32 CHAPTER 11. TAUTOLOGY (LOGIC)

sentence SA is chosen. Then the sentence obtained by replacing each variable A in S with the corresponding sentenceSA is also a tautology.For example, let S be (A ^ B) _ (:A) _ (:B) , a tautology. Let SA be C _D and let SB be C ! E . It followsfrom the substitution rule that the sentence

((C _ D) ^ (C ! E)) _ (:(C _ D)) _ (:(C ! E)) is a tautology. In turn, a tautology may besubstituted for the truth value "true": for instance, when true is symbolized by 1, a tautology may besubstituted for 1.

11.7 Ecient verication and the Boolean satisability problemThe problem of constructing practical algorithms to determine whether sentences with large numbers of propositionalvariables are tautologies is an area of contemporary research in the area of automated theorem proving.The method of truth tables illustrated above is provably correct the truth table for a tautology will end in a columnwith only T, while the truth table for a sentence that is not a tautology will contain a row whose nal column is F, andthe valuation corresponding to that row is a valuation that does not satisfy the sentence being tested. This methodfor verifying tautologies is an eective procedure, which means that given unlimited computational resources it canalways be used to mechanistically determine whether a sentence is a tautology. This means, in particular, the set oftautologies over a xed nite or countable alphabet is a decidable set.As an ecient procedure, however, truth tables are constrained by the fact that the number of valuations that must bechecked increases as 2k, where k is the number of variables in the formula. This exponential growth in the computationlength renders the truth table method useless for formulas with thousands of propositional variables, as contemporarycomputing hardware cannot execute the algorithm in a feasible time period.The problem of determining whether there is any valuation that makes a formula true is the Boolean satisabilityproblem; the problem of checking tautologies is equivalent to this problem, because verifying that a sentence S is atautology is equivalent to verifying that there is no valuation satisfying :S . It is known that the Boolean satisabilityproblem is NP complete, and widely believed that there is no polynomial-time algorithm that can perform it. Currentresearch focuses on nding algorithms that perform well on special classes of formulas, or terminate quickly onaverage even though some inputs may cause them to take much longer.

11.8 Tautologies versus validities in rst-order logicThe fundamental denition of a tautology is in the context of propositional logic. The denition can be extended,however, to sentences in rst-order logic (see Enderton (2002, p. 114) and Kleene (1967 secs. 1718)). These sen-tences may contain quantiers, unlike sentences of propositional logic. In the context of rst-order logic, a distinctionis maintained between logical validities, sentences that are true in every model, and tautologies, which are a propersubset of the rst-order logical validities. In the context of propositional logic, these two terms coincide.A tautology in rst-order logic is a sentence that can be obtained by taking a tautology of propositional logic anduniformly replacing each propositional variable by a rst-order formula (one formula per propositional variable). Forexample, because A _ :A is a tautology of propositional logic, (8x(x = x)) _ (:8x(x = x)) is a tautology inrst order logic. Similarly, in a rst-order language with a unary relation symbols R,S,T, the following sentence is atautology:

(((9xRx) ^ :(9xSx))! 8xTx), ((9xRx)! ((:9xSx)! 8xTx)):

It is obtained by replacing A with 9xRx , B with :9xSx , and C with 8xTx in the propositional tautology ((A ^B)! C), (A! (B ! C)) .Not all logical validities are tautologies in rst-order logic. For example, the sentence

(8xRx)! :9x:Rx

11.9. SEE ALSO 33

is true in any rst-order interpretation, but it corresponds to the propositional sentenceA! B which is not a tautologyof propositional logic.

11.9 See also

11.9.1 Normal forms Algebraic normal form Conjunctive normal form Disjunctive normal form Logic optimization

11.9.2 Related logical topics

11.10 References Bocheski, J. M. (1959) Prcis of Mathematical Logic, translated from the French and German editions byOtto Bird, Dordrecht, South Holland: D. Reidel.

Enderton, H. B. (2002)AMathematical Introduction to Logic, Harcourt/Academic Press, ISBN 0-12-238452-0. Kleene, S. C. (1967) Mathematical Logic, reprinted 2002, Dover Publications, ISBN 0-486-42533-9. Reichenbach, H. (1947). Elements of Symbolic Logic, reprinted 1980, Dover, ISBN 0-486-24004-5 Wittgenstein, L. (1921). Logisch-philosophiche Abhandlung, Annalen der Naturphilosophie (Leipzig), v.14, pp. 185262, reprinted in English translation as Tractatus logico-philosophicus, New York and London,1922.

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

Weisstein, Eric W., Tautology, MathWorld.

Chapter 12

Tautology (rhetoric)

In rhetoric, a tautology (from Greek , the same and , word/idea) is a logical argument constructedin such a way, generally by repeating the same concept or assertion using dierent phrasing or terminology, that theproposition as stated is logically irrefutable, while obscuring the lack of evidence or valid reasoning supporting thestated conclusion. (A rhetorical tautology should not be confused with a tautology in propositional logic.)[lower-alpha 1]

12.1 Rhetorical tautology vs. circular reasoningCircular reasoning diers from tautologies in that circular reasoning restates the premise as the conclusion, insteadof deriving the conclusion from the premise. (This is often conated with begging the question, in which the premiserelies on the assumption of the conclusion). A tautology simply states the same thing twice.Rhetorical tautologies typically present themselves as redundancies only comprising part of a statement.

12.1.1 ExampleCircular reasoning: God exists because the Bible says so, and the Bible is Gods word.Rhetorical tautology: A tautology is anything that is tautological.

12.2 See also Figure of speech Hyperbole Law of identity List of tautonyms List of tautological place names No true Scotsman Oxymoron Pleonasm Redundancy (linguistics) Rhetoric Tautology (logic) Vacuous truth

34

12.3. NOTES 35

12.3 Notes[1] Rhetorical tautologies state the same thing twice, while appearing to state two or more dierent things; logical tautologies

state the same thing twice, and must do so by logical necessity. The inherent meanings and subsequent conclusions inrhetorical and logical tautologies or logical necessities are very dierent. By axiomatic necessity, logical tautologies areneither refutable nor veriable under any condition.

12.4 External links Figures of Speech: Tautology

Chapter 13

Two Dogmas of Empiricism

Two Dogmas of Empiricism is a paper by analytic philosopher Willard Van Orman Quine published in 1951. Ac-cording to City University of New York professor of philosophy Peter Godfrey-Smith, this paper [is] sometimesregarded as the most important in all of twentieth-century philosophy".[1] The paper is an attack on two central aspectsof the logical positivists philosophy. One is the analytic-synthetic distinction between analytic truths and synthetictruths, explained by Quine as truths grounded only in meanings and independent of facts, and truths grounded in facts.The other is reductionism, the theory that each meaningful statement gets its meaning from some logical constructionof terms that refers exclusively to immediate experience.Two Dogmas is divided into six sections. The rst four sections are focused on analyticity, the last two sections onreductionism. There, Quine turns the focus to the logical positivists theory of meaning. He also presents his ownholistic theory of meaning.

13.1 Analyticity and circularityMost of Quines argument against analyticity in the rst four sections is focused on showing that dierent explanationsof analyticity are circular. The main purpose is to show that no satisfactory explanation of analyticity has been given.Quine begins by making a distinction between two dierent classes of analytic statements. The rst one is calledlogically true and has the form:

(1) No unmarried man is married

A sentence with that form is true independent of the interpretation of man and married, so long as the logicalparticles no, un-" and is have their ordinary English meaning.The statements in the second class have the form:

(2) No bachelor is married.

A statement with this form can be turned into a statement with form (1) by exchanging synonyms with synonyms, inthis case bachelor with unmarried man. It is the second class of statements that lack characterization accordingto Quine. The notion of the second form of analyticity leans on the notion of synonymy, which Quine believes isin as much need of clarication as analyticity. Most of Quines following arguments are focused on showing howexplanations of synonymy end up being dependent on the notions of analyticity, necessity, or even synonymy itself.How do we reduce sentences from the second class to a sentence of class (1)? Some might propose denitions. Nobachelor is married can be turned into No unmarried man is married because bachelor is dened as unmarriedman. But, Quine asks: how do we nd out that bachelor is dened as unmarried man"? Clearly, a dictionarywould not solve the problem, as a dictionary is a report of already known synonyms, and thus is dependent on thenotion of synonymy, which Quine holds as unexplained.A second suggestion Quine considers is an explanation of synonymy in terms of interchangeability. Two linguisticforms are (according to this view) synonymous if they are interchangeable without changing the truth-value. That is,in all contexts without change of truth value. But consider the following example:

36

13.2. REDUCTIONISM 37

(3) Bachelor has fewer than ten letters.

Obviously bachelor and unmarried man are not interchangeable in that sentence. To exclude that example andsome other obvious counterexamples, such as poetic quality, Quine introduces the notion of cognitive synonymy. Butdoes interchangeability hold as an explanation of cognitive synonymy? Suppose we have a language without modaladverbs like necessarily. Such a language would be extensional, in the way that two predicates which are true aboutthe same objects are interchangeable again without altering the truth-value. Thus, there is no assurance that two termsthat are interchangeable without the truth-value changing are interchangeable because of meaning, and not becauseof chance. For example, creature with a heart and creature with kidneys share extension.In a language with the modal adverb necessarily the problem is solved, as salva veritate holds in the following case:

(4) Necessarily all and only bachelors are unmarried men

while it does not hold for

(5) Necessarily all and only creatures with a heart are creatures with kidneys.

Presuming that 'creature with a heart' and 'creature with kidneys have the same extension, they will be interchangeablesalva veritate. But this interchangeability rests upon both empirical features of the language itself and the degree towhich extension is empirically found to be identical for the two concepts, and not upon the sought for principle ofcognitive synonymy.It seems that the only way to assert the synonymy is by supposing that the terms 'bachelor' and 'unmarried man' aresynonymous and that the sentence All and only all bachelors are unmarried men is analytic. But for salva veritateto hold as a denition of something more than extensional agreement, i.e., cognitive synonymy, we need a notion ofnecessity and thus of analyticity.So, from the above example, it can be seen that in order for us to distinguish between analytic and synthetic we mustappeal to synonymy; at the same time, we should also understand synonymy with interchangeability salva veritate.However, such a condition to understand synonymy is not enough so we not only argue that the terms should beinterchangeable, but necessarily so. And to explain this logical necess