# LOGIC I General methodology and introduction to formal logic DRAFT.

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LOGIC I General methodology and introduction to formal logic DRAFT

Lectures on informal logic are based on books by Fogelin & Sinnot-Armstrong, Fisher and Thomson and on lectures by Jesper Kallestrup.

Lecture I Informal reasoning Skills: - recognizing reasoning (arguments) - identifying conclusions - identifying reasons (premises) and assumptions - evaluating reasoning

Logical theory Semantic relationships (relationships between true or false propositions) Argument = set of propositions Context irrelevant Logical pragmatics Use of propositions by an arguer to carry out the goal of dialogue (e.g. convince or persuade the second arguer)

A seaman drafted to our ship just before we sailed form Halifaxhad never seen his new captain, who at sea often went hatless and wore a nondescript jacket. The new man had just begun a forenoon watch on the gun deck when the captain came along. The skipper suddennly stooped and picked up a butted cigarette. He trust the butt at the seaman and demanded: „I want to know who the hell owns this damned thing” The new hand considered for a moment, then said slowly to the rankless, hatless officer: „I’d say you do, mate. You found it.”

Context of dialoge Seaman: the ownership of the cigarette butt Captain: the issue of keeping the ship clean

Types of argumentative dialogue Dialogue – a sequence of exchanges of messages or speech acts (typically questions and replies) between two (or more) participants Every dialogue has a goal and requires cooperation between the participants to fulfill the goal Each participant has an obligation to work toward fulfilling his own goal in the dialogue and also an obligation to cooperate with the other participant’s fulfillment of his own goal

One context of dialogue is the personal quarrel. A quarrel is a dialogue in which one tries to trick, cheat or even attack one’s opponent directly, rather than one’s opponent’s views, using abusive language, appeal to emotions, intimidation, one-sided criticism, etc. Any means are available no matter whether they are fair or reasonable. Aggressive personal attack Appeal to emotions Desire to win the argument at all costs

A second context of dialogue is the (forensic) debate. In debates there are judges or referees who decide, maybe by voting, which side has the better argument. There are rules of procedure that determine who may speak and when and for how long. There are also some rules that disallow the more severe forms of personal attack, but many fallacious arguments may still be tolerated.

Audience Rules are often very permissive and may allow fallacious arguments Goal: win a verbal victory to impress the audience

A third context of dialogue is the persuasion dialogue (critical discussion). There are two participants each of whom has a thesis to prove. Internal proof by a participant means proof by inferring a proposition from the other participant’s concession in the dialogue. External proof is the introduction of new facts into the argument by appealing to scientific evidence or expert opinion. The best one can hope for is plausible commitment to an opinion based on reasoned evidence. My goal is to persuade you of my thesis; hence I should prove that thesis from premises that you accept or are committed to Your goal is to prove your thesis from the premises that I accept or am committed to Goal – persuade the other party of your thesis Method – prove your thesis

A forth context of dialogue is the inquiry in which premises can only be propositions that are known to be true, that have been established to the satisfaction of all parties to the inquiry. The inquiry seeks out as much certainty as can be obtained by the given evidence. The goal is to accumulate knowledge. The participants are neutral investigators of an objective truth. The inquiry is cooperative rather than adversarial.

In negotiation dialogue, the primary goal is self-interest and the method is to bargain. Bargaining makes no pretensions to be an objective inquiry into the truth of the matter. Logical proof is not important Frankly based on personal gain Not neutral, not objective Interest-based conflict

Information-seeking dialogue – one party has the goal of finding infromation that the other party is believed to possess Action-seeking dialogue – one party has the goal to bring about a specific course of action by the other party Educational dialogue – one party (the teacher) has the goal of imparting knowledge to the other party (the student)

dialogueinitial situationmethod goal quarrelEmotional disquietPersonal attack„Hit” out at other debateForensic contestVerbal victoryImpress audience Persuasion (critical discussion) Difference of opinion Internal and external proof Persuade other inquiryLack of proofKnowledge-based argumentation Establish proof negotiationDifference of interests bargainingPersonal gain Info-seekingLacking information questioningFind information Action-seekingNeed for actionIssue imperativesProduce action educationalignoranceteachingImparting knowledge

Arguments Socrates is a man. All men are mortal. Socrates is mortal.

Since Socrates is a man and all men are mortal, Socrates is mortal. Socrates is a man, since all men are mortal and Socrates is mortal.

Socrates is a man. All men are mortal. Therefore Socrates is mortal. Since Socrates is a man, all men are mortal and Socrates is mortal.

An argument is a train of reasoning aimed at establishing a particular claim, the conclusion, from a number of other claims, the premises. The premises are offered as reasons to believe or accept the conclusion. Arguments attempt to persuade others to accept a claim by offering reasons or evidence in support of that claim. One must do two things in propounding an argument: justify the premises by providing reasons or evidence, and show how the conclusion follows from the premises.

The bus is late. It must have broke down. That bird can’t be a robin. It doesn’t have a red breast. You should try to appear confident in your job interview. The employers are looking for someone who can speak confidently in public.

He must be older than he says. He told us he was forty-two, but he has a daughter who is at least thirty years old.

She didn’t turn up for their date. She obviously doesn’t really want to be his girlfriend. If she’d wanted a serious relationship with him she wouldn’t have missed the date. The engine won’t fire. The carburettor must be blocked.

Reason indicators Because …. For …. Since …. Follows from the fact that ….. The reason being ….. Firstly, ….secondly, May be inferred from the fact that ….

Conclusion indicators Therefore So Hence Thus Accordingly Consequently Which proves that Justifies the belief that I conclude that Which implies that Which allows us to infer that It follows that Establishes the fact that Demonstrates that

(1)People who diet lose weight. Wojciech Mann cannot have dieted. He hasn’t lost weight. (1*)People who diet lose weight. But Wojciech Mann hasn’t lost weight. (Therefore), he cannot have dieted. (1**)People who diet lose weight. But Wojciech Mann cannot have dieted. Therefore, he hasn’t lost weight.

People who diet lose weight. Wojciech Mann is a good journalist despite his weight. Wojciech Mann would be a better journalist if he dieted.

(1***)People who diet lose weight. Since Wojciech Mann hasn’t lost weight, he cannot have dieted.

John broke the window because he tripped. John broke the window because he has forgotten his key. John must have broken the window because he was the only person in the house.

Indicative conditional vs. argument If international terrorism continues to grow, there will be a worldwide crisis. Since international terrorism continues to grow, there will be a worldwide crisis.

Standard form of arguments Socrates is a man. All men are mortal. _____________________ Socrates is mortal.

Since Chicago is north of Boston, and Boston is north of Charleston, Chicago is north of Charleston. Toward evening, clouds formed and the sky grew darker; then the storm broke. Both houses of Congress may pass the bill, but the President may still veto it.

Texas has a greater area than Topeka, and Topeka has a greater area than the Bronx Zoo, so Texas has a greater area than the Bronx Zoo. Other airlines will carry more passengers because United Airlines is on strike. Since Jesse James left town, taking his gang with him, things have been a lot quieter.

Things are a lot quieter because Jesse James left town, taking his gang with him. Witches float, because witches are made of wood, and wood floats. The hour is up, so you must hand in your exams. Joe quit because his boss was giving him so much grief.

Red squirrels can eat yellow berries, hawthorn berries and rosehips. Grey squirrels can eat none of these. However, grey squirrels eat acorns which red squirrels cannot eat. In recent years, the demand for computer-literate personnel has increased. More students are graduating in computing science than before. Some companies find that these graduates require further training before embarking on a career in computing.

The North American Wildlife Federation, which sponsors an annual watch for endangered species, reports that sightings of the bald eagle between 1978 and 1979 increased by 35 per cent. In 1979, 13,127 sightings were reported, 3,400 over the 1978 count. This indicates considerable growth in the bald eagle population. To make an assessment of modern art is an impossible task. For one can assess a work of art only when there are accepted rules and conventions. Modern art has no rules and conventions.

If the money supply were to increase at less than 5% the rate of inflation would come down. Since the money supply is increasing at about 10% inflation will not come down. If Russia were unsure about American reactions to an attack on Western Europe, and if her intentions were to conquer Western Europe, she would create local casus belli (causes of war) but since she has not done this, she cannot intend to conquer Western Europe.

If the civil population cannot be defended in the event of nuclear war, we do not need a civil defence policy. But, we do need a civil defence policy if ‘deterrence’ is to be a convincing strategy. Therefore deterrence is not a convincing strategy. The materials of nature (air, earth, water) that remain untouched by human effort belong to no-one and are not property. It follows that a thing can become someone’s property only if he works and labours on it to change its natural state. From this I conclude that whatever a man improves by the labour of his hand and brain belongs to him and to him alone.

The only freedom which deserves the name is that of pursuing our own good in our own way, so long as we do not attempt to deprive others of theirs, or impede their efforts to obtain it. Each is the proper guardian of his own health, whether bodily, or mental and spiritual. Mankind are greater gainers suffering each other to live as seems good to themselves, rather than by compelling each to live as seems good to the rest.

Radioactive elements disintegrate and eventually turn into lead. If matter has always existed there should be no radioactive elements left. The presence of uranium etc. is scientific proof that matter has not always existed.

If the ‘nuclear winter’ scientists are right the population of Britain would be virtually eliminated in a nuclear war between the superpowers even if Britain suffered no direct nuclear attack. Quite apart from the radioactive fall-out, we would suffer the darkness, the subfreezing temperatures and the mass starvation of a nuclear winter.

Some people have solved their own unemployment problem by great ingenuity in hunting for a job or by willingness to work for less, so all the unemployed could do this.

Evaluating arguments Validity An argument is valid if and only if (iff) it is not possible for the premises to be true and the conclusion false.

All ministers are paid. Radosław Sikorski is a minister. _________________ Radosław Sikorski is paid.

All ministers are paid. Radosław Sikorski is paid. _________________ Radosław Sikorski is a ministrer.

All MPs are paid. Bronisław Komorowski is paid. ___________________ Bronisław Komorowski is an MP.

Truth All MPs are paid. Bronisław Komorowski is an MP. ___________________ Bronisław Komorowski is paid.

Soundness An argument is sound iff it is valid and all of its premises are true.

All premises true At least one false premise VALIDSOUNDUNSOUND INVALIDUNSOUND

Truth and falsity are properties of claims, propositions or statements Validity and soundness are properties of arguments Valid arguments are truth-preserving.

If the argument is invalid, it cannot establish its conclusion. But it may still be a reasonable or persuasive argument by some other standards. It may be that the premises lend inductive support to the conclusion. Inductive arguments do not guarantee the truth of their conclusion, but yield more or less highly probable conclusions. Or it may be that the truth of the conclusion is the best explanation of the truth of the premises. When we talk about validity, we typically mean deductive validity.

Valid or invalid? If John has drunken seven pints of lager, then he can’t ride his bike. John can’t ride his bike. So, John has drunken seven pints of lager. If it rains tomorrow, then the match will be cancelled. If the trains are not running tomorrow, then the match will be cancelled. Therefore, if it rains tomorrow, then the trains are not running tomorrow.

Most electors prefer A to B Most electors prefer B to C _________________________ Most electors prefer A to C

1/3 1ACB 2BAC 3CBA

Sound or unsound? If George Bush will invade Russia, then Tony Blair is not a conservative MP. Tony Blair is a conservative MP. So, George Bush will not invade Russia. Either Hugh Grant is German or PO won the last election. PO won the last election. So, Hugh Grant is not German.

Any argument must take something for granted. Basic reasons or basic premises are those that are presented without themselves being supported by other reasons or premises.

When a passage contains more than one premise we need find out whether joint reasons or independent reasons are being offered. It is right to ban cigarette advertising because it encourages young people to start smoking. But even if it had no such influence on young people, it would be right to ban smoking because it could give existing smokers the mistaken impression that their habit is socially acceptable.

If the money supply were to increase at less than 5% the rate of inflation would come down. Since the money supply is increasing at about 10% inflation will not come down.

Some arguments are very complex in that they may contain an intermediate conclusion and a main conclusion. So, first premises are advanced in support of an intermediate conclusion, and then that conclusion itself features as a premise that is advanced together with other premises in support of the main conclusion.

A majority of prospective parents would prefer to have sons rather than daughters. So, if people can choose the sex of their child, it is likely that eventually there will be many more males than females in the population. A preponderance of males in the population is likely to produce serious social problems. Therefore, we should discourage the use of techniques which enable people to choose the sex of their child.

There is no way I can finish my paper before the 9 o’clock show, since I have to do the reading first, so I won’t even start writing until at least 9 o’clock.

I have to do the reading first. I won’t even start writing until at least 9 o’clock. ________________________ There is no way I can finish my paper before the 9 o’clock show.

I have to do the reading first. _______________________ I won’t even start writing until at least 9 o’clock. __________________________ There is no way I can finish my paper before the 9 o’clock show

Assumptions = suppressed (unstated) premises We need to distinguish between premises, conclusions and assumptions. An assumption is something that is taken for granted in an argument without being explicitly stated. Assumptions function in arguments either by giving support to the basic premises, or as a missing step within the argument—maybe as an additional premise that needs to be added in order for the conclusion to follow from the existent premises.

The burglar must have left by the fire escape. This person is not in the building now, but has not been seen leaving the building, and there are guards posted at each entrance.

Unstated premises: Shared facts Linguistic principles Harriet is in New York with her son. _____________________ Harriet’s son is in New York.  Donald Tusk cannot become president of the United States because he was born in Poland.

Other kinds You shouldn’t buy pornography, because it leads to violence toward women.

An argument based on suppressed premises is called an enthymeme and is said to be enthymematic.

argument Some people say that the depiction of violence on television has no effect on viewer’s behaviour. However, if what was shown on television did not affect behaviour, television advertising would never influence viewers to buy certain products. But we know that it does. So it cannot be true that television violence does not affect behaviour.

Interpretation I Basic reason:Television advertising affect viewers’ behaviour Assumption:Television advertising and television violence are similar in that if one affects the behaviour of viewers, so does the other Conclusion: Television violence affects behaviour

Interpretation 2 Basic reason 1:Television advertising affect viewers’ behaviour Intermediate conclusion:What is shown on television affects viewers’ behaviour Main conclusion:Television violence affects viewers’ behaviour

Assert, assume, suppose To assert is to claim that something is true, or to present something as true. Assertions are expressed by means of assertoric sentences, e.g. ‘the window is closed’. Premises and conclusions of arguments are asserted propositions. To assert that p is typically to express the belief that p. To assume is to take something for granted without actually mentioning or asserting it. Assumptions (suppressed premises) are typically implicit, but can be made explicit by means of assertoric sentences. To suppose is to take something for granted for the sake of argument. Suppositions are explicit, but are not asserted. To suppose that p needn’t express belief or acceptance that p. P isn’t presented as being true, but is put forward so that we may consider its implications. The supposition that p is often made in order to conclude that p is false.

Supposing for the sake of argument that.... suppositional arguments In suppositional arguments, p is supposed to be true by the arguer. But the arguer doesn’t have to believe that p. Suppose Darwin’s theory of evolution is true. Then there should be fossil evidence which shows species changing and evolving, but this evidence simply doesn’t exist so Darwin’s theory must be wrong.

In suppositional arguments the arguer often believes or knows that p is false. The arguer asks us to consider p with a view to drawing out its implications—implications which he takes to be implausible.

Supposition indicators: Suppose that... Let us assume that... Imagine that... Consider the hypothesis/theory that... Let us postulate that...

Conditionalisation Suppose we have an argument which proceeds from some supposition R to the conclusion C by logically valid steps (i.e. the conclusion at each step follows from the reasons given for it) then the validity of the argument entitles us to infer the conditional (hence the name ‘conditionalisation’), if R then C.

Rule If we have an argument which proceeds from a supposition R to a conclusion C and then conditionalises to the conclusion „if R then C”, whether this conditional conclusion is established does not depend on the truth of R. If other basic reasons are true and the argument is sound, „if R then C” is established whether R is true or false.

Suppose that Einstein’s theory of relativity is true. Then it follows that everything is relative. So, it follows that there are no absolute moral values. So, if Einstein’s theory of relativity is true, then there are no absolute moral values.

Suppose that u. Then it follows that u. So, it follows that u. So, if Einstein’s theory of relativity is true, then there are no absolute moral values.

Galileo’s argument Suppose (as Aristotle believed) that the heavier a body is, the faster it falls to the ground and suppose that we have two bodies, a heavy one called H and a light one called L. Under our initial assumption, H will fall faster than L. Now suppose that H and L are joined together thus L/H. Now what happens? Well, L/H is heavier than H, so by our initial assumption it should fall faster than H alone. But in the joined body L/H L and H will each tend to fall just as fast as before they were joined, so L will act as a brake on H and L/H will fall slower than H alone. Hence it follows from our initial assumption that L/H will fall both faster and slower than M alone. Since this is absurd, our initial assumption must be false.

This style of reasoning is called reductio ad absurdum: a particular claim is supposed for the sake of argument. Then an absurdity - a contradiction for instance - is deduced via logically valid steps.

Induction versus deduction All ravens are black. ______________ If there is a raven on top of Mount Blanc, then it is black.

All ravens we have observed are black. _________________________ All ravens are black.

Deductive: P1,..., Pn; therefore Q Inductive: P1,..., Pn; therefore Q is very likely to be true Strong/weak

All ravens are black. ______________ If there is a raven on top of Mount Blanc, it is black. All observed ravens have been black. ______________ If there is a raven on top of Mount Blanc, it is black.

Deductive – monotonic; conclusive support; hard to establish the truth of the premise Inductive – nonmonotonic; partial support

The sun is coming out, so the rain should stop soon. It’s going to rain tomorrow, so it is either going to rain or going to be clear tomorrow.

Inductive generalizations In the past, when I tried to use 1 Euro coins in Polish supermarket trolleys, they have not worked. _________________ 1 Euro coins do not work in Polish supermarket trolleys. Opinion polls; samples

samples Unfair/biased Large enough (not affected by runs of luck) Representative Phrasing of questions may influence the outcome

Slanted questions Which do you favour: (a) preserving a citizen’s constitutional right to bear arms or (b) leaving honest citizens defenseless against armed criminals?

Which do you favour: (a) restricting the sale of assault weapons or (b) knuckling under to the demands of the well- financed gun lobby?

Informal judgmental heuristics General strategy for solving a problem; we rely too heavily on them

The representative heuristics You are randomly dealt five-card hands from a standard deck. Which of the following two hands is more likely to come up?

(1) Three of clubs Seven of diamonds Nine of diamonds Queen of hearts King of spades (2) Ace of spades Ace of hearts Ace of clubs Ace of diamonds King of spades

Unimpressive strikes us as representative

Linda is 31 years old, single, outspoken and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

Rank the following statements with respect to the probability that they are also true of Linda: Linda is a teacher in elementary school. Linda works in a bookstore and takes Yoga classes. Linda is active in the feminist movement. Linda is a bank teller. Linda is an insurance sales person. Linda is a bank teller and is active in the feminist movement.

The availability heuristic In four pages of a novel (about 2000 words), how many words would you expect to find that have the form ----ing (seven-letter word that end with ing)? In four pages of a novel (about 2000 words), how many words would you expect to find that have the form -----n- The median estimates were 13.4 for ----ing word and 4.7 for -----n- word.

More available to our memory, naturally come to mind; easier to think of Is the situation sufficiently standard to allow the use of informal judgmental heuristics?

Inductive/statistical generalisation: sample  population Statistical syllogism: population  member/subset

Statistical Syllogisms Ninety-seven percent of the Republicans in California voted for Bush. Marvin is a Republican from California. ______________________ Marvin voted for Bush. STRONG

Three percent of the socialists in California voted for Bush. Maureen is a socialist from California. _______________ Maureen didn’t vote for Bush. STRONG

X percent of Fs have the feature G. a is an F. __________________ a has/ doesn’t have the feature G. F – the reference class

Ninety-seven percent of the Republicans in California voted for Bush. Marvin is a Republican from California. ______________________ Marvin voted for Bush. Three percent of Clinton’s relatives voted for Bush. Marvin is a relative of Clinton. ______________________ Marvin didn’t vote for Bush. 

Forty-two percent of Republicans from California who were relatives of Clinton voted for Bush. Marvin is a Republican from Cliafornia who is a relative of Clinton. _________________ Marvin voted for Bush.WEAK

Less than 1 percent of the people in the world voted for Bush. Gale is a person in the world. _______________________ Gale didn’t vote for Bush.

Reasoning about causes General conditional: For all x, if x has the feature F, then x has the feature G. If something is a square, then it is a rectangle.

Causal generalizations For all x, if x has the feature F, then x has the feature G. X’s having the feature F is a sufficient condition for its having the feature G, and x’s having the feature G is a necessary condition for its having the feature F.

Causal correlation? At one time there was a strong negative correlation between the number of mules in a state and the salaries paid to professors at the state university. In other words, the more mules, the lower professorial salaries. There is a high positive correlation between the number of fire engines in a particular borough in New York City and the number of fires that occur there.

That F is a sufficient condition for G means that whenever F is present G is present. That F is a necessary condition for G means that whenever F is absent G is absent.

The sufficient-condition test (Mill’s Method of Difference) A, B, C, D – candidates for sufficient conditions G – target feature ABC D G  A B C  D  G A  B  C  D  G SCT: Any candidate that is present when G is absent is eliminated as a possible sufficient condition of G.

The necessary-condition test (Mill’s Method of Agreement) A B C D  G  A B C D G A  B C  D G NCT: Any candidate that is absent when G is present is eliminated as a possible necessary condition of G.

A B  C D  G  A B C D G A  B C D G Which of the candidates is not eliminated by SCT? Which of the candidates is not eliminated by NCT? Which of the candidates is not eliminated by either test?

A B  C D  G  A B C D G A  B C D G Which of the candidates is not eliminated by SCT? C Which of the candidates is not eliminated by NCT? C, D Which of the candidates is not eliminated by either test? C

Inferences to the best explanation The idea is that a hypothesis gains inductive support if, when it is added to our stock of previously accepted beliefs, it increases our ability to make reliable predictions and illuminating explanations.

You come back home and discover that the lock on your front door is broken and several valuable objects are missing Hypotheses: your house was burglarized; there was a drug bust and the authorities had the wrong address; your friends are playing a strange joke on you; a meteorite struck the door and then vaporized your valuables.

Your house begins to shake so violently that pictures fall off walls. Your key will not open the door.

Which explanation is best? Should contain only true claims Shouldn’t be obscure Should be simple Should be powerfull Should be conservative

Arguments from analogy Object A has properties P, Q, R. Objects B, C, D also have properties P, Q, R. Objects B, C, D have property X. _________________ Therefore, object A probably also has the property X. P, Q, R, X must be relevant and important

A – minivan Honda Odyssey B – Honda Civic sedan P – the same manufacturer Q – similar engine R – similar gearbox X – is reliable

A – minivan Honda Odyssey B – Ford P – has a sun roof Q – has four doors R – is red X – is reliable

Strong or weak? The premises must be true The cited similarities must be relevant and important The presence of relevant dissimilarities? The weaker the conclusion, the stronger the argument: - The Odyssey will run for ten years without any repairs. - The Odyssey will probably run for ten years without any repairs. - The Odyssey will probably run for at least five years without any repairs. - The Odyssey will be very reliable. - The Odyssey will be reliable.

This landscape by Cezanne is beautiful. He did another painting of a similar scene around the same time. So it is probably beautiful too. My aunt had a Siamese cat that bit me, so this Siamese cat will probably bite me too.

The paradox of the raven (confirmation) (R) All ravens are black. (R-) Nothing which is not black is a raven. (R) and (R-) are logically equivalent, i.e. they are true or false together. What would confirm (R)? What would confirm (R-)?

Confirmation is not a simple matter of enumerative induction, that is the mere accumulation of confirming instances.

Hempel’s solution It doesn’t have to be absurd. Let’s suppose that someone observed a white object. He thinks that it is a raven. Further observation reveals that it is a shoe, not a raven. So it supports (R) in a sense. Conclusion: when you determine whether a given information confirms the given hypothesis you have to take context into account. The mere logical form does not settle the issue. Carl Gustav Hempel (1905-1997)

Enumerative induction 1st observed swan was white. 2nd observed swan was white. 3rd observed swan was white. ….. __________________ All swans are white.

Russell’s chicken 1st day – The chicken is fed by the farmer. 2nd day – The chicken is fed by the farmer. 3rd day – The chicken is fed by the farmer. ….. ------------------- The farmer comes and wrings its neck.

Grue (New riddle of induction) All examined emeralds are green. _________________ All emeralds are green. Grue: x is grue if it is green and examined (by now) or blue and unexamined (by now). All examined emeralds are green. _________________ All emeralds are grue.

Only well-entrenched predicates are projectible.

Green: x is green if it is grue if examined and bleen if not.

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