For each thing: For each and every thing in the universe: For all things: For anything:

1. Important note. All references to ‘For At Least One Thing:’ FALOT’ have been replaced with ‘There is something such that:’ ‘TISS’ Important Change

2. For each thing: For each and every thing in the universe: For all things: For anything: For everything in the universe (call it x): For every x: For anything you pick: These are some of the ways to express a universal statement. When a statement starts like one of these it takes many specific objects to prove it true but only one object to prove it false. We will use the “For Each Thing:” phrase as our standard form (and abbreviate it “FET”), but if another phrase sounds better in expressing a particular statement, feel free to think of the statement in those terms. You can put “in the universe” on the end of anything that doesn’t already have it, if you want. Think about this and then go forward with the right arrow. Explanation

3. For at least one thing: For something: There is something such that: There is at least one thing in the universe such that: There is at least one thing (call it x): For some x: I can find something such that: These are some of the ways to express an existential statement. When a statement starts like one of these it takes many specific objects to prove it false but only one object to prove it true. We will use the “There is something such that:” phrase as our standard form (which we will abbreviate “TISS”) , but if another sounds better in expressing a particular statement, feel free to think in those terms. You may put “in the universe such that” on the end of anything above that doesn’t already have it, if you want. Think about this and then go forward for some other hints. Explanation

4. Basic Form for 'All A's are B's': (x)(Ax  Bx). • Some examples: • All cats are friendly. (Equivalents: Every cat is friendly; any cat is friendly; cats are all friendly; cats are always friendly.) • For every x, if x is a cat then x is friendly. • For every x, if Cx then Fx. • (x)(Cx  Fx) . • All domesticated cats are friendly. • For every x, if x is domesticated and x is a cat, then x is friendly. • For every x, if Dx and Cx, then Fx. • (x)[(Dx & Cx)  Fx] . • All cats that are treated well are friendly. • For every x, if x is a cat and x is treated well, then x is friendly. • (x)[(Cx & Tx)  Fx] . • All cats are either friendly or aloof. • For every x, if x is a cat then either x is friendly or x is aloof. • (x)[Cx  (Fx v Ax)] . • Basic Form for 'No A's are B's': (x)(Ax  ~Bx) or {Equivalent: ~(x)(Ax & Bx) } • Some examples: • No tigers are friendly. {Equivalents: All tigers are unfriendly; tigers are never friendly.) • For every x, if x is a tiger then x is not friendly. • For every x, if Tx then ~Fx. • (x)(Tx  ~Fx) . {Equivalent: ~(x)(Tx & Fx) } • Forward for more Ron McIntyre's Basic Forms

5. No wild tigers are friendly. • For every x, if x is wild and x is a tiger, then x is not friendly. • (x)[(Wx & Tx)  ~Fx] . {Equivalent: ~(x)((Wx & Tx) & Fx) .} • No tigers are contented pets. • For every x, if x is a tiger then it is not the case that x is contented and x is a pet. • (x)[Tx  ~(Cx & Px)] .{Equivalent: ~(x)(Tx & (Cx & Px)) } • No tigers are safe if they are uncaged. • For every x, if x is a tiger, then if x is not caged x is not safe. • (x)[Tx  (~Cx  ~Sx)] {Equivalent: ~(x)(Tx & (Sx & ~Cx)) } • Basic Form for 'Some A's are B's': (x)(Ax & Bx) . • Some examples: • Some cats are friendly. {Equivalents: There are friendly cats; cats are sometimes friendly; not all cats are unfriendly.) • For at least one x, x is a cat and x is friendly. • For at least one x, Cx and Fx. • (x)(Cx & Fx) . • Some domesticated cats are friendly. • For at least one x, x is domesticated and x is a cat and x is friendly. • (x)[(Dx & Cx) & Fx] . • Some cats are not friendly. • For at least one x, x is a cat and x is not friendly. • (x)(Cx & ~Fx) . • Forward for more. Ron McIntyre's Basic Forms

6. Some cats are neither friendly nor pretty. • For at least one x, x is a cat and x is neither friendly nor pretty. • For at least one x, x is a cat and x is not friendly and x is not pretty. • (x)[Cx & (~Fx & ~Px)] . • Basic Form for 'Only A's are B's': (x)(Bx  Ax) . {Equivalent form: (x)(~Ax  ~Bx) } • Some examples: • Only women are mothers. (Equivalent: None but women are mothers.} • For every x, x is a mother only if x is a woman. • For every x, if x is a mother then x is a woman [or: if x is not a woman then x is not a mother]. • (x)(Mx  Wx) . {Equivalent: (x)(~Wx  ~Mx) .} • Only adult women are mothers. • For every x, x is a mother only if x is an adult and x is a woman. • (x)[Mx  (Ax & Wx)] . {Equivalent: (x)[~(Ax & Wx)  ~Mx]. or (x)[(~Ax v ~Wx)  ~Mx] .} • Only women are mothers or sisters. • For every x, x is a mother or x is a sister only if x is a woman. • (x)[(Mx v Sx)  Wx] . {Equivalent: (x)[~Wx  ~(Mx v Sx)]. or (x)[~Wx  (~Mx & ~Sx)] .} • NOTE: 'The only A's are B's' is not equivalent to 'Only A's are B's' but to 'All A's are B's'. Thus, the correct translation of 'The only people who are mothers are women' is '(x)[(Px & Mx)  Wx]'. • NOTE: The basic form for 'A's and B's are C's' is not'(x)[(Ax & Bx)  Cx]', but '(x)[(Ax v Bx)  Cx]'. Thus, a correct translation of 'Dogs and cats are animals' is '(x)[(Dx v Cx)  Ax]'. But it could also be handled as an ampersand statement. i.e.; All Dogs are animals AND all cats are animals: ((x)(Dx  Ax) & (x)(Cx  Ax)) . • Go forward for first example. Ron McIntyre's Basic Forms

7. Doctors are Professionals First we should try to figure out what this means. Does it mean All Doctors are professionals or some Doctors? It seems to be “All”. So that gives us a hint as to what standard form to use Think about that and go forward with the right arrow. Problem 1

8. Doctors are Professionals For Each Thing:…. It is a “For Each Thing” kind of statement. But what should happen now? It cannot be “For Each Thing: it is a doctor and it is a professional” for that would mean that everything in the universe has the property of being both a doctor and a professional, including you. But ask yourself: if this statement were true and you found a Doctor, then what would you be able to conclude about him? If he were a doctor then he would have to be a professional. So imagine completing the red line and then hit the right arrow. Problem 1

9. Doctors are Professionals For Each Thing: If it is a Doctor then it is a Professional. Now that our target statement is in the ‘For Each Thing:” standard form, we use an individual’s name to represent any old thing. Let us choose “Alan” and let us delete the major connector “For Each Thing”. Hit the forward arrow to continue. Problem 1

10. Doctors are Professionals For Each Thing: If it is a Doctor then it is a Professional. If Alan is a Doctor then Alan is a Professional. Now we see that the new statement is also in standard form, so we can break it into its parts. Imagine and then go forward. Problem 1

11. Doctors are Professionals For Each Thing: If it is a Doctor then it is a Professional. If Alan is a Doctor then Alan is a Professional. Alan is a Doctor.Alan is a Professional. “Alan is a Doctor” is not in standard form and has no standard form synonym and neither does “Alan is a Professional” and so let us use “D_” for “_ is a doctor” and “P_” for “_ is a professional” If that were the dictionary given, it would be an extra clue that we were now to represent atomics. Hit right arrow to go forward. Problem 1

12. Doctors are Professionals For Each Thing: If it is a Doctor then it is a Professional. If Alan is a Doctor then Alan is a Professional. Alan is a Doctor. Alan is a Professional. Da Pa So if “Da” represents one side of an “if..then” standard form and “Pa” the other then how will We represent the statement in green? Go forward. Problem 1

13. Doctors are Professionals For Each Thing: If it is a Doctor then it is a Professional. If Alan is a Doctor then Alan is a Professional. (Da  Pa) Alan is a Doctor. Alan is a Professional. Da Pa Right, all “If… Then” standard forms get represented with a horseshoe. But the statement in red came from the “For Each Thing” standard form, so how will we represent the statement In green? Hit forward Problem 1

14. Doctors are Professionals For Each Thing: If it is a Doctor then it is a Professional. (x)(Dx  Px) If Alan is a Doctor then Alan is a Professional. (Da É Pa) Alan is a Doctor. Alan is a Professional. Da Pa All “For Each Thing” statements are represented by universal quantifiers. “Alan” was our representative individual. So now “a” will be replaced by “x”. Reading it back into English we have: For Each thing (call it ‘x’) if x (it) has the property of being a Doctor then x has the property of being a professional. Variables take the role of “it” in English. The English statement in red is synonymous with the statement in green, so how will we represent that? Go forward Problem 1

15. Doctors are Professionals (x)(Dx  Px) For Each Thing: If it is a Doctor then it is a Professional. (x)(Dx  Px) If Alan is a Doctor then Alan is a Professional. (Da  Pa) Alan is a Doctor. Alan is a Professional. Da Pa Synonymous statements are represented the same way. So there you have it. In the future we shall Shorthand “For Each Thing” statements as “FET”. Let us make that substitution now. Hit forward. Problem 1

16. Doctors are Professionals (x)(Dx  Px) FET: If it is a Doctor then it is a Professional. (x)(Dx  Px) If Alan is a Doctor then Alan is a Professional. (Da  Pa) Alan is a Doctor. Alan is a Professional. Da Pa So “(•x)(Dx É Px)” is the answer. Hit forward for new problem. Problem 1

17. Doctors are in the Audience. New Problem First we should try to figure out what this means. Does it mean “All Doctors are in the audience” or just Some Doctors? It seems to be “Some”. So that gives us a hint as to what standard form to use Think about that and go forward with the right arrow. Problem 2

18. Doctors are in the Audience. There is something such that:…. It is a “There is something such that:” kind of statement. But what should happen now? Think about what it would take to prove that “Doctors are in the audience.”? What kind of thing would I have to bring into your life to show it for certain? I’d have to find something, it would have to be a doctor and it would have to be in the audience. (Let us abbreviate “There is something such that” as “TISS”. That will be sort of a TISS symbol.) So imagine completing the red line and then hit the right arrow. Problem 2

19. Doctors are in the Audience. TISS: it is a Doctor and it is in the Audience. Now that our target statement is in the ‘TISS:” standard form, we use an individual’s name to represent any old thing. Let us choose “Alan” again and let us delete the major connector. “TISS”. Hit the forward arrow to continue. Problem 2

20. Doctors are in the Audience. TISS: it is a Doctor and it is in the Audience. Alan is a Doctor and Alan is in the Audience. So now we see that the new statement is also in standard form, so we can break it into its parts. Imagine and hit forward. Problem 2

21. Doctors are in the Audience. TISS: it is a Doctor and it is in the Audience. Alan is a Doctor and Alan is in the Audience. Alan is a Doctor.Alan is in the Audience. “Alan is a Doctor” is not in standard form and has no standard form synonym and neither does “Alan is in the Audience” and so let us use “D_” for “_ is a doctor” and “A_” for “_is in the Audience” If that were the dictionary given, it would be an extra clue that we were now to represent atomics. Imagine and hit forward Problem 2

22. Doctors are in the Audience. TISS: it is a Doctor and it is in the Audience. Alan is a Doctor and Alan is in the Audience. Alan is a Doctor.Alan is in the Audience. Da Aa So if “Da” represents one side of an “And” standard form and “Aa” the other then how will we represent the statement in green? Go forward. Problem 2

23. Doctors are in the Audience. TISS: it is a Doctor and it is in the Audience. Alan is a Doctor and Alan is in the Audience. (Da & Aa) Alan is a Doctor. Alan is in the Audience. Da Aa Right, all “And” standard forms get represented with an ampersand. But the statement in red came from the “TISS” standard form. So how will we represent the statement in green? Go forward. Problem 2

24. Doctors are in the Audience. TISS: it is a Doctor and it is in the Audience. (x)(Da & Aa) Alan is a Doctor and Alan is in the Audience. (Da & Aa) Alan is a Doctor. Alan is in the Audience. Da Aa All “TISS” statements are represented by existential quantifiers. “Alan” was our representative individual. So now “a” will be replaced by “x”. Reading it back into English we have: There is something: (call it ‘x’) x (it) has the property of being a Doctor and x has the property of being in the audience. Variables take the role of “it” in English. “There is something such that” could also be read as “There is at least one thing” or “I can find something such that”. The English statement in red is synonymous with the statement in green, so how will we represent that? Imagine and go forward. Problem 2

25. Doctors are in the Audience. (x)(Da & Aa) TISS: it is a Doctor and it is in the Audience. (x)(Da & Aa) Alan is a Doctor and Alan is in the Audience. (Da & Aa) Alan is a Doctor. Alan is in the Audience. Da Aa Synonymous statements are represented the same way. So there you have it. The answer to turn in would be: “($x)(Da & Aa)” Hit forward for new problem. Problem 2

26. Only Doctors are in the Audience. New Problem Consider the target statement in red. Is it in standard form? No. Can we restate it in standard form? Sure, but which form. If it could be shown true by one thing, then we would say that it is in the TISS form. But if it takes many things to prove it true or if one thing would prove it false, then it is a FET kind of statement. We can see that finding something in the audience that is not a Doctor would show it false. So it is a FET kind of statement. We know that FET statements usually have an “if then” form as the next connector. So how shall we proceed? Should we say “FET: if it is a Doctor then it is in the Audience.”, or should we say “FET: If it is in the Audience then it is a doctor.”? Hit forward to continue. Problem 3

27. Only Doctors are in the Audience. FET: if it is in the Audience then it is a Doctor It cannot be “FET: If it is a Doctor then it is in the Audience.”, because that would mean “All Doctors in the world are in the Audience.” which is improbable, but we see that “Only Doctors are in the Audience” is not the kind of statement that is improbable. So let us use A_, and D_ as our dictionary and jump to how this statement in complete standard form should be represented. Hit Forward. Problem 3

28. Only Doctors are in the Audience. FET: if it is in the Audience then it is a Doctor (x)(Dx  Ax) There you have the standard structure of a FET statement. A universal statement whose next major connector is a horseshoe. Since the statement in green is synonymous with the English statement in red then they will be represented the same way. Hit forward for new example. Problem 3

29. A doctor is devoted to the relief of suffering. New Problem With this one again we should ask if it means to talk of all doctors or some doctors. From the ‘A’ it might seem clear that it is one doctor. But notice that this is a common way of referring to “all” things. Note that “A whale is a mammal” is not just talking about one whale, it is more to suggest that whales by their very nature are mammals. So we should take this to mean “All doctors are devoted to the relief of suffering.” So how would we say that as a “FET” statement? Think and then hit forward. Problem 4

30. A doctor is devoted to the relief of suffering. FET:if it is a doctor then it is devoted to the relief of suffering This statement may seem odd since “devoted to the relief of suffering” may seem like a rather long property. But it’s ok for properties to be long. “Devoted to the relief of suffering” is something that an object can be. This form can now be turned into symbols with our level of experience. Think and then hit forward. Problem 4

31. A doctor is devoted to the relief of suffering. FET:if it is a doctor then it is devoted to the relief of suffering (x)(Dx  Rx) An since this is synonymous with the green statement, it will be represented the same way. Think and then hit forward. Problem 4

32. A doctor is devoted to the relief of suffering. (x)(Dx  Rx) FET:if it is a doctor then it is devoted to the relief of suffering (x)(Dx  Rx) So again we the classic form of a universal quantifier going with a horseshoe Hit forward for new problem. Problem 4

33. A doctor is devoted to my wife. New Problem We won’t make the mistake of thinking that this is about all doctors, for it’s clearly not the case that doctors are by their very nature devoted to my wife. In a way we are using what is called ‘the principle of charity’. If there is an ambiguity and the statement could be true interpreted one way but is unlike to be true the other way of interpretation then choose the interpretation that could be true. Very charitable. So this says “there is at least one thing. It is a doctor and it is devoted to my wife”, for that indicates the kind of object it would take to prove the statement true. Think and then hit forward. Problem 5

34. A doctor is devoted to my wife. TISS:it is a doctor and it is devoted to my wife. The us assume we can turn this into symbols in one step. Think and then hit forward. Problem 5

35. A doctor is devoted to my wife. TISS:it is a doctor and it is devoted to my wife. (x)(Dx & Wx) And since it is synonymous with the green statement, the green will be the same. Think and then hit forward. Problem 5

36. A doctor is devoted to my wife. (x)(Dx & Wx) TISS:it is a doctor and it is devoted to my wife. (x)(Dx & Wx) And since it is synonymous with the green statement, the green will be the same. Think and then hit forward. Problem 5

37. Doctors are not all rich. New Problem Consider the target statement in red. What does it mean. In particular how many things would it take to prove it true or false? It looks like I could prove it true by finding one thing that was both a Doctor and not rich. So that tells me that it should be a TISS kind of statement. Hit forward. Problem 6

38. Doctors are not all rich. TISS: How should we finish this statement? Because we realize that we can prove this statement true with one thing, and we know what kind of thing it is. Spell out to yourself what kind of thing it is. Hit forward Problem 6

39. Doctors are not all rich. TISS: it is a Doctor and it is not rich. It is a thing that is a Doctor and not rich. To say it is not rich is to say that it is not the case that it has the property of being rich. So we should now try to turn this into symbols since it is a familiar form. Hit forward. Problem 6

40. Doctors are not all rich. TISS: it is a Doctor and it is not rich. (x)(Dx&~Rx) So this says “I can find something such that it is a doctor and it is not rich. But where did the “all” go, from the original statement? Well it turns out that there is another natural standard form synonym of “Doctors are not all rich”. T hink about what someone is denying when they say the statement in green. Hit forward. Problem 6

41. Doctors are not all rich. The statement in green denies that all doctors are rich. It is as if someone said “all doctors are rich” and the speaker said. “No, that’s not true.” So let us see the standard form version of the green as a “It is not the case that:” statement. Hit forward. Problem 6

42. Doctors are not all rich. NOT: All doctors are rich. But since it is now in standard form, we can consider the statement it denies. Hit forward Problem 6

43. Doctors are not all rich. NOT: All doctors are rich. All doctors are rich. But the statement in red is in a form we have seen many times now. We can turn it into symbols immediately. (I hope). Hit forward. Problem 6

44. Doctors are not all rich. • NOT: All doctors are rich. • All doctors are rich. • (x)(Dx  Rx) • So, since this represents the English statement in red, how will be represent the “NOT” statement in green? • Hit forward Problem 6

45. Doctors are not all rich. • NOT: All doctors are rich. • ~(x)(Dx  Rx) • All doctors are rich. • (x)(Dx  Rx) • Exactly right. And since the green statement is synonymous with the red English statement, then it can be represented the same way. So now we see that there are two ways to represent “Doctors are not all rich”. • Hit forward. Problem 6

46. Doctors are not all rich. • NOT: All doctors are rich. • ~(x)(Dx  Rx) • TISS: it is a Doctor and it is not rich. • (x)(Dx & ~Rx) • How can these both be correct? Because they are logically equivalent. We will be able to show this when we get to quantified trees and derivations. • Hit forward for new example. Problem 6

47. No doctors are rich. • New Problem • Consider the statement in green. It is not in a standard form but could be restated in standard for. In fact there are two natural ways to approach this. First let us assess what it would take to show it true or false. If it takes one thing to prove it true then it would be an existential. If it takes one thing to prove it false then it would be a universal. It looks like one thing would prove it false. So let us try that. • Imagine and hit forward. Problem 7

48. No doctors are rich. • FET:… • If take this statement as true and we found a doctor could we conclude anything about his richness? It looks like we could. We could conclude that he was not rich. So that’s the way we should proceed. If he’s a doctor then he’s not rich. • Hit forward. Problem 7

49. No doctors are rich. • FET:if it is a doctor then it is not rich. • This we should be able to turn into symbols because it is much like other “FET”statements. • Think about it and hit forward. Problem 7

50. No doctors are rich. • FET:if it is a doctor then it is not rich. • (x)(Dx  ~Rx) • But as in earlier examples. No doctors are rich denies another statement. So it could also be treated as a “NOT:” • Think about it and hit forward. Problem 7