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In this video I'm going to talk about translating sentences ...

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from English into our symbolic notation.

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To talk about translation

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first we'll talk about the basic connectives,

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our five main symbols.

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Then about other words that don't map exactly onto the ...

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five main connectives. "Unless"

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and "neither" are two such words.

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In a later video, we'll talk about complex ...

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sentences that combine more than one connective.

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First let's talk about "not."

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If B stands for "Brad shaves"

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then if we have "Brad doesn't shave"

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the "not" is there in the middle of the sentence

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and we get negation B.

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Next let's talk about "and."

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Let A be Angie is happy and B be "Brad shaves."

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Then "Angie is happy and Brad shaves" is

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A & B.

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Let's look at another sentence,

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"Angie is happy, though Brad shaves."

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One thing we can say for sure about this is that it tells us two ...

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facts, that Angie is happy and Brad ...

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shaves. Both these things are true.

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So here we also have A&B.

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This statement also suggests that it's surprising that Angie ...

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is happy, given that Brad shaves.

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But we won't concern ourselves with the suggestion!

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We just translate what's literally entailed by the ...

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sentence, which comes out as A & B.

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So "but," "though,"

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"however," and all similar words can be ...

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translated as &.

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We also need to be on the lookout for hidden compound sentences.

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Let's start with these sentences.

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If we have "Brad and George shave"

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that's the same as "Brad shaves and George shaves," which is B&G.

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If we have "George shaves and dresses well"

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that's the same as "George shaves and George dresses ...

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well" which is G&D.

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When a connective like "and" joins things that aren't ...

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complete sentences, you have to expand it out to ...

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complete sentences before you translate it.

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Let's look at "or" now.

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In this class, "or"

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is always inclusive; it always means "or both."

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With these abbrevations

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"Brad or George shaves"

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becomes B v G.

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If you want to say the exclusive or,

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"but not both," you have to spell it out.

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So you would explicitly say "Brad or George shaves, but not both."

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The "but" becomes &,

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and we put a negation on "both Brad and George shave"

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and we get (B v G) & ~(B & G)

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Now let's look at "if," which is going to have some ...

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more complications.

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Say A is "Angie is happy" and B is "Brad shaves"

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"If Brad shaves, then Angie is happy"

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becomes B -> A.

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This doesn't necessarily mean that if Angie is happy,

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then Brad shaved! There could be other things ...

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that make her happy.

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You can also have "if" in the middle,

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like "Brad shaves if Angie is happy."

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To represent that, we have to untwist the ...

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sentence so the "if" comes at the beginning.

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We get "If Angie is happy, Brad shaves."

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Which is A -> B.

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Notice that the letters A and B are in the opposite order from ...

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the original sentence, "Brad shaves if Angie is ...

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happy." When we have the word "if"

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in the middle of a sentence, we wind up switching the ...

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order of the sentences around the arrow.

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"Only if" is related to "if," but different.

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One way to translate "only if" is as "not...

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if not." You put negations on both ...

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sides, and you change the "only if" to "if."

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So if L is you win the lottery and T is you buy a ticket

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"You'll win the lottery only if you buy a ticket"

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means "You won't win the lottery if you don't buy a ticket."

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Since "if" is in the middle here,

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we have to switch the order of the sentences.

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So it's "If you don't buy a ticket you won't win the lottery"

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Or ~T -> ~L

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Note that this does not hold vice versa!

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It is certainly not necessarily true that if you do buy a ticket ...

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you will win the lottery.

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It's also possible to do "only if"

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as a single arrow without using any extra negations.

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Using the same abbreviations

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let's look again at "You will win the lottery only if you buy a ticket."

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This amonts to saying that if you won the lottery,

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then you must have bought a ticket.

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There's no other way for you to have won the lottery.

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That becomes L -> T.

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which is equivalent to ~T -> ~L.

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(You can do a truth table to check that if you want!)

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So when we have "only if" in the middle of a sentence,

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you can translate it with an arrow and keep the sentences ...

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in the same order! No extra negations are ...

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needed if you do this.

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Note again that this does not work in reverse.

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This does not say that if you buy a ticket you win the ...

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lottery. It must be L -> T, not T -> L.

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We can organize this into a two-by-two table.

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Whether you have "if" or "only if,"

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you can translate it with a single arrow.

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If it's "if" at the beginning,

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you keep the sentences in the same order.

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If it's "if" in the middle,

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you switch the order. If you happen to see "Only if"

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at the beginning, switch the order.

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If you have "only if" in the middle,

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it's the same order.

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For a bit more on conditionals

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there are some other synonyms for "if,"

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like "Provided that," "Assuming that,"

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and "On condition that."

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All these behave like "if" no matter where they are in ...

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the sentence, at the beginning or in the middle.

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So "George shaves provided Brad shaves"

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is just "George shaves if Brad shaves"--you turn the ...

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"provided" into "if"

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in the middle, where it is.

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And since "if" is in the middle,

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we switch the order, and get B -> G!

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Our last basic connective is "if and only if"

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given these sentences

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"Brad washes his hair if and only if he showers"

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becomes W <-> S. Unlike with the arrow,

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the order doesn't really matter here.

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In order to have a double arrow,

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you're going to need "if and only if"

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in full in your sentence.

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If you just have "if" by itself,

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or "only if" by itself,

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you use a single arrow (and maybe some negations,

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if you're using that way of translating "only if.")

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Let's look at a couple of other phrases now.

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First, "neither... nor."

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With these abbreviations

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"Neither Brad nor George shaves"

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could be translated by taking the negation of Brad shaves ...

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or George shaves

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which would be negation parentheses B v G.

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There's another equivalent formulation for this.

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If neither one of them shaves, that means Brad doesn't ...

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shave, and also George doesn't shave.

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So that's ~B & ~G.

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These are not equivalent to ~B v ~G,

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though. That would mean at least one ...

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of them doesn't shave, but it would leave open the ...

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possibility that one of them does shave.

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Finally, let's talk about "unless."

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With these abbreviations

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let's translate "You'll be cold unless you wear a jacket."

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One way to handle it is by turning "unless" into "if not."

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So we get "You'll be cold if you don't wear a jacket."

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Since this has "if" in the middle,

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we switch the sides, and we get ~J -> C.

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Note that this isn't an if and only if.

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We haven't literally said that you won't be cold if you do ...

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wear a jacket. Maybe you wear a jacket but it's too thin.

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We can also, surprisingly,

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translate "unless" as "or."

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To say that you'll be cold unless you wear a jacket is to ...

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say that either you wear a jacket or you'll be cold.

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This can be translated as J v C

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or, since order doesn't matter ...

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around a wedge, as C v J.

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That's also what you'd get if you just put "or"

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where the original "unless" was.

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So, translating "unless"

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as "if not" means you have to worry ...

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about where the extra negation goes and which ...

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order things go around the arrow.

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If you just translate it as "or" you don't have to worry about ...

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that. So that might be easier for you.