~F Logic: Logicians Guide to Philosophy

26 May

~F Logic is another page on this site where  I plan to put all of the symbols used in sentential and predicate logic. In posts in the future and a few already done, sentential and predicate logic is used to logically perceive arguments philosophers make. I want to create a guide on this website to the symbols of logic and what things mean when they are logically laid out. After this is posted, its going to go on the ~F Logic page on this site. There is a link for it above ^.

Symbols of Logic I am going to show the truth table to each connection because to do truth tables, you have to know how each connection gives truth and false values between letter variables.

1)   ~      this squiggly symbol means the negation of something. So if  P is true  what is ~P?  False. Here is the truth table for propositions just containing the ~ symbol.

P                  ~  P

T                 F T

F                T F

Looking at this truth table, the ~ symbol just reverses the truth of falsity of the object/thought discussed. I call this ~F Logic page    ~F because this reads “the negation of falsity”. In philosophy and logic, we want to negate falsities to get to the truth right? That is my reasoning for calling this logicians guide  ~F.

2)  If…then Symbol(s)-  –>  or  ⊃

The arrow or the sideways u signifies that if theres something, the next thing concludes. If this then that. Whenever you are doing a truth table, fill in all the T’s and F’s of the letter variables themselves, then decide what the connection symbol truth is. Heres the truth table for the if/then.

P       ⊃          Q

T       T T

T       F F

F      T T

F      T F

The only time the inference is false in a if/then proposition is when the first variable is true and then the second is false.

*Side note*

Heres a truth table of the first two symbols explained here, combined. Read: If P then not Q.

P    ⊃     ~      Q

T    F F      T

T    T T      F

F     T F      T

F    T T      F

And also, sometimes people use 1 and 0 for T and F. I dislike that method. It confuses me.

3)   Or –     v (for either one)      v (for one of the two)

v is hard to do a truth table for because it requires that only one of the two things be true. v is easy to do truth tables for because if there is two variables and one is true, the inference will be a true statement. The only time the or connection will yield falsity is when both variables are false. Truth table:

P       v        Q

T     T T

T      T F

F      T T

F      F F

Think about this in terms of the algebraic connection of Union. This is just like that but this is logic.

4)  And-    .    or &

Think about this in terms of the algebraic connection of Intersection. This is also like that. Truth table:

P        .         Q

T       T T

T       F F

F       F T

F       F F

The and and or logic connections are similar because, if its or, one truth in the bunch makes the whole proposition true, while if its and, one falsity in the bunch makes the whole proposition false.

Heres a truth table of everything so far:

P       v       ~Q           ⊃          ~P       .     Y

T        T F T           F F T      F T

T       T T F          F F T      F T

F      F F T          T T F      T T

T      T F T          F F T     F F

F      T T F          T T F      T T

F      F F T          T T F      F F

T      T T F          F F T     F F

F      T T F          F T F      F F

5) If/Then  both ways – ↔

If you have P –> Q you know what is if and what is the then.↔ indicates that this if then relationship can be used both ways   so    P↔Q  signifies:   P –>Q and/or  Q –>P. Truth table:

P       ↔      Q

T       T T

T       F F

F       F T

F       T F

6) Therefore- ⊢

This symbol of a turnstile signifies that theres this, therefore, theres this too. The turnstile does not have a truth value, and cannot be represented for its truth or falsity in a truth table. It just says this therefore this.

these are often put back to back the vertical parts facing each other to represent two turnstiles. This back to back turnstile represents that P therefore Q and/or Q therefore P. I dont use this symbol much in logic.

On the ~F Logic page, I will  do a lot more examples of truth tables like the big one above.

Predicate based Logic

This form of logic involves expressions where the existence of objects and their tendencies are shown.

Two kinds of quantifiers begin each expression

A universal quantifier is shown by the symbol ∀ and a variable is next to it, then the expression follows. This upside down A signifies the word ‘every’.

An existential quantifier is shown by the symbol ∃ and a variable next to it, then the expression follows. This backwards E signifies the word ‘some’.  ∃!   signifies that exactly one thing in existence qualifies in this expression.

Heres some examples of this predicate based logic:

∀xPx              read  all things are P

∀x(Px –>Qx)      read  All P’s are Q’s/  Every P is a Q/  If its a P, its a Q/  Only P’s are Q’s/ Any P is a Q/ Everything that is                                            P is Q

∀xAx(Dx . Cx)      All A’s are D’s and C’s

∀xAx(Dx v Cx)   All A’s are  either D’s or C’s

∃xAx               Some things are A/ Something is an A

∃x(Bx . Cx)  Some B’s are C’s

∃x(Bx –>Cx)    Some B’s are C’s

x is put by the existential quantifier and the upper case descriptive variables because x signifies the actual object in discussion.

∃x(Bx v Cx)      Some things are B’s or C’s

∃!x(Bx –>Cx)  Exactly 1 B is a C

∃!xFx              Exactly one thing is an F

Predicate based logic is best learned by learning to take English and turning it into a predicate based logic expression and taking the expressions and making scenarios to fit them.

Heres a few from English to predicate logic:

1)   All people are good or bad

2) All things are created.

3)   All whales are not fish.

4)  Some people are schizophrenic

5)  Most people are men or women

6) Exactly one reptile has white scales and a blue tongue

7) If some things are good, then some things are bad

8) Some things are finite, therefore at least one thing is infinite

9) If all things come into existence, then all will come out of existence.

10) All birds are blue, therefore a jaybird is blue.

Do n0t scroll down anymore if you want to figure out the above 10 questions. All of the answers are below this sentence.

Answers: random variables are assigned to random things

1) ∀xPx(Gx v Bx)

2) ∀x(Ex –> Cx)

3) ∀x( Wx –> ~Fx)

4) ∃x( Px –> Sx)

5)  ∃xPx( Mx v Wx)

6) ∃!xRx( Wx . Bx)

7) ∃x( Gx ⊃ Bx)

8) ∃x( Fx ⊢ Ix)

9) ∀x( Ix ⊃ Ox)

10) ∀xBx(Blx ⊢ Jbx)

I think In logic that you should be able to put lowercase letters next to your uppercase variables so thats what I did for #10. B signifies birds, Bl signifies blue  and Jb signifies jaybirds and blue.  The variable B was already used so I picked variables with extra letters. Is that okay  to do ? I think so.

Lets do that process vice versa

I will give a predicate logic expression and  you  figure out some scenarios that fit the expression.

1)  ∀xMx( (Px v Qx)⊃ ~Zx)

2) ∃!x(Px –> Yx)

3)∃xAx( (Bx . Qx) ⊃ ~( Bx . Px))

Okay so 1 and 3 might be a little difficult but 2 is easy. Heres my scenarios for these logical expressions:

1) If all men (Mx) are either stupid (Px) or shallow (Qx), then all men are not compassionate ( ~Zx ).

2) Exactly one North American colubrid snake (Px) is venomous (Yx).

3)  In some shoes (Ax), if a shoe is sporty (Bx) and girlie (Qx), then the shoe cannot be sporty (Bx) and high heeled (Px)

More examples of truth tables, sentential logic, predicate logic, and going from English to logic expressions coming to the ~F Logic page on this site.

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