Justified True Belief

(●) Symbolic logic is a subdiscipline of philosophy akin to mathematics that deals with the rules of reasoning. In symbolic logic, letters and symbols are used to stand for sentences and the words that connect them. This approach helps to make the logical form of a sentence clear without being distracted by its grammatical form, as sentences with different grammatical structures may still have the same logical form.

Here is a legend of some common symbols:

Letters (P, Q, R, S, etc.)
Meaning: These capital letters stand for any arbitrary sentences.
Example: In the argument "If today is Sunday, the library is closed. Today is Sunday. Therefore, the library is closed," we can let P = "Today is Sunday" and Q = "the library is closed".

Arrow (→)
Meaning: The arrow stands for the connecting words, "if . . . , then . . ." or it can be read as "implies". In a sentence of the form P → Q, P is the antecedent clause and states a sufficient condition of the consequent clause Q. Q is the consequent clause and states a necessary condition of the antecedent clause P. The clause that follows a simple "if" is symbolized P (sufficient condition), and the clause that follows "only if" is symbolized Q (necessary condition).
Example: The sentence "If John studies hard, then he will get a good grade in logic" can be symbolized as P → Q, where P = "John studies hard" and Q = "he will get a good grade in logic". The sentence "Extra credit will be permitted only if you have completed all the required work" can be symbolized as P → Q, where P = "You may do extra credit work" and Q = "You have completed the required work".

Negation (¬)
Meaning: This symbol stands for "not" and is the sign of negation.
Example: ¬Q is read as "Not-Q". If Q is the sentence "My roommate is sleeping in," then ¬Q is "My roommate is not sleeping in". ¬¬Q is logically equivalent to Q.

Conjunction (&)
Meaning: This symbol is read as "and". It symbolizes any conjunction, including words like but, while, although, whereas, and many other words when they function as conjunctions. For a conjunction P & Q to be true, both P and Q must be true.
Example: The sentence "Charity is playing the piano, and Jimmy is trying to play the piano" can be symbolized as P & Q, where P = "Charity is playing the piano" and Q = "Jimmy is trying to play the piano". The sentence "They ate their spinach, even though they didn’t like it" would be symbolized P & Q, where P symbolizes "They ate their spinach" and Q symbolizes "they didn’t like it".

Disjunction (v)
Meaning: This symbol is read as "or". A sentence composed of two sentences connected by "or" is called a disjunction. In order for a disjunction to be true, only one part has to be true (or both).
Example: The sentence "Either Mallory will carefully work on decorating their new apartment, or she will allow it to degenerate into a pigsty" can be symbolized as P v Q, where P = "Mallory will carefully work on decorating their new apartment" and Q = "she will allow it to degenerate into a pigsty". Note that in logic, both parts of a disjunction can be true.

Universal Quantification ((x))
Meaning: This symbol is used in first-order predicate logic to deal with quantified sentences, specifically those about all or none of a group. It can be read as "For any x, . . .". Universally quantified statements turn out to be disguised "if . . . , then . . ." statements. The variable 'x' can be replaced by any individual thing.
Example: The statement "All bears are mammals" can be symbolized as (x) (Bx → Mx), where Bx = "x is a bear" and Mx = "x is a mammal". This is read as "For any x, if x is a bear, then x is a mammal". A negative universal statement like "No goose is hairy" is symbolized by negating the consequent: (x) (Gx → ¬Hx), read as "For any x, if x is a goose, then x is not hairy".

Existential Quantification (∃x)
Meaning: This symbol is used in first-order predicate logic for statements about only some members of a group. It tells us that there really exists at least one thing that has the property in question. It may be read as "There is at least one ___ such that . . .". Existentially quantified statements are typically symbolized using & (conjunction), not → (conditional).
Example: The statement "Some bears are white" can be symbolized as (∃x) (Bx & Wx), where Bx = "x is a bear" and Wx = "x is white". This is read as "There is at least one x such that x is a bear and x is white". The statement "Some bears are not white" is symbolized as (∃x) (Bx & ¬Wx).

Necessity (□)
Meaning: This symbol is used in modal logic to stand for the mode of necessity. □P is read as "Necessarily, P" and indicates that the statement P is necessarily true (true in every possible world). □¬P indicates that P is necessarily false (false in every possible world).
Example: □P is read as "Necessarily, P". □¬P is read as "Necessarily, not-P".

Possibility (◊)
Meaning: This symbol is used in modal logic to stand for the mode of possibility. ◊P is read as "Possibly, P" and indicates that the statement P is possible (true in at least one possible world). ¬◊P is read as "Not-possibly, P," meaning it is impossible for P to be true.
Example: ◊P is read as "Possibly, P".

"Would" Counterfactual (□→)
Meaning: This symbol is used in counterfactual logic for conditional statements in the subjunctive mood that state what would happen if the antecedent were true. P □→ Q is read as "If it were the case that P, then it would be the case that Q".
Example: The conditional "If Oswald hadn’t shot Kennedy, then somebody else would have" is a "would" counterfactual. It would be symbolized using □→.

"Might" Counterfactual (◊→)
Meaning: This symbol is used in counterfactual logic for conditional statements in the subjunctive mood that state what might happen if the antecedent were true. P ◊→ Q is read as "If it were the case that P, then it might be the case that Q". It is defined as the contradictory of P □→ ¬Q. "Might" indicates a genuine, live option under the circumstances.